Optimal. Leaf size=65 \[ \frac{c x^4 \sqrt{c+d x^3} F_1\left (\frac{4}{3};1,-\frac{3}{2};\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{4 a \sqrt{\frac{d x^3}{c}+1}} \]
[Out]
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Rubi [A] time = 0.187155, antiderivative size = 65, 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 \[ \frac{c x^4 \sqrt{c+d x^3} F_1\left (\frac{4}{3};1,-\frac{3}{2};\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{4 a \sqrt{\frac{d x^3}{c}+1}} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(c + d*x^3)^(3/2))/(a + b*x^3),x]
[Out]
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Rubi in Sympy [A] time = 28.1699, size = 53, normalized size = 0.82 \[ \frac{c x^{4} \sqrt{c + d x^{3}} \operatorname{appellf_{1}}{\left (\frac{4}{3},- \frac{3}{2},1,\frac{7}{3},- \frac{d x^{3}}{c},- \frac{b x^{3}}{a} \right )}}{4 a \sqrt{1 + \frac{d x^{3}}{c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(d*x**3+c)**(3/2)/(b*x**3+a),x)
[Out]
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Mathematica [B] time = 1.01815, size = 382, normalized size = 5.88 \[ \frac{x \left (-\frac{7 a c x^3 \left (55 a^2 d^2-88 a b c d+27 b^2 c^2\right ) F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{\left (a+b x^3\right ) \left (3 x^3 \left (2 b c F_1\left (\frac{7}{3};\frac{1}{2},2;\frac{10}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (\frac{7}{3};\frac{3}{2},1;\frac{10}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-14 a c F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )}+\frac{32 a^2 c^2 (14 b c-11 a d) F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{\left (a+b x^3\right ) \left (3 x^3 \left (2 b c F_1\left (\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-8 a c F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )}+4 \left (c+d x^3\right ) \left (-11 a d+14 b c+5 b d x^3\right )\right )}{110 b^2 \sqrt{c+d x^3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(x^3*(c + d*x^3)^(3/2))/(a + b*x^3),x]
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Maple [C] time = 0.06, size = 1101, normalized size = 16.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(d*x^3+c)^(3/2)/(b*x^3+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{3} + c\right )}^{\frac{3}{2}} x^{3}}{b x^{3} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)^(3/2)*x^3/(b*x^3 + a),x, algorithm="maxima")
[Out]
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)^(3/2)*x^3/(b*x^3 + a),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(x**3*(d*x**3+c)**(3/2)/(b*x**3+a),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{3} + c\right )}^{\frac{3}{2}} x^{3}}{b x^{3} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)^(3/2)*x^3/(b*x^3 + a),x, algorithm="giac")
[Out]