Optimal. Leaf size=149 \[ -\frac {2 b^2 \sqrt {e+f x} (-3 a d f+b c f+2 b d e)}{d^2 f^3}+\frac {2 (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{5/2} (d e-c f)^{3/2}}-\frac {2 (b e-a f)^3}{f^3 \sqrt {e+f x} (d e-c f)}+\frac {2 b^3 (e+f x)^{3/2}}{3 d f^3} \]
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Rubi [A] time = 0.21, antiderivative size = 169, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {87, 43, 63, 208} \[ -\frac {2 b^2 \sqrt {e+f x} (-3 a d f+b c f+b d e)}{d^2 f^3}+\frac {2 (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{5/2} (d e-c f)^{3/2}}-\frac {2 (b e-a f)^3}{f^3 \sqrt {e+f x} (d e-c f)}+\frac {2 b^3 (e+f x)^{3/2}}{3 d f^3}-\frac {2 b^3 e \sqrt {e+f x}}{d f^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 63
Rule 87
Rule 208
Rubi steps
\begin {align*} \int \frac {(a+b x)^3}{(c+d x) (e+f x)^{3/2}} \, dx &=\int \left (\frac {(-b e+a f)^3}{f^2 (-d e+c f) (e+f x)^{3/2}}-\frac {b^2 (b d e+b c f-3 a d f)}{d^2 f^2 \sqrt {e+f x}}+\frac {b^3 x}{d f \sqrt {e+f x}}+\frac {(-b c+a d)^3}{d^2 (d e-c f) (c+d x) \sqrt {e+f x}}\right ) \, dx\\ &=-\frac {2 (b e-a f)^3}{f^3 (d e-c f) \sqrt {e+f x}}-\frac {2 b^2 (b d e+b c f-3 a d f) \sqrt {e+f x}}{d^2 f^3}+\frac {b^3 \int \frac {x}{\sqrt {e+f x}} \, dx}{d f}-\frac {(b c-a d)^3 \int \frac {1}{(c+d x) \sqrt {e+f x}} \, dx}{d^2 (d e-c f)}\\ &=-\frac {2 (b e-a f)^3}{f^3 (d e-c f) \sqrt {e+f x}}-\frac {2 b^2 (b d e+b c f-3 a d f) \sqrt {e+f x}}{d^2 f^3}+\frac {b^3 \int \left (-\frac {e}{f \sqrt {e+f x}}+\frac {\sqrt {e+f x}}{f}\right ) \, dx}{d f}-\frac {\left (2 (b c-a d)^3\right ) \operatorname {Subst}\left (\int \frac {1}{c-\frac {d e}{f}+\frac {d x^2}{f}} \, dx,x,\sqrt {e+f x}\right )}{d^2 f (d e-c f)}\\ &=-\frac {2 (b e-a f)^3}{f^3 (d e-c f) \sqrt {e+f x}}-\frac {2 b^3 e \sqrt {e+f x}}{d f^3}-\frac {2 b^2 (b d e+b c f-3 a d f) \sqrt {e+f x}}{d^2 f^3}+\frac {2 b^3 (e+f x)^{3/2}}{3 d f^3}+\frac {2 (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{5/2} (d e-c f)^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.18, size = 165, normalized size = 1.11 \[ \frac {2 \left (-\frac {3 b \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{f^3}-\frac {3 b^2 d (e+f x) (-3 a d f+b c f+2 b d e)}{f^3}+\frac {3 (b c-a d)^3 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {d (e+f x)}{d e-c f}\right )}{c f-d e}+\frac {b^3 d^2 (e+f x)^2}{f^3}\right )}{3 d^3 \sqrt {e+f x}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.71, size = 978, normalized size = 6.56 \[ \left [\frac {3 \, {\left ({\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{4} x + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} e f^{3}\right )} \sqrt {d^{2} e - c d f} \log \left (\frac {d f x + 2 \, d e - c f + 2 \, \sqrt {d^{2} e - c d f} \sqrt {f x + e}}{d x + c}\right ) - 2 \, {\left (8 \, b^{3} d^{4} e^{4} + 3 \, a^{3} c d^{3} f^{4} - 2 \, {\left (5 \, b^{3} c d^{3} + 9 \, a b^{2} d^{4}\right )} e^{3} f - {\left (b^{3} c^{2} d^{2} - 27 \, a b^{2} c d^{3} - 9 \, a^{2} b d^{4}\right )} e^{2} f^{2} + 3 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} - 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} e f^{3} - {\left (b^{3} d^{4} e^{2} f^{2} - 2 \, b^{3} c d^{3} e f^{3} + b^{3} c^{2} d^{2} f^{4}\right )} x^{2} + {\left (4 \, b^{3} d^{4} e^{3} f - {\left (5 \, b^{3} c d^{3} + 9 \, a b^{2} d^{4}\right )} e^{2} f^{2} - 2 \, {\left (b^{3} c^{2} d^{2} - 9 \, a b^{2} c d^{3}\right )} e f^{3} + 3 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2}\right )} f^{4}\right )} x\right )} \sqrt {f x + e}}{3 \, {\left (d^{5} e^{3} f^{3} - 2 \, c d^{4} e^{2} f^{4} + c^{2} d^{3} e f^{5} + {\left (d^{5} e^{2} f^{4} - 2 \, c d^{4} e f^{5} + c^{2} d^{3} f^{6}\right )} x\right )}}, -\frac {2 \, {\left (3 \, {\left ({\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{4} x + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} e f^{3}\right )} \sqrt {-d^{2} e + c d f} \arctan \left (\frac {\sqrt {-d^{2} e + c d f} \sqrt {f x + e}}{d f x + d e}\right ) + {\left (8 \, b^{3} d^{4} e^{4} + 3 \, a^{3} c d^{3} f^{4} - 2 \, {\left (5 \, b^{3} c d^{3} + 9 \, a b^{2} d^{4}\right )} e^{3} f - {\left (b^{3} c^{2} d^{2} - 27 \, a b^{2} c d^{3} - 9 \, a^{2} b d^{4}\right )} e^{2} f^{2} + 3 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} - 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} e f^{3} - {\left (b^{3} d^{4} e^{2} f^{2} - 2 \, b^{3} c d^{3} e f^{3} + b^{3} c^{2} d^{2} f^{4}\right )} x^{2} + {\left (4 \, b^{3} d^{4} e^{3} f - {\left (5 \, b^{3} c d^{3} + 9 \, a b^{2} d^{4}\right )} e^{2} f^{2} - 2 \, {\left (b^{3} c^{2} d^{2} - 9 \, a b^{2} c d^{3}\right )} e f^{3} + 3 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2}\right )} f^{4}\right )} x\right )} \sqrt {f x + e}\right )}}{3 \, {\left (d^{5} e^{3} f^{3} - 2 \, c d^{4} e^{2} f^{4} + c^{2} d^{3} e f^{5} + {\left (d^{5} e^{2} f^{4} - 2 \, c d^{4} e f^{5} + c^{2} d^{3} f^{6}\right )} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.29, size = 241, normalized size = 1.62 \[ \frac {2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {c d f - d^{2} e}}\right )}{{\left (c d^{2} f - d^{3} e\right )} \sqrt {c d f - d^{2} e}} - \frac {2 \, {\left (a^{3} f^{3} - 3 \, a^{2} b f^{2} e + 3 \, a b^{2} f e^{2} - b^{3} e^{3}\right )}}{{\left (c f^{4} - d f^{3} e\right )} \sqrt {f x + e}} + \frac {2 \, {\left ({\left (f x + e\right )}^{\frac {3}{2}} b^{3} d^{2} f^{6} - 3 \, \sqrt {f x + e} b^{3} c d f^{7} + 9 \, \sqrt {f x + e} a b^{2} d^{2} f^{7} - 6 \, \sqrt {f x + e} b^{3} d^{2} f^{6} e\right )}}{3 \, d^{3} f^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 395, normalized size = 2.65 \[ -\frac {2 a^{3} d \arctan \left (\frac {\sqrt {f x +e}\, d}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right ) \sqrt {\left (c f -d e \right ) d}}+\frac {6 a^{2} b c \arctan \left (\frac {\sqrt {f x +e}\, d}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right ) \sqrt {\left (c f -d e \right ) d}}-\frac {6 a \,b^{2} c^{2} \arctan \left (\frac {\sqrt {f x +e}\, d}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right ) \sqrt {\left (c f -d e \right ) d}\, d}+\frac {2 b^{3} c^{3} \arctan \left (\frac {\sqrt {f x +e}\, d}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right ) \sqrt {\left (c f -d e \right ) d}\, d^{2}}-\frac {2 a^{3}}{\left (c f -d e \right ) \sqrt {f x +e}}+\frac {6 a^{2} b e}{\left (c f -d e \right ) \sqrt {f x +e}\, f}-\frac {6 a \,b^{2} e^{2}}{\left (c f -d e \right ) \sqrt {f x +e}\, f^{2}}+\frac {2 b^{3} e^{3}}{\left (c f -d e \right ) \sqrt {f x +e}\, f^{3}}+\frac {6 \sqrt {f x +e}\, a \,b^{2}}{d \,f^{2}}-\frac {2 \sqrt {f x +e}\, b^{3} c}{d^{2} f^{2}}-\frac {4 \sqrt {f x +e}\, b^{3} e}{d \,f^{3}}+\frac {2 \left (f x +e \right )^{\frac {3}{2}} b^{3}}{3 d \,f^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.32, size = 257, normalized size = 1.72 \[ \frac {2\,b^3\,{\left (e+f\,x\right )}^{3/2}}{3\,d\,f^3}-\sqrt {e+f\,x}\,\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{d^2\,f^6}\right )+\frac {2\,\mathrm {atan}\left (\frac {2\,\sqrt {e+f\,x}\,{\left (a\,d-b\,c\right )}^3\,\left (d^3\,e-c\,d^2\,f\right )}{d^{3/2}\,{\left (c\,f-d\,e\right )}^{3/2}\,\left (2\,a^3\,d^3-6\,a^2\,b\,c\,d^2+6\,a\,b^2\,c^2\,d-2\,b^3\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{d^{5/2}\,{\left (c\,f-d\,e\right )}^{3/2}}-\frac {2\,\left (a^3\,d^2\,f^3-3\,a^2\,b\,d^2\,e\,f^2+3\,a\,b^2\,d^2\,e^2\,f-b^3\,d^2\,e^3\right )}{d^2\,f^3\,\sqrt {e+f\,x}\,\left (c\,f-d\,e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 93.73, size = 144, normalized size = 0.97 \[ \frac {2 b^{3} \left (e + f x\right )^{\frac {3}{2}}}{3 d f^{3}} - \frac {2 \left (a f - b e\right )^{3}}{f^{3} \sqrt {e + f x} \left (c f - d e\right )} + \frac {\sqrt {e + f x} \left (6 a b^{2} d f - 2 b^{3} c f - 4 b^{3} d e\right )}{d^{2} f^{3}} - \frac {2 \left (a d - b c\right )^{3} \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{d^{3} \sqrt {\frac {c f - d e}{d}} \left (c f - d e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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