Optimal. Leaf size=116 \[ -\frac{2 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a^2 \sqrt{b}}+\frac{\sqrt{c} (2 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a^2}-\frac{c \sqrt{c+d x^3}}{3 a x^3} \]
[Out]
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Rubi [A] time = 0.401954, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{2 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a^2 \sqrt{b}}+\frac{\sqrt{c} (2 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a^2}-\frac{c \sqrt{c+d x^3}}{3 a x^3} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^3)^(3/2)/(x^4*(a + b*x^3)),x]
[Out]
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Rubi in Sympy [A] time = 43.1943, size = 104, normalized size = 0.9 \[ - \frac{c \sqrt{c + d x^{3}}}{3 a x^{3}} - \frac{\sqrt{c} \left (3 a d - 2 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{3 a^{2}} + \frac{2 \left (a d - b c\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{3 a^{2} \sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**3+c)**(3/2)/x**4/(b*x**3+a),x)
[Out]
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Mathematica [C] time = 0.62109, size = 414, normalized size = 3.57 \[ \frac{c \left (\frac{5 b d x^3 \left (3 a \left (c+2 d x^3\right )+b x^3 \left (c+3 d x^3\right )\right ) F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )-3 \left (a+b x^3\right ) \left (c+d x^3\right ) \left (2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )\right )}{a \left (-5 b d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )\right )}+\frac{6 d x^6 (b c-2 a d) F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{x^3 \left (2 b c F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-4 a c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}\right )}{9 x^3 \left (a+b x^3\right ) \sqrt{c+d x^3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(c + d*x^3)^(3/2)/(x^4*(a + b*x^3)),x]
[Out]
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Maple [C] time = 0.013, size = 620, normalized size = 5.3 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^3+c)^(3/2)/x^4/(b*x^3+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{3} + c\right )}^{\frac{3}{2}}}{{\left (b x^{3} + a\right )} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)^(3/2)/((b*x^3 + a)*x^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.285111, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (b c - a d\right )} x^{3} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x^{3} + 2 \, b c - a d + 2 \, \sqrt{d x^{3} + c} b \sqrt{\frac{b c - a d}{b}}}{b x^{3} + a}\right ) +{\left (2 \, b c - 3 \, a d\right )} \sqrt{c} x^{3} \log \left (\frac{d x^{3} - 2 \, \sqrt{d x^{3} + c} \sqrt{c} + 2 \, c}{x^{3}}\right ) + 2 \, \sqrt{d x^{3} + c} a c}{6 \, a^{2} x^{3}}, -\frac{4 \,{\left (b c - a d\right )} x^{3} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) +{\left (2 \, b c - 3 \, a d\right )} \sqrt{c} x^{3} \log \left (\frac{d x^{3} - 2 \, \sqrt{d x^{3} + c} \sqrt{c} + 2 \, c}{x^{3}}\right ) + 2 \, \sqrt{d x^{3} + c} a c}{6 \, a^{2} x^{3}}, \frac{{\left (2 \, b c - 3 \, a d\right )} \sqrt{-c} x^{3} \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right ) -{\left (b c - a d\right )} x^{3} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x^{3} + 2 \, b c - a d + 2 \, \sqrt{d x^{3} + c} b \sqrt{\frac{b c - a d}{b}}}{b x^{3} + a}\right ) - \sqrt{d x^{3} + c} a c}{3 \, a^{2} x^{3}}, \frac{{\left (2 \, b c - 3 \, a d\right )} \sqrt{-c} x^{3} \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right ) - 2 \,{\left (b c - a d\right )} x^{3} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) - \sqrt{d x^{3} + c} a c}{3 \, a^{2} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)^(3/2)/((b*x^3 + a)*x^4),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((d*x**3+c)**(3/2)/x**4/(b*x**3+a),x)
[Out]
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GIAC/XCAS [A] time = 0.219405, size = 182, normalized size = 1.57 \[ \frac{1}{3} \, d^{2}{\left (\frac{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{2} d^{2}} - \frac{{\left (2 \, b c^{2} - 3 \, a c d\right )} \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} d^{2}} - \frac{\sqrt{d x^{3} + c} c}{a d^{2} x^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)^(3/2)/((b*x^3 + a)*x^4),x, algorithm="giac")
[Out]