3.5.60 \(\int \frac {(c+d x^2)^3}{x^{9/2} (a+b x^2)^2} \, dx\) [460]

Optimal. Leaf size=376 \[ -\frac {c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac {c \left (11 b^2 c^2-21 a b c d+6 a^2 d^2\right )}{6 a^3 b x^{3/2}}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{7/2} \left (a+b x^2\right )}-\frac {(b c-a d)^2 (11 b c+a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{15/4} b^{5/4}}+\frac {(b c-a d)^2 (11 b c+a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{15/4} b^{5/4}}-\frac {(b c-a d)^2 (11 b c+a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{15/4} b^{5/4}}+\frac {(b c-a d)^2 (11 b c+a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{15/4} b^{5/4}} \]

[Out]

-1/14*c^2*(-7*a*d+11*b*c)/a^2/b/x^(7/2)+1/6*c*(6*a^2*d^2-21*a*b*c*d+11*b^2*c^2)/a^3/b/x^(3/2)+1/2*(-a*d+b*c)*(
d*x^2+c)^2/a/b/x^(7/2)/(b*x^2+a)-1/8*(-a*d+b*c)^2*(a*d+11*b*c)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(15
/4)/b^(5/4)*2^(1/2)+1/8*(-a*d+b*c)^2*(a*d+11*b*c)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(15/4)/b^(5/4)*2
^(1/2)-1/16*(-a*d+b*c)^2*(a*d+11*b*c)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(15/4)/b^(5/4)*2
^(1/2)+1/16*(-a*d+b*c)^2*(a*d+11*b*c)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(15/4)/b^(5/4)*2
^(1/2)

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Rubi [A]
time = 0.29, antiderivative size = 376, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {477, 479, 584, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-a d)^2 (a d+11 b c)}{4 \sqrt {2} a^{15/4} b^{5/4}}+\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-a d)^2 (a d+11 b c)}{4 \sqrt {2} a^{15/4} b^{5/4}}-\frac {(b c-a d)^2 (a d+11 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{15/4} b^{5/4}}+\frac {(b c-a d)^2 (a d+11 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{15/4} b^{5/4}}-\frac {c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac {c \left (6 a^2 d^2-21 a b c d+11 b^2 c^2\right )}{6 a^3 b x^{3/2}}+\frac {\left (c+d x^2\right )^2 (b c-a d)}{2 a b x^{7/2} \left (a+b x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^3/(x^(9/2)*(a + b*x^2)^2),x]

[Out]

-1/14*(c^2*(11*b*c - 7*a*d))/(a^2*b*x^(7/2)) + (c*(11*b^2*c^2 - 21*a*b*c*d + 6*a^2*d^2))/(6*a^3*b*x^(3/2)) + (
(b*c - a*d)*(c + d*x^2)^2)/(2*a*b*x^(7/2)*(a + b*x^2)) - ((b*c - a*d)^2*(11*b*c + a*d)*ArcTan[1 - (Sqrt[2]*b^(
1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(15/4)*b^(5/4)) + ((b*c - a*d)^2*(11*b*c + a*d)*ArcTan[1 + (Sqrt[2]*b^(1/
4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(15/4)*b^(5/4)) - ((b*c - a*d)^2*(11*b*c + a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/
4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(15/4)*b^(5/4)) + ((b*c - a*d)^2*(11*b*c + a*d)*Log[Sqrt[a] + Sq
rt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(15/4)*b^(5/4))

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 479

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-(c*b -
a*d))*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(a*b*e*n*(p + 1))), x] + Dist[1/(a*b*n*(p + 1)),
Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(c*b*n*(p + 1) + (c*b - a*d)*(m + 1)) + d*(c*b*n*(
p + 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0]
&& IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 584

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^
(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a,
b, c, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps

\begin {align*} \int \frac {\left (c+d x^2\right )^3}{x^{9/2} \left (a+b x^2\right )^2} \, dx &=2 \text {Subst}\left (\int \frac {\left (c+d x^4\right )^3}{x^8 \left (a+b x^4\right )^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{7/2} \left (a+b x^2\right )}-\frac {\text {Subst}\left (\int \frac {\left (c+d x^4\right ) \left (-c (11 b c-7 a d)-d (3 b c+a d) x^4\right )}{x^8 \left (a+b x^4\right )} \, dx,x,\sqrt {x}\right )}{2 a b}\\ &=\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{7/2} \left (a+b x^2\right )}-\frac {\text {Subst}\left (\int \left (\frac {c^2 (-11 b c+7 a d)}{a x^8}+\frac {c \left (11 b^2 c^2-21 a b c d+6 a^2 d^2\right )}{a^2 x^4}-\frac {(-b c+a d)^2 (11 b c+a d)}{a^2 \left (a+b x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{2 a b}\\ &=-\frac {c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac {c \left (11 b^2 c^2-21 a b c d+6 a^2 d^2\right )}{6 a^3 b x^{3/2}}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{7/2} \left (a+b x^2\right )}+\frac {\left ((b c-a d)^2 (11 b c+a d)\right ) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a^3 b}\\ &=-\frac {c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac {c \left (11 b^2 c^2-21 a b c d+6 a^2 d^2\right )}{6 a^3 b x^{3/2}}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{7/2} \left (a+b x^2\right )}+\frac {\left ((b c-a d)^2 (11 b c+a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^{7/2} b}+\frac {\left ((b c-a d)^2 (11 b c+a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^{7/2} b}\\ &=-\frac {c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac {c \left (11 b^2 c^2-21 a b c d+6 a^2 d^2\right )}{6 a^3 b x^{3/2}}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{7/2} \left (a+b x^2\right )}+\frac {\left ((b c-a d)^2 (11 b c+a d)\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 a^{7/2} b^{3/2}}+\frac {\left ((b c-a d)^2 (11 b c+a d)\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 a^{7/2} b^{3/2}}-\frac {\left ((b c-a d)^2 (11 b c+a d)\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} a^{15/4} b^{5/4}}-\frac {\left ((b c-a d)^2 (11 b c+a d)\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} a^{15/4} b^{5/4}}\\ &=-\frac {c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac {c \left (11 b^2 c^2-21 a b c d+6 a^2 d^2\right )}{6 a^3 b x^{3/2}}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{7/2} \left (a+b x^2\right )}-\frac {(b c-a d)^2 (11 b c+a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{15/4} b^{5/4}}+\frac {(b c-a d)^2 (11 b c+a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{15/4} b^{5/4}}+\frac {\left ((b c-a d)^2 (11 b c+a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{15/4} b^{5/4}}-\frac {\left ((b c-a d)^2 (11 b c+a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{15/4} b^{5/4}}\\ &=-\frac {c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac {c \left (11 b^2 c^2-21 a b c d+6 a^2 d^2\right )}{6 a^3 b x^{3/2}}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{7/2} \left (a+b x^2\right )}-\frac {(b c-a d)^2 (11 b c+a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{15/4} b^{5/4}}+\frac {(b c-a d)^2 (11 b c+a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{15/4} b^{5/4}}-\frac {(b c-a d)^2 (11 b c+a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{15/4} b^{5/4}}+\frac {(b c-a d)^2 (11 b c+a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{15/4} b^{5/4}}\\ \end {align*}

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Mathematica [A]
time = 0.55, size = 236, normalized size = 0.63 \begin {gather*} \frac {-\frac {4 a^{3/4} \sqrt [4]{b} \left (-77 b^3 c^3 x^4+21 a^3 d^3 x^4+a b^2 c^2 x^2 \left (-44 c+147 d x^2\right )+3 a^2 b c \left (4 c^2+28 c d x^2-21 d^2 x^4\right )\right )}{x^{7/2} \left (a+b x^2\right )}-21 \sqrt {2} (b c-a d)^2 (11 b c+a d) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+21 \sqrt {2} (b c-a d)^2 (11 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{168 a^{15/4} b^{5/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)^3/(x^(9/2)*(a + b*x^2)^2),x]

[Out]

((-4*a^(3/4)*b^(1/4)*(-77*b^3*c^3*x^4 + 21*a^3*d^3*x^4 + a*b^2*c^2*x^2*(-44*c + 147*d*x^2) + 3*a^2*b*c*(4*c^2
+ 28*c*d*x^2 - 21*d^2*x^4)))/(x^(7/2)*(a + b*x^2)) - 21*Sqrt[2]*(b*c - a*d)^2*(11*b*c + a*d)*ArcTan[(Sqrt[a] -
Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 21*Sqrt[2]*(b*c - a*d)^2*(11*b*c + a*d)*ArcTanh[(Sqrt[2]*a^(1
/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(168*a^(15/4)*b^(5/4))

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Maple [A]
time = 0.12, size = 236, normalized size = 0.63

method	result	size
derivativedivides	\(\frac {-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {x}}{2 b \left (b \,x^{2}+a \right )}+\frac {\left (a^{3} d^{3}+9 a^{2} b c \,d^{2}-21 a \,b^{2} c^{2} d +11 b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 b a}}{a^{3}}-\frac {2 c^{3}}{7 a^{2} x^{\frac {7}{2}}}-\frac {2 c^{2} \left (3 a d -2 b c \right )}{3 a^{3} x^{\frac {3}{2}}}\)	\(236\)
default	\(\frac {-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {x}}{2 b \left (b \,x^{2}+a \right )}+\frac {\left (a^{3} d^{3}+9 a^{2} b c \,d^{2}-21 a \,b^{2} c^{2} d +11 b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 b a}}{a^{3}}-\frac {2 c^{3}}{7 a^{2} x^{\frac {7}{2}}}-\frac {2 c^{2} \left (3 a d -2 b c \right )}{3 a^{3} x^{\frac {3}{2}}}\)	\(236\)
risch	\(-\frac {2 c^{2} \left (21 a d \,x^{2}-14 c \,x^{2} b +3 a c \right )}{21 a^{3} x^{\frac {7}{2}}}-\frac {\sqrt {x}\, d^{3}}{2 b \left (b \,x^{2}+a \right )}+\frac {3 \sqrt {x}\, d^{2} c}{2 a \left (b \,x^{2}+a \right )}-\frac {3 b \sqrt {x}\, d \,c^{2}}{2 a^{2} \left (b \,x^{2}+a \right )}+\frac {\sqrt {x}\, c^{3} b^{2}}{2 a^{3} \left (b \,x^{2}+a \right )}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) d^{3}}{8 a b}+\frac {9 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) d^{2} c}{8 a^{2}}-\frac {21 b \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) d \,c^{2}}{8 a^{3}}+\frac {11 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) c^{3} b^{2}}{8 a^{4}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) d^{3}}{8 a b}+\frac {9 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) d^{2} c}{8 a^{2}}-\frac {21 b \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) d \,c^{2}}{8 a^{3}}+\frac {11 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) c^{3} b^{2}}{8 a^{4}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right ) d^{3}}{16 a b}+\frac {9 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right ) d^{2} c}{16 a^{2}}-\frac {21 b \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right ) d \,c^{2}}{16 a^{3}}+\frac {11 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right ) c^{3} b^{2}}{16 a^{4}}\)	\(701\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)^3/x^(9/2)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

2/a^3*(-1/4*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/b*x^(1/2)/(b*x^2+a)+1/32*(a^3*d^3+9*a^2*b*c*d^2-21*a
*b^2*c^2*d+11*b^3*c^3)/b*(a/b)^(1/4)/a*2^(1/2)*(ln((x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*
x^(1/2)*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1))
)-2/7*c^3/a^2/x^(7/2)-2/3*c^2*(3*a*d-2*b*c)/a^3/x^(3/2)

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Maxima [A]
time = 0.55, size = 424, normalized size = 1.13 \begin {gather*} -\frac {12 \, a^{2} b c^{3} - 7 \, {\left (11 \, b^{3} c^{3} - 21 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 3 \, a^{3} d^{3}\right )} x^{4} - 4 \, {\left (11 \, a b^{2} c^{3} - 21 \, a^{2} b c^{2} d\right )} x^{2}}{42 \, {\left (a^{3} b^{2} x^{\frac {11}{2}} + a^{4} b x^{\frac {7}{2}}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (11 \, b^{3} c^{3} - 21 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (11 \, b^{3} c^{3} - 21 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (11 \, b^{3} c^{3} - 21 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (11 \, b^{3} c^{3} - 21 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{16 \, a^{3} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^3/x^(9/2)/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

-1/42*(12*a^2*b*c^3 - 7*(11*b^3*c^3 - 21*a*b^2*c^2*d + 9*a^2*b*c*d^2 - 3*a^3*d^3)*x^4 - 4*(11*a*b^2*c^3 - 21*a
^2*b*c^2*d)*x^2)/(a^3*b^2*x^(11/2) + a^4*b*x^(7/2)) + 1/16*(2*sqrt(2)*(11*b^3*c^3 - 21*a*b^2*c^2*d + 9*a^2*b*c
*d^2 + a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(
a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*(11*b^3*c^3 - 21*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*arctan(-1/2*sqrt
(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqr
t(2)*(11*b^3*c^3 - 21*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x +
sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2)*(11*b^3*c^3 - 21*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*log(-sqrt(2)*a^(
1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))/(a^3*b)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1955 vs. \(2 (292) = 584\).
time = 0.51, size = 1955, normalized size = 5.20 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^3/x^(9/2)/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

1/168*(84*(a^3*b^2*x^6 + a^4*b*x^4)*(-(14641*b^12*c^12 - 111804*a*b^11*c^11*d + 368082*a^2*b^10*c^10*d^2 - 676
588*a^3*b^9*c^9*d^3 + 746703*a^4*b^8*c^8*d^4 - 486648*a^5*b^7*c^7*d^5 + 160188*a^6*b^6*c^6*d^6 - 5688*a^7*b^5*
c^5*d^7 - 10017*a^8*b^4*c^4*d^8 + 692*a^9*b^3*c^3*d^9 + 402*a^10*b^2*c^2*d^10 + 36*a^11*b*c*d^11 + a^12*d^12)/
(a^15*b^5))^(1/4)*arctan((sqrt(a^8*b^2*sqrt(-(14641*b^12*c^12 - 111804*a*b^11*c^11*d + 368082*a^2*b^10*c^10*d^
2 - 676588*a^3*b^9*c^9*d^3 + 746703*a^4*b^8*c^8*d^4 - 486648*a^5*b^7*c^7*d^5 + 160188*a^6*b^6*c^6*d^6 - 5688*a
^7*b^5*c^5*d^7 - 10017*a^8*b^4*c^4*d^8 + 692*a^9*b^3*c^3*d^9 + 402*a^10*b^2*c^2*d^10 + 36*a^11*b*c*d^11 + a^12
*d^12)/(a^15*b^5)) + (121*b^6*c^6 - 462*a*b^5*c^5*d + 639*a^2*b^4*c^4*d^2 - 356*a^3*b^3*c^3*d^3 + 39*a^4*b^2*c
^2*d^4 + 18*a^5*b*c*d^5 + a^6*d^6)*x)*a^11*b^4*(-(14641*b^12*c^12 - 111804*a*b^11*c^11*d + 368082*a^2*b^10*c^1
0*d^2 - 676588*a^3*b^9*c^9*d^3 + 746703*a^4*b^8*c^8*d^4 - 486648*a^5*b^7*c^7*d^5 + 160188*a^6*b^6*c^6*d^6 - 56
88*a^7*b^5*c^5*d^7 - 10017*a^8*b^4*c^4*d^8 + 692*a^9*b^3*c^3*d^9 + 402*a^10*b^2*c^2*d^10 + 36*a^11*b*c*d^11 +
a^12*d^12)/(a^15*b^5))^(3/4) - (11*a^11*b^7*c^3 - 21*a^12*b^6*c^2*d + 9*a^13*b^5*c*d^2 + a^14*b^4*d^3)*sqrt(x)
*(-(14641*b^12*c^12 - 111804*a*b^11*c^11*d + 368082*a^2*b^10*c^10*d^2 - 676588*a^3*b^9*c^9*d^3 + 746703*a^4*b^
8*c^8*d^4 - 486648*a^5*b^7*c^7*d^5 + 160188*a^6*b^6*c^6*d^6 - 5688*a^7*b^5*c^5*d^7 - 10017*a^8*b^4*c^4*d^8 + 6
92*a^9*b^3*c^3*d^9 + 402*a^10*b^2*c^2*d^10 + 36*a^11*b*c*d^11 + a^12*d^12)/(a^15*b^5))^(3/4))/(14641*b^12*c^12
- 111804*a*b^11*c^11*d + 368082*a^2*b^10*c^10*d^2 - 676588*a^3*b^9*c^9*d^3 + 746703*a^4*b^8*c^8*d^4 - 486648*
a^5*b^7*c^7*d^5 + 160188*a^6*b^6*c^6*d^6 - 5688*a^7*b^5*c^5*d^7 - 10017*a^8*b^4*c^4*d^8 + 692*a^9*b^3*c^3*d^9
+ 402*a^10*b^2*c^2*d^10 + 36*a^11*b*c*d^11 + a^12*d^12)) + 21*(a^3*b^2*x^6 + a^4*b*x^4)*(-(14641*b^12*c^12 - 1
11804*a*b^11*c^11*d + 368082*a^2*b^10*c^10*d^2 - 676588*a^3*b^9*c^9*d^3 + 746703*a^4*b^8*c^8*d^4 - 486648*a^5*
b^7*c^7*d^5 + 160188*a^6*b^6*c^6*d^6 - 5688*a^7*b^5*c^5*d^7 - 10017*a^8*b^4*c^4*d^8 + 692*a^9*b^3*c^3*d^9 + 40
2*a^10*b^2*c^2*d^10 + 36*a^11*b*c*d^11 + a^12*d^12)/(a^15*b^5))^(1/4)*log(a^4*b*(-(14641*b^12*c^12 - 111804*a*
b^11*c^11*d + 368082*a^2*b^10*c^10*d^2 - 676588*a^3*b^9*c^9*d^3 + 746703*a^4*b^8*c^8*d^4 - 486648*a^5*b^7*c^7*
d^5 + 160188*a^6*b^6*c^6*d^6 - 5688*a^7*b^5*c^5*d^7 - 10017*a^8*b^4*c^4*d^8 + 692*a^9*b^3*c^3*d^9 + 402*a^10*b
^2*c^2*d^10 + 36*a^11*b*c*d^11 + a^12*d^12)/(a^15*b^5))^(1/4) + (11*b^3*c^3 - 21*a*b^2*c^2*d + 9*a^2*b*c*d^2 +
a^3*d^3)*sqrt(x)) - 21*(a^3*b^2*x^6 + a^4*b*x^4)*(-(14641*b^12*c^12 - 111804*a*b^11*c^11*d + 368082*a^2*b^10*
c^10*d^2 - 676588*a^3*b^9*c^9*d^3 + 746703*a^4*b^8*c^8*d^4 - 486648*a^5*b^7*c^7*d^5 + 160188*a^6*b^6*c^6*d^6 -
5688*a^7*b^5*c^5*d^7 - 10017*a^8*b^4*c^4*d^8 + 692*a^9*b^3*c^3*d^9 + 402*a^10*b^2*c^2*d^10 + 36*a^11*b*c*d^11
+ a^12*d^12)/(a^15*b^5))^(1/4)*log(-a^4*b*(-(14641*b^12*c^12 - 111804*a*b^11*c^11*d + 368082*a^2*b^10*c^10*d^
2 - 676588*a^3*b^9*c^9*d^3 + 746703*a^4*b^8*c^8*d^4 - 486648*a^5*b^7*c^7*d^5 + 160188*a^6*b^6*c^6*d^6 - 5688*a
^7*b^5*c^5*d^7 - 10017*a^8*b^4*c^4*d^8 + 692*a^9*b^3*c^3*d^9 + 402*a^10*b^2*c^2*d^10 + 36*a^11*b*c*d^11 + a^12
*d^12)/(a^15*b^5))^(1/4) + (11*b^3*c^3 - 21*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*sqrt(x)) - 4*(12*a^2*b*c^3
- 7*(11*b^3*c^3 - 21*a*b^2*c^2*d + 9*a^2*b*c*d^2 - 3*a^3*d^3)*x^4 - 4*(11*a*b^2*c^3 - 21*a^2*b*c^2*d)*x^2)*sqr
t(x))/(a^3*b^2*x^6 + a^4*b*x^4)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)**3/x**(9/2)/(b*x**2+a)**2,x)

[Out]

Timed out

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Giac [A]
time = 1.02, size = 509, normalized size = 1.35 \begin {gather*} \frac {\sqrt {2} {\left (11 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 21 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 9 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{4} b^{2}} + \frac {\sqrt {2} {\left (11 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 21 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 9 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{4} b^{2}} + \frac {\sqrt {2} {\left (11 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 21 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 9 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{4} b^{2}} - \frac {\sqrt {2} {\left (11 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 21 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 9 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{4} b^{2}} + \frac {b^{3} c^{3} \sqrt {x} - 3 \, a b^{2} c^{2} d \sqrt {x} + 3 \, a^{2} b c d^{2} \sqrt {x} - a^{3} d^{3} \sqrt {x}}{2 \, {\left (b x^{2} + a\right )} a^{3} b} + \frac {2 \, {\left (14 \, b c^{3} x^{2} - 21 \, a c^{2} d x^{2} - 3 \, a c^{3}\right )}}{21 \, a^{3} x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^3/x^(9/2)/(b*x^2+a)^2,x, algorithm="giac")

[Out]

1/8*sqrt(2)*(11*(a*b^3)^(1/4)*b^3*c^3 - 21*(a*b^3)^(1/4)*a*b^2*c^2*d + 9*(a*b^3)^(1/4)*a^2*b*c*d^2 + (a*b^3)^(
1/4)*a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a^4*b^2) + 1/8*sqrt(2)*(11*(a
*b^3)^(1/4)*b^3*c^3 - 21*(a*b^3)^(1/4)*a*b^2*c^2*d + 9*(a*b^3)^(1/4)*a^2*b*c*d^2 + (a*b^3)^(1/4)*a^3*d^3)*arct
an(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a^4*b^2) + 1/16*sqrt(2)*(11*(a*b^3)^(1/4)*b^3*
c^3 - 21*(a*b^3)^(1/4)*a*b^2*c^2*d + 9*(a*b^3)^(1/4)*a^2*b*c*d^2 + (a*b^3)^(1/4)*a^3*d^3)*log(sqrt(2)*sqrt(x)*
(a/b)^(1/4) + x + sqrt(a/b))/(a^4*b^2) - 1/16*sqrt(2)*(11*(a*b^3)^(1/4)*b^3*c^3 - 21*(a*b^3)^(1/4)*a*b^2*c^2*d
+ 9*(a*b^3)^(1/4)*a^2*b*c*d^2 + (a*b^3)^(1/4)*a^3*d^3)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^4
*b^2) + 1/2*(b^3*c^3*sqrt(x) - 3*a*b^2*c^2*d*sqrt(x) + 3*a^2*b*c*d^2*sqrt(x) - a^3*d^3*sqrt(x))/((b*x^2 + a)*a
^3*b) + 2/21*(14*b*c^3*x^2 - 21*a*c^2*d*x^2 - 3*a*c^3)/(a^3*x^(7/2))

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Mupad [B]
time = 0.33, size = 1746, normalized size = 4.64 \begin {gather*} -\frac {\frac {2\,c^3}{7\,a}+\frac {x^4\,\left (3\,a^3\,d^3-9\,a^2\,b\,c\,d^2+21\,a\,b^2\,c^2\,d-11\,b^3\,c^3\right )}{6\,a^3\,b}+\frac {2\,c^2\,x^2\,\left (21\,a\,d-11\,b\,c\right )}{21\,a^2}}{a\,x^{7/2}+b\,x^{11/2}}+\frac {\mathrm {atan}\left (\frac {\frac {\left (\sqrt {x}\,\left (32\,a^{15}\,b^6\,d^6+576\,a^{14}\,b^7\,c\,d^5+1248\,a^{13}\,b^8\,c^2\,d^4-11392\,a^{12}\,b^9\,c^3\,d^3+20448\,a^{11}\,b^{10}\,c^4\,d^2-14784\,a^{10}\,b^{11}\,c^5\,d+3872\,a^9\,b^{12}\,c^6\right )-\frac {{\left (a\,d-b\,c\right )}^2\,\left (a\,d+11\,b\,c\right )\,\left (256\,a^{16}\,b^7\,d^3+2304\,a^{15}\,b^8\,c\,d^2-5376\,a^{14}\,b^9\,c^2\,d+2816\,a^{13}\,b^{10}\,c^3\right )}{8\,{\left (-a\right )}^{15/4}\,b^{5/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d+11\,b\,c\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{15/4}\,b^{5/4}}+\frac {\left (\sqrt {x}\,\left (32\,a^{15}\,b^6\,d^6+576\,a^{14}\,b^7\,c\,d^5+1248\,a^{13}\,b^8\,c^2\,d^4-11392\,a^{12}\,b^9\,c^3\,d^3+20448\,a^{11}\,b^{10}\,c^4\,d^2-14784\,a^{10}\,b^{11}\,c^5\,d+3872\,a^9\,b^{12}\,c^6\right )+\frac {{\left (a\,d-b\,c\right )}^2\,\left (a\,d+11\,b\,c\right )\,\left (256\,a^{16}\,b^7\,d^3+2304\,a^{15}\,b^8\,c\,d^2-5376\,a^{14}\,b^9\,c^2\,d+2816\,a^{13}\,b^{10}\,c^3\right )}{8\,{\left (-a\right )}^{15/4}\,b^{5/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d+11\,b\,c\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{15/4}\,b^{5/4}}}{\frac {\left (\sqrt {x}\,\left (32\,a^{15}\,b^6\,d^6+576\,a^{14}\,b^7\,c\,d^5+1248\,a^{13}\,b^8\,c^2\,d^4-11392\,a^{12}\,b^9\,c^3\,d^3+20448\,a^{11}\,b^{10}\,c^4\,d^2-14784\,a^{10}\,b^{11}\,c^5\,d+3872\,a^9\,b^{12}\,c^6\right )-\frac {{\left (a\,d-b\,c\right )}^2\,\left (a\,d+11\,b\,c\right )\,\left (256\,a^{16}\,b^7\,d^3+2304\,a^{15}\,b^8\,c\,d^2-5376\,a^{14}\,b^9\,c^2\,d+2816\,a^{13}\,b^{10}\,c^3\right )}{8\,{\left (-a\right )}^{15/4}\,b^{5/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d+11\,b\,c\right )}{8\,{\left (-a\right )}^{15/4}\,b^{5/4}}-\frac {\left (\sqrt {x}\,\left (32\,a^{15}\,b^6\,d^6+576\,a^{14}\,b^7\,c\,d^5+1248\,a^{13}\,b^8\,c^2\,d^4-11392\,a^{12}\,b^9\,c^3\,d^3+20448\,a^{11}\,b^{10}\,c^4\,d^2-14784\,a^{10}\,b^{11}\,c^5\,d+3872\,a^9\,b^{12}\,c^6\right )+\frac {{\left (a\,d-b\,c\right )}^2\,\left (a\,d+11\,b\,c\right )\,\left (256\,a^{16}\,b^7\,d^3+2304\,a^{15}\,b^8\,c\,d^2-5376\,a^{14}\,b^9\,c^2\,d+2816\,a^{13}\,b^{10}\,c^3\right )}{8\,{\left (-a\right )}^{15/4}\,b^{5/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d+11\,b\,c\right )}{8\,{\left (-a\right )}^{15/4}\,b^{5/4}}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d+11\,b\,c\right )\,1{}\mathrm {i}}{4\,{\left (-a\right )}^{15/4}\,b^{5/4}}+\frac {\mathrm {atan}\left (\frac {\frac {\left (\sqrt {x}\,\left (32\,a^{15}\,b^6\,d^6+576\,a^{14}\,b^7\,c\,d^5+1248\,a^{13}\,b^8\,c^2\,d^4-11392\,a^{12}\,b^9\,c^3\,d^3+20448\,a^{11}\,b^{10}\,c^4\,d^2-14784\,a^{10}\,b^{11}\,c^5\,d+3872\,a^9\,b^{12}\,c^6\right )-\frac {{\left (a\,d-b\,c\right )}^2\,\left (a\,d+11\,b\,c\right )\,\left (256\,a^{16}\,b^7\,d^3+2304\,a^{15}\,b^8\,c\,d^2-5376\,a^{14}\,b^9\,c^2\,d+2816\,a^{13}\,b^{10}\,c^3\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{15/4}\,b^{5/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d+11\,b\,c\right )}{8\,{\left (-a\right )}^{15/4}\,b^{5/4}}+\frac {\left (\sqrt {x}\,\left (32\,a^{15}\,b^6\,d^6+576\,a^{14}\,b^7\,c\,d^5+1248\,a^{13}\,b^8\,c^2\,d^4-11392\,a^{12}\,b^9\,c^3\,d^3+20448\,a^{11}\,b^{10}\,c^4\,d^2-14784\,a^{10}\,b^{11}\,c^5\,d+3872\,a^9\,b^{12}\,c^6\right )+\frac {{\left (a\,d-b\,c\right )}^2\,\left (a\,d+11\,b\,c\right )\,\left (256\,a^{16}\,b^7\,d^3+2304\,a^{15}\,b^8\,c\,d^2-5376\,a^{14}\,b^9\,c^2\,d+2816\,a^{13}\,b^{10}\,c^3\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{15/4}\,b^{5/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d+11\,b\,c\right )}{8\,{\left (-a\right )}^{15/4}\,b^{5/4}}}{\frac {\left (\sqrt {x}\,\left (32\,a^{15}\,b^6\,d^6+576\,a^{14}\,b^7\,c\,d^5+1248\,a^{13}\,b^8\,c^2\,d^4-11392\,a^{12}\,b^9\,c^3\,d^3+20448\,a^{11}\,b^{10}\,c^4\,d^2-14784\,a^{10}\,b^{11}\,c^5\,d+3872\,a^9\,b^{12}\,c^6\right )-\frac {{\left (a\,d-b\,c\right )}^2\,\left (a\,d+11\,b\,c\right )\,\left (256\,a^{16}\,b^7\,d^3+2304\,a^{15}\,b^8\,c\,d^2-5376\,a^{14}\,b^9\,c^2\,d+2816\,a^{13}\,b^{10}\,c^3\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{15/4}\,b^{5/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d+11\,b\,c\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{15/4}\,b^{5/4}}-\frac {\left (\sqrt {x}\,\left (32\,a^{15}\,b^6\,d^6+576\,a^{14}\,b^7\,c\,d^5+1248\,a^{13}\,b^8\,c^2\,d^4-11392\,a^{12}\,b^9\,c^3\,d^3+20448\,a^{11}\,b^{10}\,c^4\,d^2-14784\,a^{10}\,b^{11}\,c^5\,d+3872\,a^9\,b^{12}\,c^6\right )+\frac {{\left (a\,d-b\,c\right )}^2\,\left (a\,d+11\,b\,c\right )\,\left (256\,a^{16}\,b^7\,d^3+2304\,a^{15}\,b^8\,c\,d^2-5376\,a^{14}\,b^9\,c^2\,d+2816\,a^{13}\,b^{10}\,c^3\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{15/4}\,b^{5/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d+11\,b\,c\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{15/4}\,b^{5/4}}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d+11\,b\,c\right )}{4\,{\left (-a\right )}^{15/4}\,b^{5/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^2)^3/(x^(9/2)*(a + b*x^2)^2),x)

[Out]

(atan((((x^(1/2)*(3872*a^9*b^12*c^6 + 32*a^15*b^6*d^6 - 14784*a^10*b^11*c^5*d + 576*a^14*b^7*c*d^5 + 20448*a^1
1*b^10*c^4*d^2 - 11392*a^12*b^9*c^3*d^3 + 1248*a^13*b^8*c^2*d^4) - ((a*d - b*c)^2*(a*d + 11*b*c)*(2816*a^13*b^
10*c^3 + 256*a^16*b^7*d^3 - 5376*a^14*b^9*c^2*d + 2304*a^15*b^8*c*d^2))/(8*(-a)^(15/4)*b^(5/4)))*(a*d - b*c)^2
*(a*d + 11*b*c)*1i)/(8*(-a)^(15/4)*b^(5/4)) + ((x^(1/2)*(3872*a^9*b^12*c^6 + 32*a^15*b^6*d^6 - 14784*a^10*b^11
*c^5*d + 576*a^14*b^7*c*d^5 + 20448*a^11*b^10*c^4*d^2 - 11392*a^12*b^9*c^3*d^3 + 1248*a^13*b^8*c^2*d^4) + ((a*
d - b*c)^2*(a*d + 11*b*c)*(2816*a^13*b^10*c^3 + 256*a^16*b^7*d^3 - 5376*a^14*b^9*c^2*d + 2304*a^15*b^8*c*d^2))
/(8*(-a)^(15/4)*b^(5/4)))*(a*d - b*c)^2*(a*d + 11*b*c)*1i)/(8*(-a)^(15/4)*b^(5/4)))/(((x^(1/2)*(3872*a^9*b^12*
c^6 + 32*a^15*b^6*d^6 - 14784*a^10*b^11*c^5*d + 576*a^14*b^7*c*d^5 + 20448*a^11*b^10*c^4*d^2 - 11392*a^12*b^9*
c^3*d^3 + 1248*a^13*b^8*c^2*d^4) - ((a*d - b*c)^2*(a*d + 11*b*c)*(2816*a^13*b^10*c^3 + 256*a^16*b^7*d^3 - 5376
*a^14*b^9*c^2*d + 2304*a^15*b^8*c*d^2))/(8*(-a)^(15/4)*b^(5/4)))*(a*d - b*c)^2*(a*d + 11*b*c))/(8*(-a)^(15/4)*
b^(5/4)) - ((x^(1/2)*(3872*a^9*b^12*c^6 + 32*a^15*b^6*d^6 - 14784*a^10*b^11*c^5*d + 576*a^14*b^7*c*d^5 + 20448
*a^11*b^10*c^4*d^2 - 11392*a^12*b^9*c^3*d^3 + 1248*a^13*b^8*c^2*d^4) + ((a*d - b*c)^2*(a*d + 11*b*c)*(2816*a^1
3*b^10*c^3 + 256*a^16*b^7*d^3 - 5376*a^14*b^9*c^2*d + 2304*a^15*b^8*c*d^2))/(8*(-a)^(15/4)*b^(5/4)))*(a*d - b*
c)^2*(a*d + 11*b*c))/(8*(-a)^(15/4)*b^(5/4))))*(a*d - b*c)^2*(a*d + 11*b*c)*1i)/(4*(-a)^(15/4)*b^(5/4)) - ((2*
c^3)/(7*a) + (x^4*(3*a^3*d^3 - 11*b^3*c^3 + 21*a*b^2*c^2*d - 9*a^2*b*c*d^2))/(6*a^3*b) + (2*c^2*x^2*(21*a*d -
11*b*c))/(21*a^2))/(a*x^(7/2) + b*x^(11/2)) + (atan((((x^(1/2)*(3872*a^9*b^12*c^6 + 32*a^15*b^6*d^6 - 14784*a^
10*b^11*c^5*d + 576*a^14*b^7*c*d^5 + 20448*a^11*b^10*c^4*d^2 - 11392*a^12*b^9*c^3*d^3 + 1248*a^13*b^8*c^2*d^4)
- ((a*d - b*c)^2*(a*d + 11*b*c)*(2816*a^13*b^10*c^3 + 256*a^16*b^7*d^3 - 5376*a^14*b^9*c^2*d + 2304*a^15*b^8*
c*d^2)*1i)/(8*(-a)^(15/4)*b^(5/4)))*(a*d - b*c)^2*(a*d + 11*b*c))/(8*(-a)^(15/4)*b^(5/4)) + ((x^(1/2)*(3872*a^
9*b^12*c^6 + 32*a^15*b^6*d^6 - 14784*a^10*b^11*c^5*d + 576*a^14*b^7*c*d^5 + 20448*a^11*b^10*c^4*d^2 - 11392*a^
12*b^9*c^3*d^3 + 1248*a^13*b^8*c^2*d^4) + ((a*d - b*c)^2*(a*d + 11*b*c)*(2816*a^13*b^10*c^3 + 256*a^16*b^7*d^3
- 5376*a^14*b^9*c^2*d + 2304*a^15*b^8*c*d^2)*1i)/(8*(-a)^(15/4)*b^(5/4)))*(a*d - b*c)^2*(a*d + 11*b*c))/(8*(-
a)^(15/4)*b^(5/4)))/(((x^(1/2)*(3872*a^9*b^12*c^6 + 32*a^15*b^6*d^6 - 14784*a^10*b^11*c^5*d + 576*a^14*b^7*c*d
^5 + 20448*a^11*b^10*c^4*d^2 - 11392*a^12*b^9*c^3*d^3 + 1248*a^13*b^8*c^2*d^4) - ((a*d - b*c)^2*(a*d + 11*b*c)
*(2816*a^13*b^10*c^3 + 256*a^16*b^7*d^3 - 5376*a^14*b^9*c^2*d + 2304*a^15*b^8*c*d^2)*1i)/(8*(-a)^(15/4)*b^(5/4
)))*(a*d - b*c)^2*(a*d + 11*b*c)*1i)/(8*(-a)^(15/4)*b^(5/4)) - ((x^(1/2)*(3872*a^9*b^12*c^6 + 32*a^15*b^6*d^6
- 14784*a^10*b^11*c^5*d + 576*a^14*b^7*c*d^5 + 20448*a^11*b^10*c^4*d^2 - 11392*a^12*b^9*c^3*d^3 + 1248*a^13*b^
8*c^2*d^4) + ((a*d - b*c)^2*(a*d + 11*b*c)*(2816*a^13*b^10*c^3 + 256*a^16*b^7*d^3 - 5376*a^14*b^9*c^2*d + 2304
*a^15*b^8*c*d^2)*1i)/(8*(-a)^(15/4)*b^(5/4)))*(a*d - b*c)^2*(a*d + 11*b*c)*1i)/(8*(-a)^(15/4)*b^(5/4))))*(a*d
- b*c)^2*(a*d + 11*b*c))/(4*(-a)^(15/4)*b^(5/4))

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