Optimal. Leaf size=59 \[ \frac{x \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{1}{3};2,\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{a^2 \sqrt{c+d x^3}} \]
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Rubi [A] time = 0.0963542, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{x \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{1}{3};2,\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{a^2 \sqrt{c+d x^3}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x^3)^2*Sqrt[c + d*x^3]),x]
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Rubi in Sympy [A] time = 19.8986, size = 49, normalized size = 0.83 \[ \frac{x \sqrt{c + d x^{3}} \operatorname{appellf_{1}}{\left (\frac{1}{3},\frac{1}{2},2,\frac{4}{3},- \frac{d x^{3}}{c},- \frac{b x^{3}}{a} \right )}}{a^{2} c \sqrt{1 + \frac{d x^{3}}{c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x**3+a)**2/(d*x**3+c)**(1/2),x)
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Mathematica [B] time = 0.530552, size = 341, normalized size = 5.78 \[ \frac{x \left (\frac{7 b c d x^3 F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{3 x^3 \left (2 b c F_1\left (\frac{7}{3};\frac{1}{2},2;\frac{10}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (\frac{7}{3};\frac{3}{2},1;\frac{10}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-14 a c F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}+\frac{32 c (2 b c-3 a d) F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{3 x^3 \left (2 b c F_1\left (\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-8 a c F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}-\frac{4 b \left (c+d x^3\right )}{a}\right )}{12 \left (a+b x^3\right ) \sqrt{c+d x^3} (a d-b c)} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/((a + b*x^3)^2*Sqrt[c + d*x^3]),x]
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Maple [C] time = 0.007, size = 769, normalized size = 13. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x^3+a)^2/(d*x^3+c)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{3} + a\right )}^{2} \sqrt{d x^{3} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^3 + a)^2*sqrt(d*x^3 + c)),x, algorithm="maxima")
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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*sqrt(d*x^3 + c)),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/(b*x**3+a)**2/(d*x**3+c)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{3} + a\right )}^{2} \sqrt{d x^{3} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^3 + a)^2*sqrt(d*x^3 + c)),x, algorithm="giac")
[Out]