Optimal. Leaf size=184 \[ \frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) \sqrt {e+f x}}{d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{3/2}}{3 d^2 f^3}+\frac {2 b^3 (e+f x)^{5/2}}{5 d f^3}+\frac {2 (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{7/2} \sqrt {d e-c f}} \]
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Rubi [A]
time = 0.09, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {90, 65, 214}
\begin {gather*} \frac {2 b \sqrt {e+f x} \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 )}{d^3 f^3}-\frac {2 b^2 (e+f x)^{3/2} (-3 a d f+b c f+2 b d e)}{3 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^{7/2} \sqrt {d e-c f}}+\frac {2 b^3 (e+f x)^{5/2}}{5 d f^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 90
Rule 214
Rubi steps
\begin {align*} \int \frac {(a+b x)^3}{(c+d x) \sqrt {e+f x}} \, dx &=\int \left (\frac {b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right )}{d^3 f^2 \sqrt {e+f x}}+\frac {(-b c+a d)^3}{d^3 (c+d x) \sqrt {e+f x}}-\frac {b^2 (2 b d e+b c f-3 a d f) \sqrt {e+f x}}{d^2 f^2}+\frac {b^3 (e+f x)^{3/2}}{d f^2}\right ) \, dx\\ &=\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) \sqrt {e+f x}}{d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{3/2}}{3 d^2 f^3}+\frac {2 b^3 (e+f x)^{5/2}}{5 d f^3}-\frac {(b c-a d)^3 \int \frac {1}{(c+d x) \sqrt {e+f x}} \, dx}{d^3}\\ &=\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) \sqrt {e+f x}}{d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{3/2}}{3 d^2 f^3}+\frac {2 b^3 (e+f x)^{5/2}}{5 d f^3}-\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^3 f}\\ &=\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) \sqrt {e+f x}}{d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{3/2}}{3 d^2 f^3}+\frac {2 b^3 (e+f x)^{5/2}}{5 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^{7/2} \sqrt {d e-c f}}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 157, normalized size = 0.85 \begin {gather*} \frac {2 b \sqrt {e+f x} \left (45 a^2 d^2 f^2+15 a b d f (-2 d e-3 c f+d f x)+b^2 \left (15 c^2 f^2-5 c d f (-2 e+f x)+d^2 \left (8 e^2-4 e f x+3 f^2 x^2\right )\right )\right )}{15 d^3 f^3}+\frac {2 (-b c+a d)^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{7/2} \sqrt {-d e+c f}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 249, normalized size = 1.35
method | result | size |
derivativedivides | \(\frac {\frac {2 b \left (\frac {b^{2} \left (f x +e \right )^{\frac {5}{2}} d^{2}}{5}+a b \,d^{2} f \left (f x +e \right )^{\frac {3}{2}}-\frac {b^{2} c d f \left (f x +e \right )^{\frac {3}{2}}}{3}-\frac {2 b^{2} d^{2} e \left (f x +e \right )^{\frac {3}{2}}}{3}+3 a^{2} d^{2} f^{2} \sqrt {f x +e}-3 a b c d \,f^{2} \sqrt {f x +e}-3 f \,d^{2} e a b \sqrt {f x +e}+b^{2} c^{2} f^{2} \sqrt {f x +e}+b^{2} c d e f \sqrt {f x +e}+b^{2} d^{2} e^{2} \sqrt {f x +e}\right )}{d^{3}}+\frac {2 f^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{3} \sqrt {\left (c f -d e \right ) d}}}{f^{3}}\) | \(249\) |
default | \(\frac {\frac {2 b \left (\frac {b^{2} \left (f x +e \right )^{\frac {5}{2}} d^{2}}{5}+a b \,d^{2} f \left (f x +e \right )^{\frac {3}{2}}-\frac {b^{2} c d f \left (f x +e \right )^{\frac {3}{2}}}{3}-\frac {2 b^{2} d^{2} e \left (f x +e \right )^{\frac {3}{2}}}{3}+3 a^{2} d^{2} f^{2} \sqrt {f x +e}-3 a b c d \,f^{2} \sqrt {f x +e}-3 f \,d^{2} e a b \sqrt {f x +e}+b^{2} c^{2} f^{2} \sqrt {f x +e}+b^{2} c d e f \sqrt {f x +e}+b^{2} d^{2} e^{2} \sqrt {f x +e}\right )}{d^{3}}+\frac {2 f^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{3} \sqrt {\left (c f -d e \right ) d}}}{f^{3}}\) | \(249\) |
risch | \(\frac {2 b \left (3 b^{2} d^{2} f^{2} x^{2}+15 a b \,d^{2} f^{2} x -5 b^{2} c d \,f^{2} x -4 b^{2} d^{2} e f x +45 a^{2} d^{2} f^{2}-45 a b c d \,f^{2}-30 a b \,d^{2} e f +15 b^{2} c^{2} f^{2}+10 b^{2} c d e f +8 b^{2} d^{2} e^{2}\right ) \sqrt {f x +e}}{15 f^{3} d^{3}}+\frac {2 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a^{3}}{\sqrt {\left (c f -d e \right ) d}}-\frac {6 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a^{2} b c}{d \sqrt {\left (c f -d e \right ) d}}+\frac {6 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a \,b^{2} c^{2}}{d^{2} \sqrt {\left (c f -d e \right ) d}}-\frac {2 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) b^{3} c^{3}}{d^{3} \sqrt {\left (c f -d e \right ) d}}\) | \(300\) |
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 [A]
time = 1.22, size = 666, normalized size = 3.62 \begin {gather*} \left [\frac {15 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-c d f + d^{2} e} f^{3} \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 (3 \, b^{3} c d^{3} f^{3} x^{2} - 8 \, b^{3} d^{4} e^{3} - 5 \, {\left (b^{3} c^{2} d^{2} - 3 \, a b^{2} c d^{3}\right )} f^{3} x + 15 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3}\right )} f^{3} + 2 \, {\left (2 \, b^{3} d^{4} f x - {\left (b^{3} c d^{3} - 15 \, a b^{2} d^{4}\right )} f\right )} e^{2} - {\left (3 \, b^{3} d^{4} f^{2} x^{2} - {\left (b^{3} c d^{3} - 15 \, a b^{2} d^{4}\right )} f^{2} x + 5 \, {\left (b^{3} c^{2} d^{2} - 3 \, a b^{2} c d^{3} + 9 \, a^{2} b d^{4}\right )} f^{2}\right )} e\right )} \sqrt {f x + e}}{15 \, {\left (c d^{4} f^{4} - d^{5} f^{3} e\right )}}, \frac {2 \, {\left (15 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {c d f - d^{2} e} f^{3} \arctan \left (\frac {\sqrt {c d f - d^{2} e} \sqrt {f x + e}}{d f x + d e}\right ) + {\left (3 \, b^{3} c d^{3} f^{3} x^{2} - 8 \, b^{3} d^{4} e^{3} - 5 \, {\left (b^{3} c^{2} d^{2} - 3 \, a b^{2} c d^{3}\right )} f^{3} x + 15 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3}\right )} f^{3} + 2 \, {\left (2 \, b^{3} d^{4} f x - {\left (b^{3} c d^{3} - 15 \, a b^{2} d^{4}\right )} f\right )} e^{2} - {\left (3 \, b^{3} d^{4} f^{2} x^{2} - {\left (b^{3} c d^{3} - 15 \, a b^{2} d^{4}\right )} f^{2} x + 5 \, {\left (b^{3} c^{2} d^{2} - 3 \, a b^{2} c d^{3} + 9 \, a^{2} b d^{4}\right )} f^{2}\right )} e\right )} \sqrt {f x + e}\right )}}{15 \, {\left (c d^{4} f^{4} - d^{5} f^{3} e\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 31.76, size = 201, normalized size = 1.09 \begin {gather*} \frac {2 b^{3} \left (e + f x\right )^{\frac {5}{2}}}{5 d f^{3}} + \frac {2 b^{2} \left (e + f x\right )^{\frac {3}{2}} \cdot \left (3 a d f - b c f - 2 b d e\right )}{3 d^{2} f^{3}} + \frac {2 b \sqrt {e + f x} \left (3 a^{2} d^{2} f^{2} - 3 a b c d f^{2} - 3 a b d^{2} e f + b^{2} c^{2} f^{2} + b^{2} c d e f + b^{2} d^{2} e^{2}\right )}{d^{3} f^{3}} - \frac {2 \left (a d - b c\right )^{3} \operatorname {atan}{\left (\frac {1}{\sqrt {\frac {d}{c f - d e}} \sqrt {e + f x}} \right )}}{d^{3} \sqrt {\frac {d}{c f - d e}} \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 = 0.63, size = 298, 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 )}{\sqrt {c d f - d^{2} e} d^{3}} + \frac {2 \, {\left (3 \, {\left (f x + e\right )}^{\frac {5}{2}} b^{3} d^{4} f^{12} - 5 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} c d^{3} f^{13} + 15 \, {\left (f x + e\right )}^{\frac {3}{2}} a b^{2} d^{4} f^{13} + 15 \, \sqrt {f x + e} b^{3} c^{2} d^{2} f^{14} - 45 \, \sqrt {f x + e} a b^{2} c d^{3} f^{14} + 45 \, \sqrt {f x + e} a^{2} b d^{4} f^{14} - 10 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} d^{4} f^{12} e + 15 \, \sqrt {f x + e} b^{3} c d^{3} f^{13} e - 45 \, \sqrt {f x + e} a b^{2} d^{4} f^{13} e + 15 \, \sqrt {f x + e} b^{3} d^{4} f^{12} e^{2}\right )}}{15 \, d^{5} f^{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 264, normalized size = 1.43 \begin {gather*} \sqrt {e+f\,x}\,\left (\frac {\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 )\,\left (c\,f^4-d\,e\,f^3\right )}{d\,f^3}+\frac {6\,b\,{\left (a\,f-b\,e\right )}^2}{d\,f^3}\right )-{\left (e+f\,x\right )}^{3/2}\,\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{3\,d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{3\,d^2\,f^6}\right )+\frac {2\,b^3\,{\left (e+f\,x\right )}^{5/2}}{5\,d\,f^3}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}\,{\left (a\,d-b\,c\right )}^3}{\sqrt {c\,f-d\,e}\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{d^{7/2}\,\sqrt {c\,f-d\,e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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