Optimal. Leaf size=184 \[ \frac {2 b \sqrt {e+f x} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{d^3 f^3}-\frac {2 b^2 (e+f x)^{3/2} (-3 a d f+b c f+2 b d e)}{3 d^2 f^3}+\frac {2 (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{7/2} \sqrt {d e-c f}}+\frac {2 b^3 (e+f x)^{5/2}}{5 d f^3} \]
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Rubi [A] time = 0.17, 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 = {88, 63, 208} \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 63
Rule 88
Rule 208
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 ) \operatorname {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 = 0.20, size = 184, normalized size = 1.00 \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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IntegrateAlgebraic [A] time = 0.25, size = 204, normalized size = 1.11 \begin {gather*} \frac {2 b \sqrt {e+f x} \left (45 a^2 d^2 f^2-45 a b c d f^2+15 a b d^2 f (e+f x)-45 a b d^2 e f+15 b^2 c^2 f^2-5 b^2 c d f (e+f x)+15 b^2 c d e f+15 b^2 d^2 e^2+3 b^2 d^2 (e+f x)^2-10 b^2 d^2 e (e+f x)\right )}{15 d^3 f^3}-\frac {2 (a d-b c)^3 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x} \sqrt {c f-d e}}{d e-c f}\right )}{d^{7/2} \sqrt {c f-d e}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.46, size = 653, normalized size = 3.55 \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 {d^{2} e - c d f} f^{3} \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^{3} + 2 \, {\left (b^{3} c d^{3} - 15 \, a b^{2} d^{4}\right )} e^{2} f + 5 \, {\left (b^{3} c^{2} d^{2} - 3 \, a b^{2} c d^{3} + 9 \, a^{2} b d^{4}\right )} e f^{2} - 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} + 3 \, {\left (b^{3} d^{4} e f^{2} - b^{3} c d^{3} f^{3}\right )} x^{2} - {\left (4 \, b^{3} d^{4} e^{2} f + {\left (b^{3} c d^{3} - 15 \, a b^{2} d^{4}\right )} e f^{2} - 5 \, {\left (b^{3} c^{2} d^{2} - 3 \, a b^{2} c d^{3}\right )} f^{3}\right )} x\right )} \sqrt {f x + e}}{15 \, {\left (d^{5} e f^{3} - c d^{4} f^{4}\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 {-d^{2} e + c d f} f^{3} \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^{3} + 2 \, {\left (b^{3} c d^{3} - 15 \, a b^{2} d^{4}\right )} e^{2} f + 5 \, {\left (b^{3} c^{2} d^{2} - 3 \, a b^{2} c d^{3} + 9 \, a^{2} b d^{4}\right )} e f^{2} - 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} + 3 \, {\left (b^{3} d^{4} e f^{2} - b^{3} c d^{3} f^{3}\right )} x^{2} - {\left (4 \, b^{3} d^{4} e^{2} f + {\left (b^{3} c d^{3} - 15 \, a b^{2} d^{4}\right )} e f^{2} - 5 \, {\left (b^{3} c^{2} d^{2} - 3 \, a b^{2} c d^{3}\right )} f^{3}\right )} x\right )} \sqrt {f x + e}\right )}}{15 \, {\left (d^{5} e f^{3} - c d^{4} f^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.20, 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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maple [B] time = 0.01, size = 372, normalized size = 2.02 \begin {gather*} \frac {2 a^{3} \arctan \left (\frac {\sqrt {f x +e}\, d}{\sqrt {\left (c f -d e \right ) d}}\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 )}{\sqrt {\left (c f -d e \right ) d}\, 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 )}{\sqrt {\left (c f -d e \right ) d}\, d^{2}}-\frac {2 b^{3} c^{3} \arctan \left (\frac {\sqrt {f x +e}\, d}{\sqrt {\left (c f -d e \right ) d}}\right )}{\sqrt {\left (c f -d e \right ) d}\, d^{3}}+\frac {6 \sqrt {f x +e}\, a^{2} b}{d f}-\frac {6 \sqrt {f x +e}\, a \,b^{2} c}{d^{2} f}-\frac {6 \sqrt {f x +e}\, a \,b^{2} e}{d \,f^{2}}+\frac {2 \sqrt {f x +e}\, b^{3} c^{2}}{d^{3} f}+\frac {2 \sqrt {f x +e}\, b^{3} c e}{d^{2} f^{2}}+\frac {2 \sqrt {f x +e}\, b^{3} e^{2}}{d \,f^{3}}+\frac {2 \left (f x +e \right )^{\frac {3}{2}} a \,b^{2}}{d \,f^{2}}-\frac {2 \left (f x +e \right )^{\frac {3}{2}} b^{3} c}{3 d^{2} f^{2}}-\frac {4 \left (f x +e \right )^{\frac {3}{2}} b^{3} e}{3 d \,f^{3}}+\frac {2 \left (f x +e \right )^{\frac {5}{2}} b^{3}}{5 d \,f^{3}} \end {gather*}
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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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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sympy [A] time = 67.01, 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}} \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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