3.108 \(\int \frac{x^7}{(a+b x^3) (c+d x^3)} \, dx\)

Optimal. Leaf size=301 \[ \frac{a^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{5/3} (b c-a d)}-\frac{a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{5/3} (b c-a d)}-\frac{a^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{5/3} (b c-a d)}-\frac{c^{5/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 d^{5/3} (b c-a d)}+\frac{c^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 d^{5/3} (b c-a d)}+\frac{c^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} d^{5/3} (b c-a d)}+\frac{x^2}{2 b d} \]

[Out]

x^2/(2*b*d) - (a^(5/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*b^(5/3)*(b*c - a*d)) + (c^(
5/3)*ArcTan[(c^(1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*d^(5/3)*(b*c - a*d)) - (a^(5/3)*Log[a^(1/3) +
b^(1/3)*x])/(3*b^(5/3)*(b*c - a*d)) + (c^(5/3)*Log[c^(1/3) + d^(1/3)*x])/(3*d^(5/3)*(b*c - a*d)) + (a^(5/3)*L
og[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(5/3)*(b*c - a*d)) - (c^(5/3)*Log[c^(2/3) - c^(1/3)*d^(1/3
)*x + d^(2/3)*x^2])/(6*d^(5/3)*(b*c - a*d))

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Rubi [A]  time = 0.309725, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {479, 584, 292, 31, 634, 617, 204, 628} \[ \frac{a^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{5/3} (b c-a d)}-\frac{a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{5/3} (b c-a d)}-\frac{a^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{5/3} (b c-a d)}-\frac{c^{5/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 d^{5/3} (b c-a d)}+\frac{c^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 d^{5/3} (b c-a d)}+\frac{c^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} d^{5/3} (b c-a d)}+\frac{x^2}{2 b d} \]

Antiderivative was successfully verified.

[In]

Int[x^7/((a + b*x^3)*(c + d*x^3)),x]

[Out]

x^2/(2*b*d) - (a^(5/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*b^(5/3)*(b*c - a*d)) + (c^(
5/3)*ArcTan[(c^(1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*d^(5/3)*(b*c - a*d)) - (a^(5/3)*Log[a^(1/3) +
b^(1/3)*x])/(3*b^(5/3)*(b*c - a*d)) + (c^(5/3)*Log[c^(1/3) + d^(1/3)*x])/(3*d^(5/3)*(b*c - a*d)) + (a^(5/3)*L
og[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(5/3)*(b*c - a*d)) - (c^(5/3)*Log[c^(2/3) - c^(1/3)*d^(1/3
)*x + d^(2/3)*x^2])/(6*d^(5/3)*(b*c - a*d))

Rule 479

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

Rule 584

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

Rule 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

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 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 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]

Rubi steps

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

Mathematica [A]  time = 0.143007, size = 242, normalized size = 0.8 \[ \frac{\frac{a^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{5/3}}-\frac{2 a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{5/3}}-\frac{2 \sqrt{3} a^{5/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{5/3}}-\frac{3 a x^2}{b}-\frac{c^{5/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{d^{5/3}}+\frac{2 c^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{d^{5/3}}+\frac{2 \sqrt{3} c^{5/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{d^{5/3}}+\frac{3 c x^2}{d}}{6 b c-6 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-3*a*x^2)/b + (3*c*x^2)/d - (2*Sqrt[3]*a^(5/3)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(5/3) + (2*Sqr
t[3]*c^(5/3)*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]])/d^(5/3) - (2*a^(5/3)*Log[a^(1/3) + b^(1/3)*x])/b^(5/
3) + (2*c^(5/3)*Log[c^(1/3) + d^(1/3)*x])/d^(5/3) + (a^(5/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b
^(5/3) - (c^(5/3)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/d^(5/3))/(6*b*c - 6*a*d)

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Maple [A]  time = 0.008, size = 269, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{2\,bd}}-{\frac{{c}^{2}}{3\,{d}^{2} \left ( ad-bc \right ) }\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}+{\frac{{c}^{2}}{6\,{d}^{2} \left ( ad-bc \right ) }\ln \left ({x}^{2}-\sqrt [3]{{\frac{c}{d}}}x+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}+{\frac{{c}^{2}\sqrt{3}}{3\,{d}^{2} \left ( ad-bc \right ) }\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}+{\frac{{a}^{2}}{3\,{b}^{2} \left ( ad-bc \right ) }\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{{a}^{2}}{6\,{b}^{2} \left ( ad-bc \right ) }\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{{a}^{2}\sqrt{3}}{3\,{b}^{2} \left ( ad-bc \right ) }\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2/b/d-1/3*c^2/d^2/(a*d-b*c)/(1/d*c)^(1/3)*ln(x+(1/d*c)^(1/3))+1/6*c^2/d^2/(a*d-b*c)/(1/d*c)^(1/3)*ln(x^2
-(1/d*c)^(1/3)*x+(1/d*c)^(2/3))+1/3*c^2/d^2/(a*d-b*c)*3^(1/2)/(1/d*c)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/d*c)^(1/3
)*x-1))+1/3*a^2/b^2/(a*d-b*c)/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-1/6*a^2/b^2/(a*d-b*c)/(a/b)^(1/3)*ln(x^2-(a/b)^(1/
3)*x+(a/b)^(2/3))-1/3*a^2/b^2/(a*d-b*c)*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.93356, size = 659, normalized size = 2.19 \begin{align*} \frac{2 \, \sqrt{3} a d \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} - \sqrt{3} a}{3 \, a}\right ) - 2 \, \sqrt{3} b c \left (-\frac{c^{2}}{d^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} d x \left (-\frac{c^{2}}{d^{2}}\right )^{\frac{1}{3}} + \sqrt{3} c}{3 \, c}\right ) + a d \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x^{2} - b x \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} + a \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}}\right ) + b c \left (-\frac{c^{2}}{d^{2}}\right )^{\frac{1}{3}} \log \left (c x^{2} - d x \left (-\frac{c^{2}}{d^{2}}\right )^{\frac{2}{3}} - c \left (-\frac{c^{2}}{d^{2}}\right )^{\frac{1}{3}}\right ) - 2 \, a d \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x + b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}\right ) - 2 \, b c \left (-\frac{c^{2}}{d^{2}}\right )^{\frac{1}{3}} \log \left (c x + d \left (-\frac{c^{2}}{d^{2}}\right )^{\frac{2}{3}}\right ) + 3 \,{\left (b c - a d\right )} x^{2}}{6 \,{\left (b^{2} c d - a b d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*sqrt(3)*a*d*(a^2/b^2)^(1/3)*arctan(1/3*(2*sqrt(3)*b*x*(a^2/b^2)^(1/3) - sqrt(3)*a)/a) - 2*sqrt(3)*b*c*(
-c^2/d^2)^(1/3)*arctan(1/3*(2*sqrt(3)*d*x*(-c^2/d^2)^(1/3) + sqrt(3)*c)/c) + a*d*(a^2/b^2)^(1/3)*log(a*x^2 - b
*x*(a^2/b^2)^(2/3) + a*(a^2/b^2)^(1/3)) + b*c*(-c^2/d^2)^(1/3)*log(c*x^2 - d*x*(-c^2/d^2)^(2/3) - c*(-c^2/d^2)
^(1/3)) - 2*a*d*(a^2/b^2)^(1/3)*log(a*x + b*(a^2/b^2)^(2/3)) - 2*b*c*(-c^2/d^2)^(1/3)*log(c*x + d*(-c^2/d^2)^(
2/3)) + 3*(b*c - a*d)*x^2)/(b^2*c*d - a*b*d^2)

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Sympy [B]  time = 167.128, size = 663, normalized size = 2.2 \begin{align*} \operatorname{RootSum}{\left (t^{3} \left (27 a^{3} d^{8} - 81 a^{2} b c d^{7} + 81 a b^{2} c^{2} d^{6} - 27 b^{3} c^{3} d^{5}\right ) + c^{5}, \left ( t \mapsto t \log{\left (x + \frac{- 243 t^{5} a^{8} b^{5} d^{13} + 1215 t^{5} a^{7} b^{6} c d^{12} - 2430 t^{5} a^{6} b^{7} c^{2} d^{11} + 2673 t^{5} a^{5} b^{8} c^{3} d^{10} - 2430 t^{5} a^{4} b^{9} c^{4} d^{9} + 2673 t^{5} a^{3} b^{10} c^{5} d^{8} - 2430 t^{5} a^{2} b^{11} c^{6} d^{7} + 1215 t^{5} a b^{12} c^{7} d^{6} - 243 t^{5} b^{13} c^{8} d^{5} + 9 t^{2} a^{10} d^{10} - 18 t^{2} a^{9} b c d^{9} + 9 t^{2} a^{8} b^{2} c^{2} d^{8} + 9 t^{2} a^{2} b^{8} c^{8} d^{2} - 18 t^{2} a b^{9} c^{9} d + 9 t^{2} b^{10} c^{10}}{a^{8} c^{3} d^{5} + a^{3} b^{5} c^{8}} \right )} \right )\right )} + \operatorname{RootSum}{\left (t^{3} \left (27 a^{3} b^{5} d^{3} - 81 a^{2} b^{6} c d^{2} + 81 a b^{7} c^{2} d - 27 b^{8} c^{3}\right ) - a^{5}, \left ( t \mapsto t \log{\left (x + \frac{- 243 t^{5} a^{8} b^{5} d^{13} + 1215 t^{5} a^{7} b^{6} c d^{12} - 2430 t^{5} a^{6} b^{7} c^{2} d^{11} + 2673 t^{5} a^{5} b^{8} c^{3} d^{10} - 2430 t^{5} a^{4} b^{9} c^{4} d^{9} + 2673 t^{5} a^{3} b^{10} c^{5} d^{8} - 2430 t^{5} a^{2} b^{11} c^{6} d^{7} + 1215 t^{5} a b^{12} c^{7} d^{6} - 243 t^{5} b^{13} c^{8} d^{5} + 9 t^{2} a^{10} d^{10} - 18 t^{2} a^{9} b c d^{9} + 9 t^{2} a^{8} b^{2} c^{2} d^{8} + 9 t^{2} a^{2} b^{8} c^{8} d^{2} - 18 t^{2} a b^{9} c^{9} d + 9 t^{2} b^{10} c^{10}}{a^{8} c^{3} d^{5} + a^{3} b^{5} c^{8}} \right )} \right )\right )} + \frac{x^{2}}{2 b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(b*x**3+a)/(d*x**3+c),x)

[Out]

RootSum(_t**3*(27*a**3*d**8 - 81*a**2*b*c*d**7 + 81*a*b**2*c**2*d**6 - 27*b**3*c**3*d**5) + c**5, Lambda(_t, _
t*log(x + (-243*_t**5*a**8*b**5*d**13 + 1215*_t**5*a**7*b**6*c*d**12 - 2430*_t**5*a**6*b**7*c**2*d**11 + 2673*
_t**5*a**5*b**8*c**3*d**10 - 2430*_t**5*a**4*b**9*c**4*d**9 + 2673*_t**5*a**3*b**10*c**5*d**8 - 2430*_t**5*a**
2*b**11*c**6*d**7 + 1215*_t**5*a*b**12*c**7*d**6 - 243*_t**5*b**13*c**8*d**5 + 9*_t**2*a**10*d**10 - 18*_t**2*
a**9*b*c*d**9 + 9*_t**2*a**8*b**2*c**2*d**8 + 9*_t**2*a**2*b**8*c**8*d**2 - 18*_t**2*a*b**9*c**9*d + 9*_t**2*b
**10*c**10)/(a**8*c**3*d**5 + a**3*b**5*c**8)))) + RootSum(_t**3*(27*a**3*b**5*d**3 - 81*a**2*b**6*c*d**2 + 81
*a*b**7*c**2*d - 27*b**8*c**3) - a**5, Lambda(_t, _t*log(x + (-243*_t**5*a**8*b**5*d**13 + 1215*_t**5*a**7*b**
6*c*d**12 - 2430*_t**5*a**6*b**7*c**2*d**11 + 2673*_t**5*a**5*b**8*c**3*d**10 - 2430*_t**5*a**4*b**9*c**4*d**9
+ 2673*_t**5*a**3*b**10*c**5*d**8 - 2430*_t**5*a**2*b**11*c**6*d**7 + 1215*_t**5*a*b**12*c**7*d**6 - 243*_t**
5*b**13*c**8*d**5 + 9*_t**2*a**10*d**10 - 18*_t**2*a**9*b*c*d**9 + 9*_t**2*a**8*b**2*c**2*d**8 + 9*_t**2*a**2*
b**8*c**8*d**2 - 18*_t**2*a*b**9*c**9*d + 9*_t**2*b**10*c**10)/(a**8*c**3*d**5 + a**3*b**5*c**8)))) + x**2/(2*
b*d)

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Giac [A]  time = 1.17978, size = 436, normalized size = 1.45 \begin{align*} \frac{\left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \,{\left (a b^{5} c - a^{2} b^{4} d\right )}} - \frac{\left (-c d^{2}\right )^{\frac{2}{3}} c^{2} d \log \left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b c^{2} d^{4} - a c d^{5}\right )}} - \frac{a^{2} \left (-\frac{a}{b}\right )^{\frac{2}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (a b^{2} c - a^{2} b d\right )}} + \frac{c^{2} \left (-\frac{c}{d}\right )^{\frac{2}{3}} \log \left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (b c^{2} d - a c d^{2}\right )}} - \frac{\left (-a b^{2}\right )^{\frac{2}{3}} a \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b^{4} c - \sqrt{3} a b^{3} d} + \frac{\left (-c d^{2}\right )^{\frac{2}{3}} c \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b c d^{3} - \sqrt{3} a d^{4}} + \frac{x^{2}}{2 \, b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(-a*b^2)^(2/3)*a^2*b*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a*b^5*c - a^2*b^4*d) - 1/6*(-c*d^2)^(2/3)*c
^2*d*log(x^2 + x*(-c/d)^(1/3) + (-c/d)^(2/3))/(b*c^2*d^4 - a*c*d^5) - 1/3*a^2*(-a/b)^(2/3)*log(abs(x - (-a/b)^
(1/3)))/(a*b^2*c - a^2*b*d) + 1/3*c^2*(-c/d)^(2/3)*log(abs(x - (-c/d)^(1/3)))/(b*c^2*d - a*c*d^2) - (-a*b^2)^(
2/3)*a*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(sqrt(3)*b^4*c - sqrt(3)*a*b^3*d) + (-c*d^2)^(2/3
)*c*arctan(1/3*sqrt(3)*(2*x + (-c/d)^(1/3))/(-c/d)^(1/3))/(sqrt(3)*b*c*d^3 - sqrt(3)*a*d^4) + 1/2*x^2/(b*d)