Optimal. Leaf size=170 \[ \frac{\sqrt{c} (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a^3}-\frac{\sqrt{b c-a d} (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a^3 \sqrt{b}}-\frac{\sqrt{c+d x^3} (2 b c-a d)}{3 a^2 \left (a+b x^3\right )}-\frac{c \sqrt{c+d x^3}}{3 a x^3 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.748304, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{\sqrt{c} (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a^3}-\frac{\sqrt{b c-a d} (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a^3 \sqrt{b}}-\frac{\sqrt{c+d x^3} (2 b c-a d)}{3 a^2 \left (a+b x^3\right )}-\frac{c \sqrt{c+d x^3}}{3 a x^3 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^3)^(3/2)/(x^4*(a + b*x^3)^2),x]
[Out]
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Rubi in Sympy [A] time = 79.1586, size = 153, normalized size = 0.9 \[ - \frac{\sqrt{c + d x^{3}} \left (a d - b c\right )}{3 a b x^{3} \left (a + b x^{3}\right )} + \frac{\sqrt{c + d x^{3}} \left (a d - 2 b c\right )}{3 a^{2} b x^{3}} - \frac{\sqrt{c} \left (3 a d - 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{3 a^{3}} + \frac{\left (a d - 4 b c\right ) \sqrt{a d - b c} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{3 a^{3} \sqrt{b}} \]
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)**2,x)
[Out]
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Mathematica [C] time = 0.808406, size = 439, normalized size = 2.58 \[ \frac{\frac{5 b d x^3 \left (3 a \left (c^2+c d x^3-d^2 x^6\right )+2 b c x^3 \left (c+3 d x^3\right )\right ) F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )-3 \left (c+d x^3\right ) \left (a \left (c-d x^3\right )+2 b c x^3\right ) \left (2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )\right )}{-5 b d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},-\frac{a}{b x^3}\right )}+\frac{6 a c d x^6 (a d-2 b c) F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{4 a c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )-x^3 \left (2 b c F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )}}{9 a^2 x^3 \left (a+b x^3\right ) \sqrt{c+d x^3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(c + d*x^3)^(3/2)/(x^4*(a + b*x^3)^2),x]
[Out]
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Maple [C] time = 0.02, size = 1093, normalized size = 6.4 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^3+c)^(3/2)/x^4/(b*x^3+a)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{3} + c\right )}^{\frac{3}{2}}}{{\left (b x^{3} + a\right )}^{2} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)^(3/2)/((b*x^3 + a)^2*x^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.255062, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)^(3/2)/((b*x^3 + a)^2*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)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.225684, size = 300, normalized size = 1.76 \[ -\frac{1}{3} \, d^{3}{\left (\frac{2 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} b c - 2 \, \sqrt{d x^{3} + c} b c^{2} -{\left (d x^{3} + c\right )}^{\frac{3}{2}} a d + 2 \, \sqrt{d x^{3} + c} a c d}{{\left ({\left (d x^{3} + c\right )}^{2} b - 2 \,{\left (d x^{3} + c\right )} b c + b c^{2} +{\left (d x^{3} + c\right )} a d - a c d\right )} a^{2} d^{2}} - \frac{{\left (4 \, b^{2} c^{2} - 5 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{3} d^{3}} + \frac{{\left (4 \, b c^{2} - 3 \, a c d\right )} \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{a^{3} \sqrt{-c} d^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)^(3/2)/((b*x^3 + a)^2*x^4),x, algorithm="giac")
[Out]