3.5.59 \(\int \frac {(c+d x^2)^3}{x^{7/2} (a+b x^2)^2} \, dx\) [459]

Optimal. Leaf size=376 \[ -\frac {c^2 (9 b c-5 a d)}{10 a^2 b x^{5/2}}+\frac {c \left (9 b^2 c^2-15 a b c d+2 a^2 d^2\right )}{2 a^3 b \sqrt {x}}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{5/2} \left (a+b x^2\right )}-\frac {3 (b c-a d)^2 (3 b c+a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{13/4} b^{7/4}}+\frac {3 (b c-a d)^2 (3 b c+a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{13/4} b^{7/4}}+\frac {3 (b c-a d)^2 (3 b c+a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4} b^{7/4}}-\frac {3 (b c-a d)^2 (3 b c+a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4} b^{7/4}} \]

[Out]

-1/10*c^2*(-5*a*d+9*b*c)/a^2/b/x^(5/2)+1/2*(-a*d+b*c)*(d*x^2+c)^2/a/b/x^(5/2)/(b*x^2+a)-3/8*(-a*d+b*c)^2*(a*d+
3*b*c)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(13/4)/b^(7/4)*2^(1/2)+3/8*(-a*d+b*c)^2*(a*d+3*b*c)*arctan(
1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(13/4)/b^(7/4)*2^(1/2)+3/16*(-a*d+b*c)^2*(a*d+3*b*c)*ln(a^(1/2)+x*b^(1/2)
-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(13/4)/b^(7/4)*2^(1/2)-3/16*(-a*d+b*c)^2*(a*d+3*b*c)*ln(a^(1/2)+x*b^(1/2)+
a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(13/4)/b^(7/4)*2^(1/2)+1/2*c*(2*a^2*d^2-15*a*b*c*d+9*b^2*c^2)/a^3/b/x^(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, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {3 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-a d)^2 (a d+3 b c)}{4 \sqrt {2} a^{13/4} b^{7/4}}+\frac {3 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-a d)^2 (a d+3 b c)}{4 \sqrt {2} a^{13/4} b^{7/4}}+\frac {3 (b c-a d)^2 (a d+3 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4} b^{7/4}}-\frac {3 (b c-a d)^2 (a d+3 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4} b^{7/4}}-\frac {c^2 (9 b c-5 a d)}{10 a^2 b x^{5/2}}+\frac {c \left (2 a^2 d^2-15 a b c d+9 b^2 c^2\right )}{2 a^3 b \sqrt {x}}+\frac {\left (c+d x^2\right )^2 (b c-a d)}{2 a b x^{5/2} \left (a+b x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^3/(x^(7/2)*(a + b*x^2)^2),x]

[Out]

-1/10*(c^2*(9*b*c - 5*a*d))/(a^2*b*x^(5/2)) + (c*(9*b^2*c^2 - 15*a*b*c*d + 2*a^2*d^2))/(2*a^3*b*Sqrt[x]) + ((b
*c - a*d)*(c + d*x^2)^2)/(2*a*b*x^(5/2)*(a + b*x^2)) - (3*(b*c - a*d)^2*(3*b*c + a*d)*ArcTan[1 - (Sqrt[2]*b^(1
/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(13/4)*b^(7/4)) + (3*(b*c - a*d)^2*(3*b*c + a*d)*ArcTan[1 + (Sqrt[2]*b^(1/
4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(13/4)*b^(7/4)) + (3*(b*c - a*d)^2*(3*b*c + a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1
/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(13/4)*b^(7/4)) - (3*(b*c - a*d)^2*(3*b*c + a*d)*Log[Sqrt[a] +
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(13/4)*b^(7/4))

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 303

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{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^{7/2} \left (a+b x^2\right )^2} \, dx &=2 \text {Subst}\left (\int \frac {\left (c+d x^4\right )^3}{x^6 \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^{5/2} \left (a+b x^2\right )}-\frac {\text {Subst}\left (\int \frac {\left (c+d x^4\right ) \left (-c (9 b c-5 a d)-d (b c+3 a d) x^4\right )}{x^6 \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^{5/2} \left (a+b x^2\right )}-\frac {\text {Subst}\left (\int \left (\frac {c^2 (-9 b c+5 a d)}{a x^6}+\frac {c \left (9 b^2 c^2-15 a b c d+2 a^2 d^2\right )}{a^2 x^2}-\frac {3 (-b c+a d)^2 (3 b c+a d) x^2}{a^2 \left (a+b x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{2 a b}\\ &=-\frac {c^2 (9 b c-5 a d)}{10 a^2 b x^{5/2}}+\frac {c \left (9 b^2 c^2-15 a b c d+2 a^2 d^2\right )}{2 a^3 b \sqrt {x}}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{5/2} \left (a+b x^2\right )}+\frac {\left (3 (b c-a d)^2 (3 b c+a d)\right ) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a^3 b}\\ &=-\frac {c^2 (9 b c-5 a d)}{10 a^2 b x^{5/2}}+\frac {c \left (9 b^2 c^2-15 a b c d+2 a^2 d^2\right )}{2 a^3 b \sqrt {x}}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{5/2} \left (a+b x^2\right )}-\frac {\left (3 (b c-a d)^2 (3 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^3 b^{3/2}}+\frac {\left (3 (b c-a d)^2 (3 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^3 b^{3/2}}\\ &=-\frac {c^2 (9 b c-5 a d)}{10 a^2 b x^{5/2}}+\frac {c \left (9 b^2 c^2-15 a b c d+2 a^2 d^2\right )}{2 a^3 b \sqrt {x}}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{5/2} \left (a+b x^2\right )}+\frac {\left (3 (b c-a d)^2 (3 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^3 b^2}+\frac {\left (3 (b c-a d)^2 (3 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^3 b^2}+\frac {\left (3 (b c-a d)^2 (3 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^{13/4} b^{7/4}}+\frac {\left (3 (b c-a d)^2 (3 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^{13/4} b^{7/4}}\\ &=-\frac {c^2 (9 b c-5 a d)}{10 a^2 b x^{5/2}}+\frac {c \left (9 b^2 c^2-15 a b c d+2 a^2 d^2\right )}{2 a^3 b \sqrt {x}}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{5/2} \left (a+b x^2\right )}+\frac {3 (b c-a d)^2 (3 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^{13/4} b^{7/4}}-\frac {3 (b c-a d)^2 (3 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^{13/4} b^{7/4}}+\frac {\left (3 (b c-a d)^2 (3 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^{13/4} b^{7/4}}-\frac {\left (3 (b c-a d)^2 (3 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^{13/4} b^{7/4}}\\ &=-\frac {c^2 (9 b c-5 a d)}{10 a^2 b x^{5/2}}+\frac {c \left (9 b^2 c^2-15 a b c d+2 a^2 d^2\right )}{2 a^3 b \sqrt {x}}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{5/2} \left (a+b x^2\right )}-\frac {3 (b c-a d)^2 (3 b c+a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{13/4} b^{7/4}}+\frac {3 (b c-a d)^2 (3 b c+a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{13/4} b^{7/4}}+\frac {3 (b c-a d)^2 (3 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^{13/4} b^{7/4}}-\frac {3 (b c-a d)^2 (3 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^{13/4} b^{7/4}}\\ \end {align*}

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Mathematica [A]
time = 0.56, size = 236, normalized size = 0.63 \begin {gather*} \frac {-\frac {4 \sqrt [4]{a} b^{3/4} \left (-45 b^3 c^3 x^4+5 a^3 d^3 x^4+3 a b^2 c^2 x^2 \left (-12 c+25 d x^2\right )+a^2 b c \left (4 c^2+60 c d x^2-15 d^2 x^4\right )\right )}{x^{5/2} \left (a+b x^2\right )}-15 \sqrt {2} (b c-a d)^2 (3 b c+a d) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-15 \sqrt {2} (b c-a d)^2 (3 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{40 a^{13/4} b^{7/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)^3/(x^(7/2)*(a + b*x^2)^2),x]

[Out]

((-4*a^(1/4)*b^(3/4)*(-45*b^3*c^3*x^4 + 5*a^3*d^3*x^4 + 3*a*b^2*c^2*x^2*(-12*c + 25*d*x^2) + a^2*b*c*(4*c^2 +
60*c*d*x^2 - 15*d^2*x^4)))/(x^(5/2)*(a + b*x^2)) - 15*Sqrt[2]*(b*c - a*d)^2*(3*b*c + a*d)*ArcTan[(Sqrt[a] - Sq
rt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - 15*Sqrt[2]*(b*c - a*d)^2*(3*b*c + a*d)*ArcTanh[(Sqrt[2]*a^(1/4)*
b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(40*a^(13/4)*b^(7/4))

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Maple [A]
time = 0.12, size = 232, normalized size = 0.62

method	result	size
derivativedivides	\(\frac {-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{\frac {3}{2}}}{2 b \left (b \,x^{2}+a \right )}+\frac {3 \left (a^{3} d^{3}+a^{2} b c \,d^{2}-5 a \,b^{2} c^{2} d +3 b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 b^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{a^{3}}-\frac {2 c^{3}}{5 a^{2} x^{\frac {5}{2}}}-\frac {2 c^{2} \left (3 a d -2 b c \right )}{a^{3} \sqrt {x}}\)	\(232\)
default	\(\frac {-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{\frac {3}{2}}}{2 b \left (b \,x^{2}+a \right )}+\frac {3 \left (a^{3} d^{3}+a^{2} b c \,d^{2}-5 a \,b^{2} c^{2} d +3 b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 b^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{a^{3}}-\frac {2 c^{3}}{5 a^{2} x^{\frac {5}{2}}}-\frac {2 c^{2} \left (3 a d -2 b c \right )}{a^{3} \sqrt {x}}\)	\(232\)
risch	\(-\frac {2 c^{2} \left (15 a d \,x^{2}-10 c \,x^{2} b +a c \right )}{5 a^{3} x^{\frac {5}{2}}}-\frac {x^{\frac {3}{2}} d^{3}}{2 b \left (b \,x^{2}+a \right )}+\frac {3 x^{\frac {3}{2}} d^{2} c}{2 a \left (b \,x^{2}+a \right )}-\frac {3 b \,x^{\frac {3}{2}} d \,c^{2}}{2 a^{2} \left (b \,x^{2}+a \right )}+\frac {x^{\frac {3}{2}} c^{3} b^{2}}{2 a^{3} \left (b \,x^{2}+a \right )}+\frac {3 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) d^{3}}{8 b^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {3 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) d^{2} c}{8 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {15 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) d \,c^{2}}{8 a^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {9 b \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) c^{3}}{8 a^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {3 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) d^{3}}{8 b^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {3 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) d^{2} c}{8 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {15 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) d \,c^{2}}{8 a^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {9 b \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) c^{3}}{8 a^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {3 \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 b^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {3 \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 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {15 \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^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {9 b \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}}{16 a^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\)	\(691\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)^3/x^(7/2)/(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^(3/2)/(b*x^2+a)+3/32*(a^3*d^3+a^2*b*c*d^2-5*a*b^
2*c^2*d+3*b^3*c^3)/b^2/(a/b)^(1/4)*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/
5*c^3/a^2/x^(5/2)-2*c^2*(3*a*d-2*b*c)/a^3/x^(1/2)

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Maxima [A]
time = 0.55, size = 315, normalized size = 0.84 \begin {gather*} -\frac {4 \, a^{2} b c^{3} - 5 \, {\left (9 \, b^{3} c^{3} - 15 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{4} - 12 \, {\left (3 \, a b^{2} c^{3} - 5 \, a^{2} b c^{2} d\right )} x^{2}}{10 \, {\left (a^{3} b^{2} x^{\frac {9}{2}} + a^{4} b x^{\frac {5}{2}}\right )}} + \frac {3 \, {\left (3 \, b^{3} c^{3} - 5 \, a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3}\right )} {\left (\frac {2 \, \sqrt {2} \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 {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \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 {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{16 \, a^{3} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^3/x^(7/2)/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

-1/10*(4*a^2*b*c^3 - 5*(9*b^3*c^3 - 15*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x^4 - 12*(3*a*b^2*c^3 - 5*a^2*b*
c^2*d)*x^2)/(a^3*b^2*x^(9/2) + a^4*b*x^(5/2)) + 3/16*(3*b^3*c^3 - 5*a*b^2*c^2*d + a^2*b*c*d^2 + a^3*d^3)*(2*sq
rt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sq
rt(b))*sqrt(b)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqr
t(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - sqrt(2)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a
^(1/4)*b^(3/4)) + sqrt(2)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)))/(a^3*
b)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2549 vs. \(2 (292) = 584\).
time = 0.58, size = 2549, normalized size = 6.78 \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^(7/2)/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

-1/40*(60*(a^3*b^2*x^5 + a^4*b*x^3)*(-(81*b^12*c^12 - 540*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 1932*a^3*b^
9*c^9*d^3 + 1039*a^4*b^8*c^8*d^4 + 328*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 136*a^7*b^5*c^5*d^7 + 127*a^8*b
^4*c^4*d^8 - 44*a^9*b^3*c^3*d^9 - 14*a^10*b^2*c^2*d^10 + 4*a^11*b*c*d^11 + a^12*d^12)/(a^13*b^7))^(1/4)*arctan
((sqrt((729*b^18*c^18 - 7290*a*b^17*c^17*d + 31833*a^2*b^16*c^16*d^2 - 78192*a^3*b^15*c^15*d^3 + 113940*a^4*b^
14*c^14*d^4 - 88920*a^5*b^13*c^13*d^5 + 10180*a^6*b^12*c^12*d^6 + 46320*a^7*b^11*c^11*d^7 - 35970*a^8*b^10*c^1
0*d^8 - 220*a^9*b^9*c^9*d^9 + 12078*a^10*b^8*c^8*d^10 - 3600*a^11*b^7*c^7*d^11 - 1884*a^12*b^6*c^6*d^12 + 936*
a^13*b^5*c^5*d^13 + 180*a^14*b^4*c^4*d^14 - 112*a^15*b^3*c^3*d^15 - 15*a^16*b^2*c^2*d^16 + 6*a^17*b*c*d^17 + a
^18*d^18)*x - (81*a^7*b^15*c^12 - 540*a^8*b^14*c^11*d + 1458*a^9*b^13*c^10*d^2 - 1932*a^10*b^12*c^9*d^3 + 1039
*a^11*b^11*c^8*d^4 + 328*a^12*b^10*c^7*d^5 - 644*a^13*b^9*c^6*d^6 + 136*a^14*b^8*c^5*d^7 + 127*a^15*b^7*c^4*d^
8 - 44*a^16*b^6*c^3*d^9 - 14*a^17*b^5*c^2*d^10 + 4*a^18*b^4*c*d^11 + a^19*b^3*d^12)*sqrt(-(81*b^12*c^12 - 540*
a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 1932*a^3*b^9*c^9*d^3 + 1039*a^4*b^8*c^8*d^4 + 328*a^5*b^7*c^7*d^5 - 6
44*a^6*b^6*c^6*d^6 + 136*a^7*b^5*c^5*d^7 + 127*a^8*b^4*c^4*d^8 - 44*a^9*b^3*c^3*d^9 - 14*a^10*b^2*c^2*d^10 + 4
*a^11*b*c*d^11 + a^12*d^12)/(a^13*b^7)))*a^3*b^2*(-(81*b^12*c^12 - 540*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2
- 1932*a^3*b^9*c^9*d^3 + 1039*a^4*b^8*c^8*d^4 + 328*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 136*a^7*b^5*c^5*d^
7 + 127*a^8*b^4*c^4*d^8 - 44*a^9*b^3*c^3*d^9 - 14*a^10*b^2*c^2*d^10 + 4*a^11*b*c*d^11 + a^12*d^12)/(a^13*b^7))
^(1/4) - (27*a^3*b^11*c^9 - 135*a^4*b^10*c^8*d + 252*a^5*b^9*c^7*d^2 - 188*a^6*b^8*c^6*d^3 - 6*a^7*b^7*c^5*d^4
+ 78*a^8*b^6*c^4*d^5 - 20*a^9*b^5*c^3*d^6 - 12*a^10*b^4*c^2*d^7 + 3*a^11*b^3*c*d^8 + a^12*b^2*d^9)*sqrt(x)*(-
(81*b^12*c^12 - 540*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 1932*a^3*b^9*c^9*d^3 + 1039*a^4*b^8*c^8*d^4 + 328
*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 136*a^7*b^5*c^5*d^7 + 127*a^8*b^4*c^4*d^8 - 44*a^9*b^3*c^3*d^9 - 14*a
^10*b^2*c^2*d^10 + 4*a^11*b*c*d^11 + a^12*d^12)/(a^13*b^7))^(1/4))/(81*b^12*c^12 - 540*a*b^11*c^11*d + 1458*a^
2*b^10*c^10*d^2 - 1932*a^3*b^9*c^9*d^3 + 1039*a^4*b^8*c^8*d^4 + 328*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 13
6*a^7*b^5*c^5*d^7 + 127*a^8*b^4*c^4*d^8 - 44*a^9*b^3*c^3*d^9 - 14*a^10*b^2*c^2*d^10 + 4*a^11*b*c*d^11 + a^12*d
^12)) - 15*(a^3*b^2*x^5 + a^4*b*x^3)*(-(81*b^12*c^12 - 540*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 1932*a^3*b
^9*c^9*d^3 + 1039*a^4*b^8*c^8*d^4 + 328*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 136*a^7*b^5*c^5*d^7 + 127*a^8*
b^4*c^4*d^8 - 44*a^9*b^3*c^3*d^9 - 14*a^10*b^2*c^2*d^10 + 4*a^11*b*c*d^11 + a^12*d^12)/(a^13*b^7))^(1/4)*log(2
7*a^10*b^5*(-(81*b^12*c^12 - 540*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 1932*a^3*b^9*c^9*d^3 + 1039*a^4*b^8*
c^8*d^4 + 328*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 136*a^7*b^5*c^5*d^7 + 127*a^8*b^4*c^4*d^8 - 44*a^9*b^3*c
^3*d^9 - 14*a^10*b^2*c^2*d^10 + 4*a^11*b*c*d^11 + a^12*d^12)/(a^13*b^7))^(3/4) + 27*(27*b^9*c^9 - 135*a*b^8*c^
8*d + 252*a^2*b^7*c^7*d^2 - 188*a^3*b^6*c^6*d^3 - 6*a^4*b^5*c^5*d^4 + 78*a^5*b^4*c^4*d^5 - 20*a^6*b^3*c^3*d^6
- 12*a^7*b^2*c^2*d^7 + 3*a^8*b*c*d^8 + a^9*d^9)*sqrt(x)) + 15*(a^3*b^2*x^5 + a^4*b*x^3)*(-(81*b^12*c^12 - 540*
a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 1932*a^3*b^9*c^9*d^3 + 1039*a^4*b^8*c^8*d^4 + 328*a^5*b^7*c^7*d^5 - 6
44*a^6*b^6*c^6*d^6 + 136*a^7*b^5*c^5*d^7 + 127*a^8*b^4*c^4*d^8 - 44*a^9*b^3*c^3*d^9 - 14*a^10*b^2*c^2*d^10 + 4
*a^11*b*c*d^11 + a^12*d^12)/(a^13*b^7))^(1/4)*log(-27*a^10*b^5*(-(81*b^12*c^12 - 540*a*b^11*c^11*d + 1458*a^2*
b^10*c^10*d^2 - 1932*a^3*b^9*c^9*d^3 + 1039*a^4*b^8*c^8*d^4 + 328*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 136*
a^7*b^5*c^5*d^7 + 127*a^8*b^4*c^4*d^8 - 44*a^9*b^3*c^3*d^9 - 14*a^10*b^2*c^2*d^10 + 4*a^11*b*c*d^11 + a^12*d^1
2)/(a^13*b^7))^(3/4) + 27*(27*b^9*c^9 - 135*a*b^8*c^8*d + 252*a^2*b^7*c^7*d^2 - 188*a^3*b^6*c^6*d^3 - 6*a^4*b^
5*c^5*d^4 + 78*a^5*b^4*c^4*d^5 - 20*a^6*b^3*c^3*d^6 - 12*a^7*b^2*c^2*d^7 + 3*a^8*b*c*d^8 + a^9*d^9)*sqrt(x)) +
4*(4*a^2*b*c^3 - 5*(9*b^3*c^3 - 15*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x^4 - 12*(3*a*b^2*c^3 - 5*a^2*b*c^2
*d)*x^2)*sqrt(x))/(a^3*b^2*x^5 + a^4*b*x^3)

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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**(7/2)/(b*x**2+a)**2,x)

[Out]

Timed out

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Giac [A]
time = 1.34, size = 505, normalized size = 1.34 \begin {gather*} \frac {b^{3} c^{3} x^{\frac {3}{2}} - 3 \, a b^{2} c^{2} d x^{\frac {3}{2}} + 3 \, a^{2} b c d^{2} x^{\frac {3}{2}} - a^{3} d^{3} x^{\frac {3}{2}}}{2 \, {\left (b x^{2} + a\right )} a^{3} b} + \frac {2 \, {\left (10 \, b c^{3} x^{2} - 15 \, a c^{2} d x^{2} - a c^{3}\right )}}{5 \, a^{3} x^{\frac {5}{2}}} + \frac {3 \, \sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + \left (a b^{3}\right )^{\frac {3}{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^{4}} + \frac {3 \, \sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + \left (a b^{3}\right )^{\frac {3}{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^{4}} - \frac {3 \, \sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + \left (a b^{3}\right )^{\frac {3}{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^{4}} + \frac {3 \, \sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + \left (a b^{3}\right )^{\frac {3}{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^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^3/x^(7/2)/(b*x^2+a)^2,x, algorithm="giac")

[Out]

1/2*(b^3*c^3*x^(3/2) - 3*a*b^2*c^2*d*x^(3/2) + 3*a^2*b*c*d^2*x^(3/2) - a^3*d^3*x^(3/2))/((b*x^2 + a)*a^3*b) +
2/5*(10*b*c^3*x^2 - 15*a*c^2*d*x^2 - a*c^3)/(a^3*x^(5/2)) + 3/8*sqrt(2)*(3*(a*b^3)^(3/4)*b^3*c^3 - 5*(a*b^3)^(
3/4)*a*b^2*c^2*d + (a*b^3)^(3/4)*a^2*b*c*d^2 + (a*b^3)^(3/4)*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^4) + 3/8*sqrt(2)*(3*(a*b^3)^(3/4)*b^3*c^3 - 5*(a*b^3)^(3/4)*a*b^2*c^2*d + (a*
b^3)^(3/4)*a^2*b*c*d^2 + (a*b^3)^(3/4)*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^4) - 3/16*sqrt(2)*(3*(a*b^3)^(3/4)*b^3*c^3 - 5*(a*b^3)^(3/4)*a*b^2*c^2*d + (a*b^3)^(3/4)*a^2*b*c*d
^2 + (a*b^3)^(3/4)*a^3*d^3)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^4*b^4) + 3/16*sqrt(2)*(3*(a*b^
3)^(3/4)*b^3*c^3 - 5*(a*b^3)^(3/4)*a*b^2*c^2*d + (a*b^3)^(3/4)*a^2*b*c*d^2 + (a*b^3)^(3/4)*a^3*d^3)*log(-sqrt(
2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^4*b^4)

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Mupad [B]
time = 0.14, size = 656, normalized size = 1.74 \begin {gather*} \frac {3\,\mathrm {atan}\left (\frac {3\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d+3\,b\,c\right )\,\left (288\,a^{16}\,b^5\,d^6+576\,a^{15}\,b^6\,c\,d^5-2592\,a^{14}\,b^7\,c^2\,d^4-1152\,a^{13}\,b^8\,c^3\,d^3+8928\,a^{12}\,b^9\,c^4\,d^2-8640\,a^{11}\,b^{10}\,c^5\,d+2592\,a^{10}\,b^{11}\,c^6\right )}{4\,{\left (-a\right )}^{13/4}\,b^{7/4}\,\left (216\,a^{16}\,b^3\,d^9+648\,a^{15}\,b^4\,c\,d^8-2592\,a^{14}\,b^5\,c^2\,d^7-4320\,a^{13}\,b^6\,c^3\,d^6+16848\,a^{12}\,b^7\,c^4\,d^5-1296\,a^{11}\,b^8\,c^5\,d^4-40608\,a^{10}\,b^9\,c^6\,d^3+54432\,a^9\,b^{10}\,c^7\,d^2-29160\,a^8\,b^{11}\,c^8\,d+5832\,a^7\,b^{12}\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d+3\,b\,c\right )}{4\,{\left (-a\right )}^{13/4}\,b^{7/4}}-\frac {\frac {2\,c^3}{5\,a}+\frac {x^4\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+15\,a\,b^2\,c^2\,d-9\,b^3\,c^3\right )}{2\,a^3\,b}+\frac {6\,c^2\,x^2\,\left (5\,a\,d-3\,b\,c\right )}{5\,a^2}}{a\,x^{5/2}+b\,x^{9/2}}-\frac {3\,\mathrm {atanh}\left (\frac {3\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d+3\,b\,c\right )\,\left (288\,a^{16}\,b^5\,d^6+576\,a^{15}\,b^6\,c\,d^5-2592\,a^{14}\,b^7\,c^2\,d^4-1152\,a^{13}\,b^8\,c^3\,d^3+8928\,a^{12}\,b^9\,c^4\,d^2-8640\,a^{11}\,b^{10}\,c^5\,d+2592\,a^{10}\,b^{11}\,c^6\right )}{4\,{\left (-a\right )}^{13/4}\,b^{7/4}\,\left (216\,a^{16}\,b^3\,d^9+648\,a^{15}\,b^4\,c\,d^8-2592\,a^{14}\,b^5\,c^2\,d^7-4320\,a^{13}\,b^6\,c^3\,d^6+16848\,a^{12}\,b^7\,c^4\,d^5-1296\,a^{11}\,b^8\,c^5\,d^4-40608\,a^{10}\,b^9\,c^6\,d^3+54432\,a^9\,b^{10}\,c^7\,d^2-29160\,a^8\,b^{11}\,c^8\,d+5832\,a^7\,b^{12}\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d+3\,b\,c\right )}{4\,{\left (-a\right )}^{13/4}\,b^{7/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^2)^3/(x^(7/2)*(a + b*x^2)^2),x)

[Out]

(3*atan((3*x^(1/2)*(a*d - b*c)^2*(a*d + 3*b*c)*(2592*a^10*b^11*c^6 + 288*a^16*b^5*d^6 - 8640*a^11*b^10*c^5*d +
576*a^15*b^6*c*d^5 + 8928*a^12*b^9*c^4*d^2 - 1152*a^13*b^8*c^3*d^3 - 2592*a^14*b^7*c^2*d^4))/(4*(-a)^(13/4)*b
^(7/4)*(5832*a^7*b^12*c^9 + 216*a^16*b^3*d^9 - 29160*a^8*b^11*c^8*d + 648*a^15*b^4*c*d^8 + 54432*a^9*b^10*c^7*
d^2 - 40608*a^10*b^9*c^6*d^3 - 1296*a^11*b^8*c^5*d^4 + 16848*a^12*b^7*c^4*d^5 - 4320*a^13*b^6*c^3*d^6 - 2592*a
^14*b^5*c^2*d^7)))*(a*d - b*c)^2*(a*d + 3*b*c))/(4*(-a)^(13/4)*b^(7/4)) - ((2*c^3)/(5*a) + (x^4*(a^3*d^3 - 9*b
^3*c^3 + 15*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(2*a^3*b) + (6*c^2*x^2*(5*a*d - 3*b*c))/(5*a^2))/(a*x^(5/2) + b*x^(9
/2)) - (3*atanh((3*x^(1/2)*(a*d - b*c)^2*(a*d + 3*b*c)*(2592*a^10*b^11*c^6 + 288*a^16*b^5*d^6 - 8640*a^11*b^10
*c^5*d + 576*a^15*b^6*c*d^5 + 8928*a^12*b^9*c^4*d^2 - 1152*a^13*b^8*c^3*d^3 - 2592*a^14*b^7*c^2*d^4))/(4*(-a)^
(13/4)*b^(7/4)*(5832*a^7*b^12*c^9 + 216*a^16*b^3*d^9 - 29160*a^8*b^11*c^8*d + 648*a^15*b^4*c*d^8 + 54432*a^9*b
^10*c^7*d^2 - 40608*a^10*b^9*c^6*d^3 - 1296*a^11*b^8*c^5*d^4 + 16848*a^12*b^7*c^4*d^5 - 4320*a^13*b^6*c^3*d^6
- 2592*a^14*b^5*c^2*d^7)))*(a*d - b*c)^2*(a*d + 3*b*c))/(4*(-a)^(13/4)*b^(7/4))

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