Optimal. Leaf size=149 \[ -\frac {2 (b e-a f)^3}{f^3 (d e-c f) \sqrt {e+f x}}-\frac {2 b^2 (2 b d e+b c f-3 a d f) \sqrt {e+f x}}{d^2 f^3}+\frac {2 b^3 (e+f x)^{3/2}}{3 d f^3}+\frac {2 (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{5/2} (d e-c f)^{3/2}} \]
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Rubi [A]
time = 0.13, 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 = {89, 45, 65, 214}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 65
Rule 89
Rule 214
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 ) \text {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 [A]
time = 0.34, size = 203, normalized size = 1.36 \begin {gather*} -\frac {2 \left (-9 a^2 b d^2 e f^2+3 a^3 d^2 f^3+9 a b^2 d f (-c f (e+f x)+d e (2 e+f x))+b^3 \left (3 c^2 f^2 (e+f x)+c d f \left (2 e^2+e f x-f^2 x^2\right )+d^2 e \left (-8 e^2-4 e f x+f^2 x^2\right )\right )\right )}{3 d^2 f^3 (-d e+c f) \sqrt {e+f x}}+\frac {2 (b c-a d)^3 \tan ^{-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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 205, normalized size = 1.38
method | result | size |
derivativedivides | \(\frac {\frac {2 b^{2} \left (\frac {b \left (f x +e \right )^{\frac {3}{2}} d}{3}+3 a d f \sqrt {f x +e}-b c f \sqrt {f x +e}-2 b d e \sqrt {f x +e}\right )}{d^{2}}-\frac {2 \left (a^{3} f^{3}-3 a^{2} b e \,f^{2}+3 a \,b^{2} e^{2} f -b^{3} e^{3}\right )}{\left (c f -d e \right ) \sqrt {f x +e}}-\frac {2 f^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right ) d^{2} \sqrt {\left (c f -d e \right ) d}}}{f^{3}}\) | \(205\) |
default | \(\frac {\frac {2 b^{2} \left (\frac {b \left (f x +e \right )^{\frac {3}{2}} d}{3}+3 a d f \sqrt {f x +e}-b c f \sqrt {f x +e}-2 b d e \sqrt {f x +e}\right )}{d^{2}}-\frac {2 \left (a^{3} f^{3}-3 a^{2} b e \,f^{2}+3 a \,b^{2} e^{2} f -b^{3} e^{3}\right )}{\left (c f -d e \right ) \sqrt {f x +e}}-\frac {2 f^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right ) d^{2} \sqrt {\left (c f -d e \right ) d}}}{f^{3}}\) | \(205\) |
risch | \(\frac {2 b^{2} \left (b d f x +9 a d f -3 b c f -5 b d e \right ) \sqrt {f x +e}}{3 f^{3} d^{2}}-\frac {2 a^{3}}{\left (c f -d e \right ) \sqrt {f x +e}}+\frac {6 a^{2} b e}{f \left (c f -d e \right ) \sqrt {f x +e}}-\frac {6 a \,b^{2} e^{2}}{f^{2} \left (c f -d e \right ) \sqrt {f x +e}}+\frac {2 b^{3} e^{3}}{f^{3} \left (c f -d e \right ) \sqrt {f x +e}}-\frac {2 d \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a^{3}}{\left (c f -d e \right ) \sqrt {\left (c f -d e \right ) d}}+\frac {6 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a^{2} b c}{\left (c f -d e \right ) \sqrt {\left (c f -d e \right ) d}}-\frac {6 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a \,b^{2} c^{2}}{d \left (c f -d e \right ) \sqrt {\left (c f -d e \right ) d}}+\frac {2 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) b^{3} c^{3}}{d^{2} \left (c f -d e \right ) \sqrt {\left (c f -d e \right ) d}}\) | \(359\) |
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 487 vs.
\(2 (140) = 280\).
time = 1.12, size = 987, normalized size = 6.62 \begin {gather*} \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 )} f^{3} e\right )} \sqrt {-c d f + d^{2} e} \log \left (\frac {d f x - c f + 2 \, d e - 2 \, \sqrt {-c d f + d^{2} e} \sqrt {f x + e}}{d x + c}\right ) - 2 \, {\left (b^{3} c^{2} d^{2} f^{4} x^{2} - 3 \, a^{3} c d^{3} f^{4} - 8 \, b^{3} d^{4} e^{4} - 3 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2}\right )} f^{4} x - 2 \, {\left (2 \, b^{3} d^{4} f x - {\left (5 \, b^{3} c d^{3} + 9 \, a b^{2} d^{4}\right )} f\right )} e^{3} + {\left (b^{3} d^{4} f^{2} x^{2} + {\left (5 \, b^{3} c d^{3} + 9 \, a b^{2} d^{4}\right )} f^{2} x + {\left (b^{3} c^{2} d^{2} - 27 \, a b^{2} c d^{3} - 9 \, a^{2} b d^{4}\right )} f^{2}\right )} e^{2} - {\left (2 \, b^{3} c d^{3} f^{3} x^{2} - 2 \, {\left (b^{3} c^{2} d^{2} - 9 \, a b^{2} c d^{3}\right )} f^{3} x + 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 )} f^{3}\right )} e\right )} \sqrt {f x + e}}{3 \, {\left (c^{2} d^{3} f^{6} x + d^{5} f^{3} e^{3} + {\left (d^{5} f^{4} x - 2 \, c d^{4} f^{4}\right )} e^{2} - {\left (2 \, c d^{4} f^{5} x - c^{2} d^{3} f^{5}\right )} e\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 )} f^{3} e\right )} \sqrt {c d f - d^{2} e} \arctan \left (\frac {\sqrt {c d f - d^{2} e} \sqrt {f x + e}}{d f x + d e}\right ) - {\left (b^{3} c^{2} d^{2} f^{4} x^{2} - 3 \, a^{3} c d^{3} f^{4} - 8 \, b^{3} d^{4} e^{4} - 3 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2}\right )} f^{4} x - 2 \, {\left (2 \, b^{3} d^{4} f x - {\left (5 \, b^{3} c d^{3} + 9 \, a b^{2} d^{4}\right )} f\right )} e^{3} + {\left (b^{3} d^{4} f^{2} x^{2} + {\left (5 \, b^{3} c d^{3} + 9 \, a b^{2} d^{4}\right )} f^{2} x + {\left (b^{3} c^{2} d^{2} - 27 \, a b^{2} c d^{3} - 9 \, a^{2} b d^{4}\right )} f^{2}\right )} e^{2} - {\left (2 \, b^{3} c d^{3} f^{3} x^{2} - 2 \, {\left (b^{3} c^{2} d^{2} - 9 \, a b^{2} c d^{3}\right )} f^{3} x + 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 )} f^{3}\right )} e\right )} \sqrt {f x + e}\right )}}{3 \, {\left (c^{2} d^{3} f^{6} x + d^{5} f^{3} e^{3} + {\left (d^{5} f^{4} x - 2 \, c d^{4} f^{4}\right )} e^{2} - {\left (2 \, c d^{4} f^{5} x - c^{2} d^{3} f^{5}\right )} e\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 40.34, size = 144, normalized size = 0.97 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.59, size = 241, normalized size = 1.62 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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