3.5.47 \(\int \frac {(c+d x^2)^3}{x^{7/2} (a+b x^2)} \, dx\) [447]

Optimal. Leaf size=283 \[ -\frac {2 c^3}{5 a x^{5/2}}+\frac {2 c^2 (b c-3 a d)}{a^2 \sqrt {x}}+\frac {2 d^3 x^{3/2}}{3 b}-\frac {(b c-a d)^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} b^{7/4}}+\frac {(b c-a d)^3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} b^{7/4}}+\frac {(b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} b^{7/4}}-\frac {(b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} b^{7/4}} \]

[Out]

-2/5*c^3/a/x^(5/2)+2/3*d^3*x^(3/2)/b-1/2*(-a*d+b*c)^3*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(9/4)/b^(7/4
)*2^(1/2)+1/2*(-a*d+b*c)^3*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(9/4)/b^(7/4)*2^(1/2)+1/4*(-a*d+b*c)^3*
ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(9/4)/b^(7/4)*2^(1/2)-1/4*(-a*d+b*c)^3*ln(a^(1/2)+x*b^
(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(9/4)/b^(7/4)*2^(1/2)+2*c^2*(-3*a*d+b*c)/a^2/x^(1/2)

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Rubi [A]
time = 0.19, antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {477, 472, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-a d)^3}{\sqrt {2} a^{9/4} b^{7/4}}+\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-a d)^3}{\sqrt {2} a^{9/4} b^{7/4}}+\frac {(b c-a d)^3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} b^{7/4}}-\frac {(b c-a d)^3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} b^{7/4}}+\frac {2 c^2 (b c-3 a d)}{a^2 \sqrt {x}}-\frac {2 c^3}{5 a x^{5/2}}+\frac {2 d^3 x^{3/2}}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*c^3)/(5*a*x^(5/2)) + (2*c^2*(b*c - 3*a*d))/(a^2*Sqrt[x]) + (2*d^3*x^(3/2))/(3*b) - ((b*c - a*d)^3*ArcTan[1
- (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(9/4)*b^(7/4)) + ((b*c - a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*
Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(9/4)*b^(7/4)) + ((b*c - a*d)^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] +
Sqrt[b]*x])/(2*Sqrt[2]*a^(9/4)*b^(7/4)) - ((b*c - a*d)^3*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[
b]*x])/(2*Sqrt[2]*a^(9/4)*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 472

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[(e*x)^m*((a + b*x^n)^p/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

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

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Mathematica [A]
time = 0.23, size = 177, normalized size = 0.63 \begin {gather*} \frac {\frac {4 \sqrt [4]{a} b^{3/4} \left (15 b^2 c^3 x^2+5 a^2 d^3 x^4-3 a b c^2 \left (c+15 d x^2\right )\right )}{x^{5/2}}+15 \sqrt {2} (-b c+a d)^3 \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)^3 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{30 a^{9/4} b^{7/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((4*a^(1/4)*b^(3/4)*(15*b^2*c^3*x^2 + 5*a^2*d^3*x^4 - 3*a*b*c^2*(c + 15*d*x^2)))/x^(5/2) + 15*Sqrt[2]*(-(b*c)
+ a*d)^3*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 15*Sqrt[2]*(-(b*c) + a*d)^3*ArcTanh
[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(30*a^(9/4)*b^(7/4))

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Maple [A]
time = 0.10, size = 188, normalized size = 0.66

method	result	size
derivativedivides	\(\frac {2 d^{3} x^{\frac {3}{2}}}{3 b}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{2} b^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {2 c^{3}}{5 a \,x^{\frac {5}{2}}}-\frac {2 c^{2} \left (3 a d -b c \right )}{a^{2} \sqrt {x}}\)	\(188\)
default	\(\frac {2 d^{3} x^{\frac {3}{2}}}{3 b}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{2} b^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {2 c^{3}}{5 a \,x^{\frac {5}{2}}}-\frac {2 c^{2} \left (3 a d -b c \right )}{a^{2} \sqrt {x}}\)	\(188\)
risch	\(\frac {-6 a b \,c^{2} d \,x^{2}+2 b^{2} c^{3} x^{2}-\frac {2}{5} a b \,c^{3}+\frac {2}{3} a^{2} d^{3} x^{4}}{a^{2} x^{\frac {5}{2}} b}-\frac {a \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) d^{3}}{2 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 ) c \,d^{2}}{2 b \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 ) c^{2} d}{2 a \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {b \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) c^{3}}{2 a^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {a \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) d^{3}}{2 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 ) c \,d^{2}}{2 b \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 ) c^{2} d}{2 a \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {b \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) c^{3}}{2 a^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {a \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}}{4 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 ) c \,d^{2}}{4 b \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 ) c^{2} d}{4 a \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {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}}{4 a^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\)	\(622\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.50, size = 259, normalized size = 0.92 \begin {gather*} \frac {2 \, d^{3} x^{\frac {3}{2}}}{3 \, b} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, 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 )}}{4 \, a^{2} b} - \frac {2 \, {\left (a c^{3} - 5 \, {\left (b c^{3} - 3 \, a c^{2} d\right )} x^{2}\right )}}{5 \, a^{2} x^{\frac {5}{2}}} \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),x, algorithm="maxima")

[Out]

2/3*d^3*x^(3/2)/b + 1/4*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqr
t(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) + 2*sqrt(2)*a
rctan(-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)*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^2*b) - 2/5*(a*c^3 - 5*(b*c^3 -
3*a*c^2*d)*x^2)/(a^2*x^(5/2))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2451 vs. \(2 (206) = 412\).
time = 0.59, size = 2451, normalized size = 8.66 \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),x, algorithm="fricas")

[Out]

1/30*(60*a^2*b*x^3*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*
c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*
c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^9*b^7))^(1/4)*arctan((sqrt((b^18*c^18 - 18*a
*b^17*c^17*d + 153*a^2*b^16*c^16*d^2 - 816*a^3*b^15*c^15*d^3 + 3060*a^4*b^14*c^14*d^4 - 8568*a^5*b^13*c^13*d^5
+ 18564*a^6*b^12*c^12*d^6 - 31824*a^7*b^11*c^11*d^7 + 43758*a^8*b^10*c^10*d^8 - 48620*a^9*b^9*c^9*d^9 + 43758
*a^10*b^8*c^8*d^10 - 31824*a^11*b^7*c^7*d^11 + 18564*a^12*b^6*c^6*d^12 - 8568*a^13*b^5*c^5*d^13 + 3060*a^14*b^
4*c^4*d^14 - 816*a^15*b^3*c^3*d^15 + 153*a^16*b^2*c^2*d^16 - 18*a^17*b*c*d^17 + a^18*d^18)*x - (a^5*b^15*c^12
- 12*a^6*b^14*c^11*d + 66*a^7*b^13*c^10*d^2 - 220*a^8*b^12*c^9*d^3 + 495*a^9*b^11*c^8*d^4 - 792*a^10*b^10*c^7*
d^5 + 924*a^11*b^9*c^6*d^6 - 792*a^12*b^8*c^5*d^7 + 495*a^13*b^7*c^4*d^8 - 220*a^14*b^6*c^3*d^9 + 66*a^15*b^5*
c^2*d^10 - 12*a^16*b^4*c*d^11 + a^17*b^3*d^12)*sqrt(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 22
0*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 49
5*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^9*b^7)))*a^2
*b^2*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*
a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a
^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^9*b^7))^(1/4) + (a^2*b^11*c^9 - 9*a^3*b^10*c^8*d + 36*a^4*
b^9*c^7*d^2 - 84*a^5*b^8*c^6*d^3 + 126*a^6*b^7*c^5*d^4 - 126*a^7*b^6*c^4*d^5 + 84*a^8*b^5*c^3*d^6 - 36*a^9*b^4
*c^2*d^7 + 9*a^10*b^3*c*d^8 - a^11*b^2*d^9)*sqrt(x)*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 2
20*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 4
95*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^9*b^7))^(1/
4))/(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5
*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10
*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)) - 15*a^2*b*x^3*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^
10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c
^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^9
*b^7))^(1/4)*log(a^7*b^5*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^
4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^
9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^9*b^7))^(3/4) - (b^9*c^9 - 9*a*b^8*c^8
*d + 36*a^2*b^7*c^7*d^2 - 84*a^3*b^6*c^6*d^3 + 126*a^4*b^5*c^5*d^4 - 126*a^5*b^4*c^4*d^5 + 84*a^6*b^3*c^3*d^6
- 36*a^7*b^2*c^2*d^7 + 9*a^8*b*c*d^8 - a^9*d^9)*sqrt(x)) + 15*a^2*b*x^3*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a
^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792
*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*
d^12)/(a^9*b^7))^(1/4)*log(-a^7*b^5*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d
^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d
^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^9*b^7))^(3/4) - (b^9*c^9 -
9*a*b^8*c^8*d + 36*a^2*b^7*c^7*d^2 - 84*a^3*b^6*c^6*d^3 + 126*a^4*b^5*c^5*d^4 - 126*a^5*b^4*c^4*d^5 + 84*a^6*b
^3*c^3*d^6 - 36*a^7*b^2*c^2*d^7 + 9*a^8*b*c*d^8 - a^9*d^9)*sqrt(x)) + 4*(5*a^2*d^3*x^4 - 3*a*b*c^3 + 15*(b^2*c
^3 - 3*a*b*c^2*d)*x^2)*sqrt(x))/(a^2*b*x^3)

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Sympy [A]
time = 59.73, size = 432, normalized size = 1.53 \begin {gather*} c^{3} \left (\begin {cases} \frac {\tilde {\infty }}{x^{\frac {9}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{9 b x^{\frac {9}{2}}} & \text {for}\: a = 0 \\- \frac {2}{5 a x^{\frac {5}{2}}} & \text {for}\: b = 0 \\- \frac {2}{5 a x^{\frac {5}{2}}} + \frac {b \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 a^{2} \sqrt [4]{- \frac {a}{b}}} - \frac {b \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 a^{2} \sqrt [4]{- \frac {a}{b}}} + \frac {b \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{a^{2} \sqrt [4]{- \frac {a}{b}}} + \frac {2 b}{a^{2} \sqrt {x}} & \text {otherwise} \end {cases}\right ) + 3 c^{2} d \left (\begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{5 b x^{\frac {5}{2}}} & \text {for}\: a = 0 \\- \frac {2}{a \sqrt {x}} & \text {for}\: b = 0 \\- \frac {\log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 a \sqrt [4]{- \frac {a}{b}}} + \frac {\log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 a \sqrt [4]{- \frac {a}{b}}} - \frac {\operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{a \sqrt [4]{- \frac {a}{b}}} - \frac {2}{a \sqrt {x}} & \text {otherwise} \end {cases}\right ) + 6 c d^{2} \operatorname {RootSum} {\left (256 t^{4} a b^{3} + 1, \left ( t \mapsto t \log {\left (64 t^{3} a b^{2} + \sqrt {x} \right )} \right )\right )} + d^{3} \left (\begin {cases} \tilde {\infty } x^{\frac {3}{2}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {7}{2}}}{7 a} & \text {for}\: b = 0 \\\frac {2 x^{\frac {3}{2}}}{3 b} & \text {for}\: a = 0 \\- \frac {a \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{2} \sqrt [4]{- \frac {a}{b}}} + \frac {a \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{2} \sqrt [4]{- \frac {a}{b}}} - \frac {a \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b^{2} \sqrt [4]{- \frac {a}{b}}} + \frac {2 x^{\frac {3}{2}}}{3 b} & \text {otherwise} \end {cases}\right ) \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),x)

[Out]

c**3*Piecewise((zoo/x**(9/2), Eq(a, 0) & Eq(b, 0)), (-2/(9*b*x**(9/2)), Eq(a, 0)), (-2/(5*a*x**(5/2)), Eq(b, 0
)), (-2/(5*a*x**(5/2)) + b*log(sqrt(x) - (-a/b)**(1/4))/(2*a**2*(-a/b)**(1/4)) - b*log(sqrt(x) + (-a/b)**(1/4)
)/(2*a**2*(-a/b)**(1/4)) + b*atan(sqrt(x)/(-a/b)**(1/4))/(a**2*(-a/b)**(1/4)) + 2*b/(a**2*sqrt(x)), True)) + 3
*c**2*d*Piecewise((zoo/x**(5/2), Eq(a, 0) & Eq(b, 0)), (-2/(5*b*x**(5/2)), Eq(a, 0)), (-2/(a*sqrt(x)), Eq(b, 0
)), (-log(sqrt(x) - (-a/b)**(1/4))/(2*a*(-a/b)**(1/4)) + log(sqrt(x) + (-a/b)**(1/4))/(2*a*(-a/b)**(1/4)) - at
an(sqrt(x)/(-a/b)**(1/4))/(a*(-a/b)**(1/4)) - 2/(a*sqrt(x)), True)) + 6*c*d**2*RootSum(256*_t**4*a*b**3 + 1, L
ambda(_t, _t*log(64*_t**3*a*b**2 + sqrt(x)))) + d**3*Piecewise((zoo*x**(3/2), Eq(a, 0) & Eq(b, 0)), (2*x**(7/2
)/(7*a), Eq(b, 0)), (2*x**(3/2)/(3*b), Eq(a, 0)), (-a*log(sqrt(x) - (-a/b)**(1/4))/(2*b**2*(-a/b)**(1/4)) + a*
log(sqrt(x) + (-a/b)**(1/4))/(2*b**2*(-a/b)**(1/4)) - a*atan(sqrt(x)/(-a/b)**(1/4))/(b**2*(-a/b)**(1/4)) + 2*x
**(3/2)/(3*b), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 455 vs. \(2 (206) = 412\).
time = 1.41, size = 455, normalized size = 1.61 \begin {gather*} \frac {2 \, d^{3} x^{\frac {3}{2}}}{3 \, b} + \frac {2 \, {\left (5 \, b c^{3} x^{2} - 15 \, a c^{2} d x^{2} - a c^{3}\right )}}{5 \, a^{2} x^{\frac {5}{2}}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \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 )}{2 \, a^{3} b^{4}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \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 )}{2 \, a^{3} b^{4}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \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 )}{4 \, a^{3} b^{4}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d + 3 \, \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 )}{4 \, a^{3} 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),x, algorithm="giac")

[Out]

2/3*d^3*x^(3/2)/b + 2/5*(5*b*c^3*x^2 - 15*a*c^2*d*x^2 - a*c^3)/(a^2*x^(5/2)) + 1/2*sqrt(2)*((a*b^3)^(3/4)*b^3*
c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d + 3*(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)*(s
qrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a^3*b^4) + 1/2*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*
a*b^2*c^2*d + 3*(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^3*b^4) - 1/4*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d + 3*(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^3*b^4) + 1/4
*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d + 3*(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^3*b^4)

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Mupad [B]
time = 0.20, size = 583, normalized size = 2.06 \begin {gather*} \frac {2\,d^3\,x^{3/2}}{3\,b}-\frac {\frac {2\,b\,c^3}{5\,a}+\frac {2\,b\,c^2\,x^2\,\left (3\,a\,d-b\,c\right )}{a^2}}{b\,x^{5/2}}+\frac {\mathrm {atan}\left (\frac {\sqrt {x}\,{\left (a\,d-b\,c\right )}^3\,\left (16\,a^{13}\,b^5\,d^6-96\,a^{12}\,b^6\,c\,d^5+240\,a^{11}\,b^7\,c^2\,d^4-320\,a^{10}\,b^8\,c^3\,d^3+240\,a^9\,b^9\,c^4\,d^2-96\,a^8\,b^{10}\,c^5\,d+16\,a^7\,b^{11}\,c^6\right )}{{\left (-a\right )}^{9/4}\,b^{7/4}\,\left (-16\,a^{14}\,b^3\,d^9+144\,a^{13}\,b^4\,c\,d^8-576\,a^{12}\,b^5\,c^2\,d^7+1344\,a^{11}\,b^6\,c^3\,d^6-2016\,a^{10}\,b^7\,c^4\,d^5+2016\,a^9\,b^8\,c^5\,d^4-1344\,a^8\,b^9\,c^6\,d^3+576\,a^7\,b^{10}\,c^7\,d^2-144\,a^6\,b^{11}\,c^8\,d+16\,a^5\,b^{12}\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{{\left (-a\right )}^{9/4}\,b^{7/4}}+\frac {\mathrm {atan}\left (\frac {\sqrt {x}\,{\left (a\,d-b\,c\right )}^3\,\left (16\,a^{13}\,b^5\,d^6-96\,a^{12}\,b^6\,c\,d^5+240\,a^{11}\,b^7\,c^2\,d^4-320\,a^{10}\,b^8\,c^3\,d^3+240\,a^9\,b^9\,c^4\,d^2-96\,a^8\,b^{10}\,c^5\,d+16\,a^7\,b^{11}\,c^6\right )\,1{}\mathrm {i}}{{\left (-a\right )}^{9/4}\,b^{7/4}\,\left (-16\,a^{14}\,b^3\,d^9+144\,a^{13}\,b^4\,c\,d^8-576\,a^{12}\,b^5\,c^2\,d^7+1344\,a^{11}\,b^6\,c^3\,d^6-2016\,a^{10}\,b^7\,c^4\,d^5+2016\,a^9\,b^8\,c^5\,d^4-1344\,a^8\,b^9\,c^6\,d^3+576\,a^7\,b^{10}\,c^7\,d^2-144\,a^6\,b^{11}\,c^8\,d+16\,a^5\,b^{12}\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^3\,1{}\mathrm {i}}{{\left (-a\right )}^{9/4}\,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)),x)

[Out]

(2*d^3*x^(3/2))/(3*b) - ((2*b*c^3)/(5*a) + (2*b*c^2*x^2*(3*a*d - b*c))/a^2)/(b*x^(5/2)) + (atan((x^(1/2)*(a*d
- b*c)^3*(16*a^7*b^11*c^6 + 16*a^13*b^5*d^6 - 96*a^8*b^10*c^5*d - 96*a^12*b^6*c*d^5 + 240*a^9*b^9*c^4*d^2 - 32
0*a^10*b^8*c^3*d^3 + 240*a^11*b^7*c^2*d^4))/((-a)^(9/4)*b^(7/4)*(16*a^5*b^12*c^9 - 16*a^14*b^3*d^9 - 144*a^6*b
^11*c^8*d + 144*a^13*b^4*c*d^8 + 576*a^7*b^10*c^7*d^2 - 1344*a^8*b^9*c^6*d^3 + 2016*a^9*b^8*c^5*d^4 - 2016*a^1
0*b^7*c^4*d^5 + 1344*a^11*b^6*c^3*d^6 - 576*a^12*b^5*c^2*d^7)))*(a*d - b*c)^3)/((-a)^(9/4)*b^(7/4)) + (atan((x
^(1/2)*(a*d - b*c)^3*(16*a^7*b^11*c^6 + 16*a^13*b^5*d^6 - 96*a^8*b^10*c^5*d - 96*a^12*b^6*c*d^5 + 240*a^9*b^9*
c^4*d^2 - 320*a^10*b^8*c^3*d^3 + 240*a^11*b^7*c^2*d^4)*1i)/((-a)^(9/4)*b^(7/4)*(16*a^5*b^12*c^9 - 16*a^14*b^3*
d^9 - 144*a^6*b^11*c^8*d + 144*a^13*b^4*c*d^8 + 576*a^7*b^10*c^7*d^2 - 1344*a^8*b^9*c^6*d^3 + 2016*a^9*b^8*c^5
*d^4 - 2016*a^10*b^7*c^4*d^5 + 1344*a^11*b^6*c^3*d^6 - 576*a^12*b^5*c^2*d^7)))*(a*d - b*c)^3*1i)/((-a)^(9/4)*b
^(7/4))

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