Optimal. Leaf size=82 \[ \frac{2 a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 \sqrt{b} (b c-a d)^{3/2}}-\frac{2 c}{3 d \sqrt{c+d x^3} (b c-a d)} \]
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Rubi [A] time = 0.200105, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{2 a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 \sqrt{b} (b c-a d)^{3/2}}-\frac{2 c}{3 d \sqrt{c+d x^3} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[x^5/((a + b*x^3)*(c + d*x^3)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 21.9324, size = 70, normalized size = 0.85 \[ \frac{2 a \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{3 \sqrt{b} \left (a d - b c\right )^{\frac{3}{2}}} + \frac{2 c}{3 d \sqrt{c + d x^{3}} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5/(b*x**3+a)/(d*x**3+c)**(3/2),x)
[Out]
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Mathematica [A] time = 0.145534, size = 80, normalized size = 0.98 \[ \frac{\frac{2 c}{d \sqrt{c+d x^3}}-\frac{2 a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{\sqrt{b} \sqrt{b c-a d}}}{3 a d-3 b c} \]
Antiderivative was successfully verified.
[In] Integrate[x^5/((a + b*x^3)*(c + d*x^3)^(3/2)),x]
[Out]
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Maple [C] time = 0.013, size = 487, normalized size = 5.9 \[ -{\frac{2}{3\,bd}{\frac{1}{\sqrt{d{x}^{3}+c}}}}-{\frac{a}{b} \left ( -{\frac{2}{3\,ad-3\,bc}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{c}{d}} \right ) d}}}}-{\frac{{\frac{i}{3}}b\sqrt{2}}{{d}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ( b{{\it \_Z}}^{3}+a \right ) }{\frac{1}{ \left ( -ad+bc \right ) \left ( ad-bc \right ) }\sqrt [3]{-c{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-c{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}d \left ( x+{\frac{1}{2\,d}\sqrt [3]{-c{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}},{\frac{b}{2\, \left ( ad-bc \right ) d} \left ( 2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-c{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5/(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^5/((b*x^3 + a)*(d*x^3 + c)^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.227296, size = 1, normalized size = 0.01 \[ \left [-\frac{\sqrt{d x^{3} + c} a d \log \left (\frac{{\left (b d x^{3} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} - 2 \, \sqrt{d x^{3} + c}{\left (b^{2} c - a b d\right )}}{b x^{3} + a}\right ) + 2 \, \sqrt{b^{2} c - a b d} c}{3 \, \sqrt{d x^{3} + c} \sqrt{b^{2} c - a b d}{\left (b c d - a d^{2}\right )}}, \frac{2 \,{\left (\sqrt{d x^{3} + c} a d \arctan \left (-\frac{b c - a d}{\sqrt{d x^{3} + c} \sqrt{-b^{2} c + a b d}}\right ) - \sqrt{-b^{2} c + a b d} c\right )}}{3 \, \sqrt{d x^{3} + c} \sqrt{-b^{2} c + a b d}{\left (b c d - a d^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/((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. \[ \int \frac{x^{5}}{\left (a + b x^{3}\right ) \left (c + d x^{3}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5/(b*x**3+a)/(d*x**3+c)**(3/2),x)
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GIAC/XCAS [A] time = 0.216982, size = 105, normalized size = 1.28 \[ -\frac{2 \,{\left (\frac{a d \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d}{\left (b c - a d\right )}} + \frac{c}{\sqrt{d x^{3} + c}{\left (b c - a d\right )}}\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/((b*x^3 + a)*(d*x^3 + c)^(3/2)),x, algorithm="giac")
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