Optimal. Leaf size=67 \[ -\frac{\sqrt{\frac{d x^3}{c}+1} F_1\left (-\frac{2}{3};2,\frac{3}{2};\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{2 a^2 c x^2 \sqrt{c+d x^3}} \]
[Out]
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Rubi [A] time = 0.205784, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{\sqrt{\frac{d x^3}{c}+1} F_1\left (-\frac{2}{3};2,\frac{3}{2};\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{2 a^2 c x^2 \sqrt{c+d x^3}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*(a + b*x^3)^2*(c + d*x^3)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 23.4225, size = 58, normalized size = 0.87 \[ - \frac{\sqrt{c + d x^{3}} \operatorname{appellf_{1}}{\left (- \frac{2}{3},\frac{3}{2},2,\frac{1}{3},- \frac{d x^{3}}{c},- \frac{b x^{3}}{a} \right )}}{2 a^{2} c^{2} x^{2} \sqrt{1 + \frac{d x^{3}}{c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(b*x**3+a)**2/(d*x**3+c)**(3/2),x)
[Out]
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Mathematica [B] time = 2.13514, size = 483, normalized size = 7.21 \[ \frac{\frac{7 a b c d x^6 \left (7 a^2 d^2-6 a b c d+5 b^2 c^2\right ) F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{3 x^3 \left (2 b c F_1\left (\frac{7}{3};\frac{1}{2},2;\frac{10}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (\frac{7}{3};\frac{3}{2},1;\frac{10}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-14 a c F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}+\frac{16 a c x^3 \left (7 a^3 d^3-6 a^2 b c d^2-33 a b^2 c^2 d+20 b^3 c^3\right ) F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{3 x^3 \left (2 b c F_1\left (\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-8 a c F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}-4 \left (a^3 d^2 \left (3 c+7 d x^3\right )+a^2 b d \left (-6 c^2-3 c d x^3+7 d^2 x^6\right )+3 a b^2 c \left (c^2-c d x^3-2 d^2 x^6\right )+5 b^3 c^2 x^3 \left (c+d x^3\right )\right )}{24 a^2 c^2 x^2 \left (a+b x^3\right ) \sqrt{c+d x^3} (b c-a d)^2} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/(x^3*(a + b*x^3)^2*(c + d*x^3)^(3/2)),x]
[Out]
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Maple [C] time = 0.017, size = 1919, normalized size = 28.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(b*x^3+a)^2/(d*x^3+c)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{3} + a\right )}^{2}{\left (d x^{3} + c\right )}^{\frac{3}{2}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^3 + a)^2*(d*x^3 + c)^(3/2)*x^3),x, algorithm="maxima")
[Out]
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^3 + a)^2*(d*x^3 + c)^(3/2)*x^3),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**3/(b*x**3+a)**2/(d*x**3+c)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{3} + a\right )}^{2}{\left (d x^{3} + c\right )}^{\frac{3}{2}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^3 + a)^2*(d*x^3 + c)^(3/2)*x^3),x, algorithm="giac")
[Out]