Optimal. Leaf size=65 \[ \frac{c x^2 \sqrt{c+d x^3} F_1\left (\frac{2}{3};1,-\frac{3}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{2 a \sqrt{\frac{d x^3}{c}+1}} \]
[Out]
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Rubi [A] time = 0.140085, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{c x^2 \sqrt{c+d x^3} F_1\left (\frac{2}{3};1,-\frac{3}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{2 a \sqrt{\frac{d x^3}{c}+1}} \]
Antiderivative was successfully verified.
[In] Int[(x*(c + d*x^3)^(3/2))/(a + b*x^3),x]
[Out]
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Rubi in Sympy [A] time = 20.3618, size = 53, normalized size = 0.82 \[ \frac{c x^{2} \sqrt{c + d x^{3}} \operatorname{appellf_{1}}{\left (\frac{2}{3},- \frac{3}{2},1,\frac{5}{3},- \frac{d x^{3}}{c},- \frac{b x^{3}}{a} \right )}}{2 a \sqrt{1 + \frac{d x^{3}}{c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(d*x**3+c)**(3/2)/(b*x**3+a),x)
[Out]
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Mathematica [B] time = 0.698085, size = 437, normalized size = 6.72 \[ \frac{x^2 \left (\frac{25 a c^2 (4 a d-7 b c) F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{3 x^3 \left (2 b c F_1\left (\frac{5}{3};\frac{1}{2},2;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (\frac{5}{3};\frac{3}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-10 a c F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}+\frac{2 d \left (15 x^3 \left (a+b x^3\right ) \left (c+d x^3\right ) \left (2 b c F_1\left (\frac{8}{3};\frac{1}{2},2;\frac{11}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (\frac{8}{3};\frac{3}{2},1;\frac{11}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-8 a c \left (10 a c+3 a d x^3+20 b c x^3+10 b d x^6\right ) F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )}{3 x^3 \left (2 b c F_1\left (\frac{8}{3};\frac{1}{2},2;\frac{11}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (\frac{8}{3};\frac{3}{2},1;\frac{11}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-16 a c F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}\right )}{35 b \left (a+b x^3\right ) \sqrt{c+d x^3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(x*(c + d*x^3)^(3/2))/(a + b*x^3),x]
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Maple [C] time = 0.05, size = 930, normalized size = 14.3 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(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. \[ \int \frac{{\left (d x^{3} + c\right )}^{\frac{3}{2}} x}{b x^{3} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)^(3/2)*x/(b*x^3 + a),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((d*x^3 + c)^(3/2)*x/(b*x^3 + a),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x \left (c + d x^{3}\right )^{\frac{3}{2}}}{a + b x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(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. \[ \int \frac{{\left (d x^{3} + c\right )}^{\frac{3}{2}} x}{b x^{3} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)^(3/2)*x/(b*x^3 + a),x, algorithm="giac")
[Out]