3.5.23 \(\int \frac {x^{5/2} (c+d x^2)^3}{a+b x^2} \, dx\)

Optimal. Leaf size=328 \[ -\frac {a^{3/4} (b c-a d)^3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{19/4}}+\frac {a^{3/4} (b c-a d)^3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{19/4}}+\frac {a^{3/4} (b c-a d)^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{19/4}}-\frac {a^{3/4} (b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} b^{19/4}}+\frac {2 d x^{7/2} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{7 b^3}+\frac {2 x^{3/2} (b c-a d)^3}{3 b^4}+\frac {2 d^2 x^{11/2} (3 b c-a d)}{11 b^2}+\frac {2 d^3 x^{15/2}}{15 b} \]

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Rubi [A]  time = 0.29, antiderivative size = 328, 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 = {461, 321, 329, 297, 1162, 617, 204, 1165, 628} \begin {gather*} \frac {2 d x^{7/2} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{7 b^3}-\frac {a^{3/4} (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} b^{19/4}}+\frac {a^{3/4} (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} b^{19/4}}+\frac {a^{3/4} (b c-a d)^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{19/4}}-\frac {a^{3/4} (b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} b^{19/4}}+\frac {2 d^2 x^{11/2} (3 b c-a d)}{11 b^2}+\frac {2 x^{3/2} (b c-a d)^3}{3 b^4}+\frac {2 d^3 x^{15/2}}{15 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^(5/2)*(c + d*x^2)^3)/(a + b*x^2),x]

[Out]

(2*(b*c - a*d)^3*x^(3/2))/(3*b^4) + (2*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^(7/2))/(7*b^3) + (2*d^2*(3*b*c -
a*d)*x^(11/2))/(11*b^2) + (2*d^3*x^(15/2))/(15*b) + (a^(3/4)*(b*c - a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x]
)/a^(1/4)])/(Sqrt[2]*b^(19/4)) - (a^(3/4)*(b*c - a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2
]*b^(19/4)) - (a^(3/4)*(b*c - a*d)^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^
(19/4)) + (a^(3/4)*(b*c - a*d)^3*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^(19/
4))

Rule 204

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

Rule 297

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 321

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

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 461

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 617

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 628

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 1162

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 1165

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 {x^{5/2} \left (c+d x^2\right )^3}{a+b x^2} \, dx &=\int \left (\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{5/2}}{b^3}+\frac {d^2 (3 b c-a d) x^{9/2}}{b^2}+\frac {d^3 x^{13/2}}{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 ) x^{5/2}}{b^3 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{7/2}}{7 b^3}+\frac {2 d^2 (3 b c-a d) x^{11/2}}{11 b^2}+\frac {2 d^3 x^{15/2}}{15 b}+\frac {(b c-a d)^3 \int \frac {x^{5/2}}{a+b x^2} \, dx}{b^3}\\ &=\frac {2 (b c-a d)^3 x^{3/2}}{3 b^4}+\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{7/2}}{7 b^3}+\frac {2 d^2 (3 b c-a d) x^{11/2}}{11 b^2}+\frac {2 d^3 x^{15/2}}{15 b}-\frac {\left (a (b c-a d)^3\right ) \int \frac {\sqrt {x}}{a+b x^2} \, dx}{b^4}\\ &=\frac {2 (b c-a d)^3 x^{3/2}}{3 b^4}+\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{7/2}}{7 b^3}+\frac {2 d^2 (3 b c-a d) x^{11/2}}{11 b^2}+\frac {2 d^3 x^{15/2}}{15 b}-\frac {\left (2 a (b c-a d)^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b^4}\\ &=\frac {2 (b c-a d)^3 x^{3/2}}{3 b^4}+\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{7/2}}{7 b^3}+\frac {2 d^2 (3 b c-a d) x^{11/2}}{11 b^2}+\frac {2 d^3 x^{15/2}}{15 b}+\frac {\left (a (b c-a d)^3\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b^{9/2}}-\frac {\left (a (b c-a d)^3\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b^{9/2}}\\ &=\frac {2 (b c-a d)^3 x^{3/2}}{3 b^4}+\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{7/2}}{7 b^3}+\frac {2 d^2 (3 b c-a d) x^{11/2}}{11 b^2}+\frac {2 d^3 x^{15/2}}{15 b}-\frac {\left (a (b c-a d)^3\right ) \operatorname {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 b^5}-\frac {\left (a (b c-a d)^3\right ) \operatorname {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 b^5}-\frac {\left (a^{3/4} (b c-a d)^3\right ) \operatorname {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} b^{19/4}}-\frac {\left (a^{3/4} (b c-a d)^3\right ) \operatorname {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} b^{19/4}}\\ &=\frac {2 (b c-a d)^3 x^{3/2}}{3 b^4}+\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{7/2}}{7 b^3}+\frac {2 d^2 (3 b c-a d) x^{11/2}}{11 b^2}+\frac {2 d^3 x^{15/2}}{15 b}-\frac {a^{3/4} (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} b^{19/4}}+\frac {a^{3/4} (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} b^{19/4}}-\frac {\left (a^{3/4} (b c-a d)^3\right ) \operatorname {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} b^{19/4}}+\frac {\left (a^{3/4} (b c-a d)^3\right ) \operatorname {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} b^{19/4}}\\ &=\frac {2 (b c-a d)^3 x^{3/2}}{3 b^4}+\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{7/2}}{7 b^3}+\frac {2 d^2 (3 b c-a d) x^{11/2}}{11 b^2}+\frac {2 d^3 x^{15/2}}{15 b}+\frac {a^{3/4} (b c-a d)^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{19/4}}-\frac {a^{3/4} (b c-a d)^3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{19/4}}-\frac {a^{3/4} (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} b^{19/4}}+\frac {a^{3/4} (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} b^{19/4}}\\ \end {align*}

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Mathematica [C]  time = 0.39, size = 132, normalized size = 0.40 \begin {gather*} \frac {2 x^{3/2} \left (-385 a^3 d^3+165 a^2 b d^2 \left (7 c+d x^2\right )-15 a b^2 d \left (77 c^2+33 c d x^2+7 d^2 x^4\right )-385 (b c-a d)^3 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\frac {b x^2}{a}\right )+b^3 \left (385 c^3+495 c^2 d x^2+315 c d^2 x^4+77 d^3 x^6\right )\right )}{1155 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^(5/2)*(c + d*x^2)^3)/(a + b*x^2),x]

[Out]

(2*x^(3/2)*(-385*a^3*d^3 + 165*a^2*b*d^2*(7*c + d*x^2) - 15*a*b^2*d*(77*c^2 + 33*c*d*x^2 + 7*d^2*x^4) + b^3*(3
85*c^3 + 495*c^2*d*x^2 + 315*c*d^2*x^4 + 77*d^3*x^6) - 385*(b*c - a*d)^3*Hypergeometric2F1[3/4, 1, 7/4, -((b*x
^2)/a)]))/(1155*b^4)

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IntegrateAlgebraic [A]  time = 0.28, size = 255, normalized size = 0.78 \begin {gather*} -\frac {a^{3/4} (a d-b c)^3 \tan ^{-1}\left (\frac {\frac {\sqrt [4]{a}}{\sqrt {2} \sqrt [4]{b}}-\frac {\sqrt [4]{b} x}{\sqrt {2} \sqrt [4]{a}}}{\sqrt {x}}\right )}{\sqrt {2} b^{19/4}}-\frac {a^{3/4} (a d-b c)^3 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {2} b^{19/4}}+\frac {2 x^{3/2} \left (-385 a^3 d^3+1155 a^2 b c d^2+165 a^2 b d^3 x^2-1155 a b^2 c^2 d-495 a b^2 c d^2 x^2-105 a b^2 d^3 x^4+385 b^3 c^3+495 b^3 c^2 d x^2+315 b^3 c d^2 x^4+77 b^3 d^3 x^6\right )}{1155 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^(5/2)*(c + d*x^2)^3)/(a + b*x^2),x]

[Out]

(2*x^(3/2)*(385*b^3*c^3 - 1155*a*b^2*c^2*d + 1155*a^2*b*c*d^2 - 385*a^3*d^3 + 495*b^3*c^2*d*x^2 - 495*a*b^2*c*
d^2*x^2 + 165*a^2*b*d^3*x^2 + 315*b^3*c*d^2*x^4 - 105*a*b^2*d^3*x^4 + 77*b^3*d^3*x^6))/(1155*b^4) - (a^(3/4)*(
-(b*c) + a*d)^3*ArcTan[(a^(1/4)/(Sqrt[2]*b^(1/4)) - (b^(1/4)*x)/(Sqrt[2]*a^(1/4)))/Sqrt[x]])/(Sqrt[2]*b^(19/4)
) - (a^(3/4)*(-(b*c) + a*d)^3*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(Sqrt[2]*b^(19
/4))

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fricas [B]  time = 1.66, size = 2528, normalized size = 7.71

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(d*x^2+c)^3/(b*x^2+a),x, algorithm="fricas")

[Out]

-1/2310*(4620*b^4*(-(a^3*b^12*c^12 - 12*a^4*b^11*c^11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 495*a^7
*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11*b^4*c^4*d^8 - 220*a
^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b*c*d^11 + a^15*d^12)/b^19)^(1/4)*arctan((sqrt((a^4*b^18*c^18
- 18*a^5*b^17*c^17*d + 153*a^6*b^16*c^16*d^2 - 816*a^7*b^15*c^15*d^3 + 3060*a^8*b^14*c^14*d^4 - 8568*a^9*b^13
*c^13*d^5 + 18564*a^10*b^12*c^12*d^6 - 31824*a^11*b^11*c^11*d^7 + 43758*a^12*b^10*c^10*d^8 - 48620*a^13*b^9*c^
9*d^9 + 43758*a^14*b^8*c^8*d^10 - 31824*a^15*b^7*c^7*d^11 + 18564*a^16*b^6*c^6*d^12 - 8568*a^17*b^5*c^5*d^13 +
3060*a^18*b^4*c^4*d^14 - 816*a^19*b^3*c^3*d^15 + 153*a^20*b^2*c^2*d^16 - 18*a^21*b*c*d^17 + a^22*d^18)*x - (a
^3*b^21*c^12 - 12*a^4*b^20*c^11*d + 66*a^5*b^19*c^10*d^2 - 220*a^6*b^18*c^9*d^3 + 495*a^7*b^17*c^8*d^4 - 792*a
^8*b^16*c^7*d^5 + 924*a^9*b^15*c^6*d^6 - 792*a^10*b^14*c^5*d^7 + 495*a^11*b^13*c^4*d^8 - 220*a^12*b^12*c^3*d^9
+ 66*a^13*b^11*c^2*d^10 - 12*a^14*b^10*c*d^11 + a^15*b^9*d^12)*sqrt(-(a^3*b^12*c^12 - 12*a^4*b^11*c^11*d + 66
*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 495*a^7*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 7
92*a^10*b^5*c^5*d^7 + 495*a^11*b^4*c^4*d^8 - 220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b*c*d^11 +
a^15*d^12)/b^19))*b^5*(-(a^3*b^12*c^12 - 12*a^4*b^11*c^11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 495
*a^7*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11*b^4*c^4*d^8 - 2
20*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b*c*d^11 + a^15*d^12)/b^19)^(1/4) + (a^2*b^14*c^9 - 9*a^3
*b^13*c^8*d + 36*a^4*b^12*c^7*d^2 - 84*a^5*b^11*c^6*d^3 + 126*a^6*b^10*c^5*d^4 - 126*a^7*b^9*c^4*d^5 + 84*a^8*
b^8*c^3*d^6 - 36*a^9*b^7*c^2*d^7 + 9*a^10*b^6*c*d^8 - a^11*b^5*d^9)*sqrt(x)*(-(a^3*b^12*c^12 - 12*a^4*b^11*c^1
1*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 495*a^7*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6
*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11*b^4*c^4*d^8 - 220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b*c
*d^11 + a^15*d^12)/b^19)^(1/4))/(a^3*b^12*c^12 - 12*a^4*b^11*c^11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d
^3 + 495*a^7*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11*b^4*c^4
*d^8 - 220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b*c*d^11 + a^15*d^12)) - 1155*b^4*(-(a^3*b^12*c^1
2 - 12*a^4*b^11*c^11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 495*a^7*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^
5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11*b^4*c^4*d^8 - 220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2
*d^10 - 12*a^14*b*c*d^11 + a^15*d^12)/b^19)^(1/4)*log(b^14*(-(a^3*b^12*c^12 - 12*a^4*b^11*c^11*d + 66*a^5*b^10
*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 495*a^7*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b
^5*c^5*d^7 + 495*a^11*b^4*c^4*d^8 - 220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b*c*d^11 + a^15*d^12
)/b^19)^(3/4) - (a^2*b^9*c^9 - 9*a^3*b^8*c^8*d + 36*a^4*b^7*c^7*d^2 - 84*a^5*b^6*c^6*d^3 + 126*a^6*b^5*c^5*d^4
- 126*a^7*b^4*c^4*d^5 + 84*a^8*b^3*c^3*d^6 - 36*a^9*b^2*c^2*d^7 + 9*a^10*b*c*d^8 - a^11*d^9)*sqrt(x)) + 1155*
b^4*(-(a^3*b^12*c^12 - 12*a^4*b^11*c^11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 495*a^7*b^8*c^8*d^4 -
792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11*b^4*c^4*d^8 - 220*a^12*b^3*c^3*d^
9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b*c*d^11 + a^15*d^12)/b^19)^(1/4)*log(-b^14*(-(a^3*b^12*c^12 - 12*a^4*b^11*
c^11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 495*a^7*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*
c^6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11*b^4*c^4*d^8 - 220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*
b*c*d^11 + a^15*d^12)/b^19)^(3/4) - (a^2*b^9*c^9 - 9*a^3*b^8*c^8*d + 36*a^4*b^7*c^7*d^2 - 84*a^5*b^6*c^6*d^3 +
126*a^6*b^5*c^5*d^4 - 126*a^7*b^4*c^4*d^5 + 84*a^8*b^3*c^3*d^6 - 36*a^9*b^2*c^2*d^7 + 9*a^10*b*c*d^8 - a^11*d
^9)*sqrt(x)) - 4*(77*b^3*d^3*x^7 + 105*(3*b^3*c*d^2 - a*b^2*d^3)*x^5 + 165*(3*b^3*c^2*d - 3*a*b^2*c*d^2 + a^2*
b*d^3)*x^3 + 385*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x)*sqrt(x))/b^4

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giac [B]  time = 0.52, size = 531, normalized size = 1.62 \begin {gather*} -\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 \, b^{7}} - \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 \, b^{7}} + \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 \, b^{7}} - \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 \, b^{7}} + \frac {2 \, {\left (77 \, b^{14} d^{3} x^{\frac {15}{2}} + 315 \, b^{14} c d^{2} x^{\frac {11}{2}} - 105 \, a b^{13} d^{3} x^{\frac {11}{2}} + 495 \, b^{14} c^{2} d x^{\frac {7}{2}} - 495 \, a b^{13} c d^{2} x^{\frac {7}{2}} + 165 \, a^{2} b^{12} d^{3} x^{\frac {7}{2}} + 385 \, b^{14} c^{3} x^{\frac {3}{2}} - 1155 \, a b^{13} c^{2} d x^{\frac {3}{2}} + 1155 \, a^{2} b^{12} c d^{2} x^{\frac {3}{2}} - 385 \, a^{3} b^{11} d^{3} x^{\frac {3}{2}}\right )}}{1155 \, b^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(d*x^2+c)^3/(b*x^2+a),x, algorithm="giac")

[Out]

-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))/b^7 - 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))/b^7 + 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))/b^7 - 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))/b^7 + 2/1155*(77*b^14*d^3*x^(15/2) + 31
5*b^14*c*d^2*x^(11/2) - 105*a*b^13*d^3*x^(11/2) + 495*b^14*c^2*d*x^(7/2) - 495*a*b^13*c*d^2*x^(7/2) + 165*a^2*
b^12*d^3*x^(7/2) + 385*b^14*c^3*x^(3/2) - 1155*a*b^13*c^2*d*x^(3/2) + 1155*a^2*b^12*c*d^2*x^(3/2) - 385*a^3*b^
11*d^3*x^(3/2))/b^15

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maple [B]  time = 0.01, size = 721, normalized size = 2.20 \begin {gather*} \frac {2 d^{3} x^{\frac {15}{2}}}{15 b}-\frac {2 a \,d^{3} x^{\frac {11}{2}}}{11 b^{2}}+\frac {6 c \,d^{2} x^{\frac {11}{2}}}{11 b}+\frac {2 a^{2} d^{3} x^{\frac {7}{2}}}{7 b^{3}}-\frac {6 a c \,d^{2} x^{\frac {7}{2}}}{7 b^{2}}+\frac {6 c^{2} d \,x^{\frac {7}{2}}}{7 b}-\frac {2 a^{3} d^{3} x^{\frac {3}{2}}}{3 b^{4}}+\frac {2 a^{2} c \,d^{2} x^{\frac {3}{2}}}{b^{3}}-\frac {2 a \,c^{2} d \,x^{\frac {3}{2}}}{b^{2}}+\frac {2 c^{3} x^{\frac {3}{2}}}{3 b}+\frac {\sqrt {2}\, a^{4} d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{5}}+\frac {\sqrt {2}\, a^{4} d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{5}}+\frac {\sqrt {2}\, a^{4} d^{3} \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{4 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{5}}-\frac {3 \sqrt {2}\, a^{3} c \,d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{4}}-\frac {3 \sqrt {2}\, a^{3} c \,d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{4}}-\frac {3 \sqrt {2}\, a^{3} c \,d^{2} \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{4 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{4}}+\frac {3 \sqrt {2}\, a^{2} c^{2} d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{3}}+\frac {3 \sqrt {2}\, a^{2} c^{2} d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{3}}+\frac {3 \sqrt {2}\, a^{2} c^{2} d \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{4 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{3}}-\frac {\sqrt {2}\, a \,c^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}}-\frac {\sqrt {2}\, a \,c^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}}-\frac {\sqrt {2}\, a \,c^{3} \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{4 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(d*x^2+c)^3/(b*x^2+a),x)

[Out]

2/15*d^3*x^(15/2)/b-2/11/b^2*x^(11/2)*a*d^3+6/11/b*x^(11/2)*c*d^2+2/7/b^3*x^(7/2)*a^2*d^3-6/7/b^2*x^(7/2)*a*c*
d^2+6/7/b*x^(7/2)*c^2*d-2/3/b^4*x^(3/2)*a^3*d^3+2/b^3*x^(3/2)*a^2*c*d^2-2/b^2*x^(3/2)*a*c^2*d+2/3/b*x^(3/2)*c^
3+1/2*a^4/b^5/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*d^3-3/2*a^3/b^4/(a/b)^(1/4)*2^(1/2)*ar
ctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*c*d^2+3/2*a^2/b^3/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1
)*c^2*d-1/2*a/b^2/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*c^3+1/4*a^4/b^5/(a/b)^(1/4)*2^(1/2
)*ln((x-(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2)))*d^3-3/4*a^3/b^4/
(a/b)^(1/4)*2^(1/2)*ln((x-(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2))
)*c*d^2+3/4*a^2/b^3/(a/b)^(1/4)*2^(1/2)*ln((x-(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*2^(1/2)*
x^(1/2)+(a/b)^(1/2)))*c^2*d-1/4*a/b^2/(a/b)^(1/4)*2^(1/2)*ln((x-(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2))/(x+(a
/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2)))*c^3+1/2*a^4/b^5/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)
+1)*d^3-3/2*a^3/b^4/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*c*d^2+3/2*a^2/b^3/(a/b)^(1/4)*2^
(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*c^2*d-1/2*a/b^2/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(
1/2)+1)*c^3

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maxima [A]  time = 2.47, size = 331, normalized size = 1.01 \begin {gather*} -\frac {{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} 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 \, b^{4}} + \frac {2 \, {\left (77 \, b^{3} d^{3} x^{\frac {15}{2}} + 105 \, {\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{\frac {11}{2}} + 165 \, {\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{\frac {7}{2}} + 385 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{\frac {3}{2}}\right )}}{1155 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(d*x^2+c)^3/(b*x^2+a),x, algorithm="maxima")

[Out]

-1/4*(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*(2*sqrt(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)*sqrt(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)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - sqr
t(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)))/b^4 + 2/1155*(77*b^3*d^3*x^(15/2) + 105*(3*b^3*c*
d^2 - a*b^2*d^3)*x^(11/2) + 165*(3*b^3*c^2*d - 3*a*b^2*c*d^2 + a^2*b*d^3)*x^(7/2) + 385*(b^3*c^3 - 3*a*b^2*c^2
*d + 3*a^2*b*c*d^2 - a^3*d^3)*x^(3/2))/b^4

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mupad [B]  time = 0.37, size = 634, normalized size = 1.93 \begin {gather*} x^{3/2}\,\left (\frac {2\,c^3}{3\,b}-\frac {a\,\left (\frac {6\,c^2\,d}{b}+\frac {a\,\left (\frac {2\,a\,d^3}{b^2}-\frac {6\,c\,d^2}{b}\right )}{b}\right )}{3\,b}\right )-x^{11/2}\,\left (\frac {2\,a\,d^3}{11\,b^2}-\frac {6\,c\,d^2}{11\,b}\right )+x^{7/2}\,\left (\frac {6\,c^2\,d}{7\,b}+\frac {a\,\left (\frac {2\,a\,d^3}{b^2}-\frac {6\,c\,d^2}{b}\right )}{7\,b}\right )+\frac {2\,d^3\,x^{15/2}}{15\,b}+\frac {{\left (-a\right )}^{3/4}\,\mathrm {atan}\left (\frac {{\left (-a\right )}^{3/4}\,b^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^3\,\left (a^9\,d^6-6\,a^8\,b\,c\,d^5+15\,a^7\,b^2\,c^2\,d^4-20\,a^6\,b^3\,c^3\,d^3+15\,a^5\,b^4\,c^4\,d^2-6\,a^4\,b^5\,c^5\,d+a^3\,b^6\,c^6\right )}{a^{13}\,d^9-9\,a^{12}\,b\,c\,d^8+36\,a^{11}\,b^2\,c^2\,d^7-84\,a^{10}\,b^3\,c^3\,d^6+126\,a^9\,b^4\,c^4\,d^5-126\,a^8\,b^5\,c^5\,d^4+84\,a^7\,b^6\,c^6\,d^3-36\,a^6\,b^7\,c^7\,d^2+9\,a^5\,b^8\,c^8\,d-a^4\,b^9\,c^9}\right )\,{\left (a\,d-b\,c\right )}^3}{b^{19/4}}+\frac {{\left (-a\right )}^{3/4}\,\mathrm {atan}\left (\frac {{\left (-a\right )}^{3/4}\,b^{1/4}\,\sqrt {x}\,{\left (a\,d-b\,c\right )}^3\,\left (a^9\,d^6-6\,a^8\,b\,c\,d^5+15\,a^7\,b^2\,c^2\,d^4-20\,a^6\,b^3\,c^3\,d^3+15\,a^5\,b^4\,c^4\,d^2-6\,a^4\,b^5\,c^5\,d+a^3\,b^6\,c^6\right )\,1{}\mathrm {i}}{a^{13}\,d^9-9\,a^{12}\,b\,c\,d^8+36\,a^{11}\,b^2\,c^2\,d^7-84\,a^{10}\,b^3\,c^3\,d^6+126\,a^9\,b^4\,c^4\,d^5-126\,a^8\,b^5\,c^5\,d^4+84\,a^7\,b^6\,c^6\,d^3-36\,a^6\,b^7\,c^7\,d^2+9\,a^5\,b^8\,c^8\,d-a^4\,b^9\,c^9}\right )\,{\left (a\,d-b\,c\right )}^3\,1{}\mathrm {i}}{b^{19/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(5/2)*(c + d*x^2)^3)/(a + b*x^2),x)

[Out]

x^(3/2)*((2*c^3)/(3*b) - (a*((6*c^2*d)/b + (a*((2*a*d^3)/b^2 - (6*c*d^2)/b))/b))/(3*b)) - x^(11/2)*((2*a*d^3)/
(11*b^2) - (6*c*d^2)/(11*b)) + x^(7/2)*((6*c^2*d)/(7*b) + (a*((2*a*d^3)/b^2 - (6*c*d^2)/b))/(7*b)) + (2*d^3*x^
(15/2))/(15*b) + ((-a)^(3/4)*atan(((-a)^(3/4)*b^(1/4)*x^(1/2)*(a*d - b*c)^3*(a^9*d^6 + a^3*b^6*c^6 - 6*a^4*b^5
*c^5*d + 15*a^5*b^4*c^4*d^2 - 20*a^6*b^3*c^3*d^3 + 15*a^7*b^2*c^2*d^4 - 6*a^8*b*c*d^5))/(a^13*d^9 - a^4*b^9*c^
9 + 9*a^5*b^8*c^8*d - 36*a^6*b^7*c^7*d^2 + 84*a^7*b^6*c^6*d^3 - 126*a^8*b^5*c^5*d^4 + 126*a^9*b^4*c^4*d^5 - 84
*a^10*b^3*c^3*d^6 + 36*a^11*b^2*c^2*d^7 - 9*a^12*b*c*d^8))*(a*d - b*c)^3)/b^(19/4) + ((-a)^(3/4)*atan(((-a)^(3
/4)*b^(1/4)*x^(1/2)*(a*d - b*c)^3*(a^9*d^6 + a^3*b^6*c^6 - 6*a^4*b^5*c^5*d + 15*a^5*b^4*c^4*d^2 - 20*a^6*b^3*c
^3*d^3 + 15*a^7*b^2*c^2*d^4 - 6*a^8*b*c*d^5)*1i)/(a^13*d^9 - a^4*b^9*c^9 + 9*a^5*b^8*c^8*d - 36*a^6*b^7*c^7*d^
2 + 84*a^7*b^6*c^6*d^3 - 126*a^8*b^5*c^5*d^4 + 126*a^9*b^4*c^4*d^5 - 84*a^10*b^3*c^3*d^6 + 36*a^11*b^2*c^2*d^7
- 9*a^12*b*c*d^8))*(a*d - b*c)^3*1i)/b^(19/4)

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sympy [F(-1)]  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(x**(5/2)*(d*x**2+c)**3/(b*x**2+a),x)

[Out]

Timed out

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