Optimal. Leaf size=107 \[ -\frac{2 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{3/2} (b c-a d)^{3/2}}+\frac{2 c^2}{3 d^2 \sqrt{c+d x^3} (b c-a d)}+\frac{2 \sqrt{c+d x^3}}{3 b d^2} \]
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Rubi [A] time = 0.346043, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{2 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{3/2} (b c-a d)^{3/2}}+\frac{2 c^2}{3 d^2 \sqrt{c+d x^3} (b c-a d)}+\frac{2 \sqrt{c+d x^3}}{3 b d^2} \]
Antiderivative was successfully verified.
[In] Int[x^8/((a + b*x^3)*(c + d*x^3)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 50.2725, size = 138, normalized size = 1.29 \[ \frac{2 a^{2} \sqrt{c + d x^{3}}}{3 b \left (a d - b c\right )^{2}} - \frac{2 a^{2} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{3 b^{\frac{3}{2}} \left (a d - b c\right )^{\frac{3}{2}}} - \frac{2 c^{2}}{3 d^{2} \sqrt{c + d x^{3}} \left (a d - b c\right )} - \frac{2 c \sqrt{c + d x^{3}} \left (2 a d - b c\right )}{3 d^{2} \left (a d - b c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**8/(b*x**3+a)/(d*x**3+c)**(3/2),x)
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Mathematica [A] time = 0.492404, size = 99, normalized size = 0.93 \[ \frac{1}{3} \left (\frac{2 \left (\frac{c^2}{b c-a d}+\frac{c+d x^3}{b}\right )}{d^2 \sqrt{c+d x^3}}-\frac{2 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{b^{3/2} (b c-a d)^{3/2}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^8/((a + b*x^3)*(c + d*x^3)^(3/2)),x]
[Out]
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Maple [C] time = 0.055, size = 527, normalized size = 4.9 \[{\frac{1}{{b}^{2}} \left ( b \left ({\frac{2\,c}{3\,{d}^{2}}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{c}{d}} \right ) d}}}}+{\frac{2}{3\,{d}^{2}}\sqrt{d{x}^{3}+c}} \right ) +{\frac{2\,a}{3\,d}{\frac{1}{\sqrt{d{x}^{3}+c}}}} \right ) }+{\frac{{a}^{2}}{{b}^{2}} \left ( -{\frac{2}{3\,ad-3\,bc}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{c}{d}} \right ) d}}}}-{\frac{{\frac{i}{3}}b\sqrt{2}}{{d}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ( b{{\it \_Z}}^{3}+a \right ) }{\frac{1}{ \left ( -ad+bc \right ) \left ( ad-bc \right ) }\sqrt [3]{-c{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-c{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{id\sqrt{3} \left ( x+{\frac{1}{2\,d}\sqrt [3]{-c{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}},{\frac{b}{2\, \left ( ad-bc \right ) d} \left ( 2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-c{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^8/(b*x^3+a)/(d*x^3+c)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^8/((b*x^3 + a)*(d*x^3 + c)^(3/2)),x, algorithm="maxima")
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Fricas [A] time = 0.229162, size = 1, normalized size = 0.01 \[ \left [-\frac{\sqrt{d x^{3} + c} a^{2} d^{2} \log \left (\frac{{\left (b d x^{3} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} + 2 \, \sqrt{d x^{3} + c}{\left (b^{2} c - a b d\right )}}{b x^{3} + a}\right ) - 2 \,{\left ({\left (b c d - a d^{2}\right )} x^{3} + 2 \, b c^{2} - a c d\right )} \sqrt{b^{2} c - a b d}}{3 \,{\left (b^{2} c d^{2} - a b d^{3}\right )} \sqrt{d x^{3} + c} \sqrt{b^{2} c - a b d}}, -\frac{2 \,{\left (\sqrt{d x^{3} + c} a^{2} d^{2} \arctan \left (-\frac{b c - a d}{\sqrt{d x^{3} + c} \sqrt{-b^{2} c + a b d}}\right ) -{\left ({\left (b c d - a d^{2}\right )} x^{3} + 2 \, b c^{2} - a c d\right )} \sqrt{-b^{2} c + a b d}\right )}}{3 \,{\left (b^{2} c d^{2} - a b d^{3}\right )} \sqrt{d x^{3} + c} \sqrt{-b^{2} c + a b d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^8/((b*x^3 + a)*(d*x^3 + c)^(3/2)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{\left (a + b x^{3}\right ) \left (c + d x^{3}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**8/(b*x**3+a)/(d*x**3+c)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.217434, size = 139, normalized size = 1.3 \[ \frac{2 \, a^{2} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{3 \,{\left (b^{2} c - a b d\right )} \sqrt{-b^{2} c + a b d}} + \frac{2 \, c^{2}}{3 \,{\left (b c d^{2} - a d^{3}\right )} \sqrt{d x^{3} + c}} + \frac{2 \, \sqrt{d x^{3} + c}}{3 \, b d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^8/((b*x^3 + a)*(d*x^3 + c)^(3/2)),x, algorithm="giac")
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