Optimal. Leaf size=138 \[ \frac {2 (b c-a d)^2 \sqrt {e+f x}}{d^3}-\frac {2 b (b d e+b c f-2 a d f) (e+f x)^{3/2}}{3 d^2 f^2}+\frac {2 b^2 (e+f x)^{5/2}}{5 d f^2}-\frac {2 (b c-a d)^2 \sqrt {d e-c f} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{7/2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {90, 52, 65, 214}
\begin {gather*} -\frac {2 (b c-a d)^2 \sqrt {d e-c f} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{7/2}}+\frac {2 \sqrt {e+f x} (b c-a d)^2}{d^3}-\frac {2 b (e+f x)^{3/2} (-2 a d f+b c f+b d e)}{3 d^2 f^2}+\frac {2 b^2 (e+f x)^{5/2}}{5 d f^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 90
Rule 214
Rubi steps
\begin {align*} \int \frac {(a+b x)^2 \sqrt {e+f x}}{c+d x} \, dx &=\int \left (-\frac {b (b d e+b c f-2 a d f) \sqrt {e+f x}}{d^2 f}+\frac {(-b c+a d)^2 \sqrt {e+f x}}{d^2 (c+d x)}+\frac {b^2 (e+f x)^{3/2}}{d f}\right ) \, dx\\ &=-\frac {2 b (b d e+b c f-2 a d f) (e+f x)^{3/2}}{3 d^2 f^2}+\frac {2 b^2 (e+f x)^{5/2}}{5 d f^2}+\frac {(b c-a d)^2 \int \frac {\sqrt {e+f x}}{c+d x} \, dx}{d^2}\\ &=\frac {2 (b c-a d)^2 \sqrt {e+f x}}{d^3}-\frac {2 b (b d e+b c f-2 a d f) (e+f x)^{3/2}}{3 d^2 f^2}+\frac {2 b^2 (e+f x)^{5/2}}{5 d f^2}+\frac {\left ((b c-a d)^2 (d e-c f)\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}} \, dx}{d^3}\\ &=\frac {2 (b c-a d)^2 \sqrt {e+f x}}{d^3}-\frac {2 b (b d e+b c f-2 a d f) (e+f x)^{3/2}}{3 d^2 f^2}+\frac {2 b^2 (e+f x)^{5/2}}{5 d f^2}+\frac {\left (2 (b c-a d)^2 (d e-c f)\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 c-a d)^2 \sqrt {e+f x}}{d^3}-\frac {2 b (b d e+b c f-2 a d f) (e+f x)^{3/2}}{3 d^2 f^2}+\frac {2 b^2 (e+f x)^{5/2}}{5 d f^2}-\frac {2 (b c-a d)^2 \sqrt {d e-c f} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 152, normalized size = 1.10 \begin {gather*} \frac {2 \sqrt {e+f x} \left (15 a^2 d^2 f^2+10 a b d f (-3 c f+d (e+f x))+b^2 \left (15 c^2 f^2-5 c d f (e+f x)+d^2 \left (-2 e^2+e f x+3 f^2 x^2\right )\right )\right )}{15 d^3 f^2}-\frac {2 (b c-a d)^2 \sqrt {-d e+c f} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 220, normalized size = 1.59
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (\frac {b^{2} \left (f x +e \right )^{\frac {5}{2}} d^{2}}{5}+\frac {2 a b \,d^{2} f \left (f x +e \right )^{\frac {3}{2}}}{3}-\frac {b^{2} c d f \left (f x +e \right )^{\frac {3}{2}}}{3}-\frac {b^{2} d^{2} e \left (f x +e \right )^{\frac {3}{2}}}{3}+a^{2} d^{2} f^{2} \sqrt {f x +e}-2 a b c d \,f^{2} \sqrt {f x +e}+b^{2} c^{2} f^{2} \sqrt {f x +e}\right )}{d^{3}}-\frac {2 f^{2} \left (a^{2} c \,d^{2} f -a^{2} d^{3} e -2 a b \,c^{2} d f +2 a b c \,d^{2} e +b^{2} c^{3} f -b^{2} c^{2} d e \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{3} \sqrt {\left (c f -d e \right ) d}}}{f^{2}}\) | \(220\) |
default | \(\frac {\frac {2 \left (\frac {b^{2} \left (f x +e \right )^{\frac {5}{2}} d^{2}}{5}+\frac {2 a b \,d^{2} f \left (f x +e \right )^{\frac {3}{2}}}{3}-\frac {b^{2} c d f \left (f x +e \right )^{\frac {3}{2}}}{3}-\frac {b^{2} d^{2} e \left (f x +e \right )^{\frac {3}{2}}}{3}+a^{2} d^{2} f^{2} \sqrt {f x +e}-2 a b c d \,f^{2} \sqrt {f x +e}+b^{2} c^{2} f^{2} \sqrt {f x +e}\right )}{d^{3}}-\frac {2 f^{2} \left (a^{2} c \,d^{2} f -a^{2} d^{3} e -2 a b \,c^{2} d f +2 a b c \,d^{2} e +b^{2} c^{3} f -b^{2} c^{2} d e \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{3} \sqrt {\left (c f -d e \right ) d}}}{f^{2}}\) | \(220\) |
risch | \(\frac {2 \left (3 b^{2} d^{2} f^{2} x^{2}+10 a b \,d^{2} f^{2} x -5 b^{2} c d \,f^{2} x +b^{2} d^{2} e f x +15 a^{2} d^{2} f^{2}-30 a b c d \,f^{2}+10 a b \,d^{2} e f +15 b^{2} c^{2} f^{2}-5 b^{2} c d e f -2 b^{2} d^{2} e^{2}\right ) \sqrt {f x +e}}{15 f^{2} d^{3}}-\frac {2 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a^{2} c f}{d \sqrt {\left (c f -d e \right ) d}}+\frac {2 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a^{2} e}{\sqrt {\left (c f -d e \right ) d}}+\frac {4 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a b \,c^{2} f}{d^{2} \sqrt {\left (c f -d e \right ) d}}-\frac {4 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a b c e}{d \sqrt {\left (c f -d e \right ) d}}-\frac {2 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) b^{2} c^{3} f}{d^{3} \sqrt {\left (c f -d e \right ) d}}+\frac {2 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) b^{2} c^{2} e}{d^{2} \sqrt {\left (c f -d e \right ) d}}\) | \(388\) |
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 = 0.98, size = 411, normalized size = 2.98 \begin {gather*} \left [\frac {15 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{2} \sqrt {-\frac {c f - d e}{d}} \log \left (\frac {d f x - c f - 2 \, \sqrt {f x + e} d \sqrt {-\frac {c f - d e}{d}} + 2 \, d e}{d x + c}\right ) + 2 \, {\left (3 \, b^{2} d^{2} f^{2} x^{2} - 2 \, b^{2} d^{2} e^{2} - 5 \, {\left (b^{2} c d - 2 \, a b d^{2}\right )} f^{2} x + 15 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{2} + {\left (b^{2} d^{2} f x - 5 \, {\left (b^{2} c d - 2 \, a b d^{2}\right )} f\right )} e\right )} \sqrt {f x + e}}{15 \, d^{3} f^{2}}, \frac {2 \, {\left (15 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{2} \sqrt {\frac {c f - d e}{d}} \arctan \left (-\frac {\sqrt {f x + e} d \sqrt {\frac {c f - d e}{d}}}{c f - d e}\right ) + {\left (3 \, b^{2} d^{2} f^{2} x^{2} - 2 \, b^{2} d^{2} e^{2} - 5 \, {\left (b^{2} c d - 2 \, a b d^{2}\right )} f^{2} x + 15 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{2} + {\left (b^{2} d^{2} f x - 5 \, {\left (b^{2} c d - 2 \, a b d^{2}\right )} f\right )} e\right )} \sqrt {f x + e}\right )}}{15 \, d^{3} f^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 7.18, size = 155, normalized size = 1.12 \begin {gather*} \frac {2 \left (\frac {b^{2} \left (e + f x\right )^{\frac {5}{2}}}{5 d f} + \frac {\left (e + f x\right )^{\frac {3}{2}} \cdot \left (2 a b d f - b^{2} c f - b^{2} d e\right )}{3 d^{2} f} + \frac {\sqrt {e + f x} \left (a^{2} d^{2} f - 2 a b c d f + b^{2} c^{2} f\right )}{d^{3}} - \frac {f \left (a d - b c\right )^{2} \left (c f - d e\right ) \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{d^{4} \sqrt {\frac {c f - d e}{d}}}\right )}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.60, size = 250, normalized size = 1.81 \begin {gather*} -\frac {2 \, {\left (b^{2} c^{3} f - 2 \, a b c^{2} d f + a^{2} c d^{2} f - b^{2} c^{2} d e + 2 \, a b c d^{2} e - a^{2} d^{3} e\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^{2} d^{4} f^{8} - 5 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{2} c d^{3} f^{9} + 10 \, {\left (f x + e\right )}^{\frac {3}{2}} a b d^{4} f^{9} + 15 \, \sqrt {f x + e} b^{2} c^{2} d^{2} f^{10} - 30 \, \sqrt {f x + e} a b c d^{3} f^{10} + 15 \, \sqrt {f x + e} a^{2} d^{4} f^{10} - 5 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{2} d^{4} f^{8} e\right )}}{15 \, d^{5} f^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 281, normalized size = 2.04 \begin {gather*} \sqrt {e+f\,x}\,\left (\frac {2\,{\left (a\,f-b\,e\right )}^2}{d\,f^2}+\frac {\left (\frac {4\,b^2\,e-4\,a\,b\,f}{d\,f^2}+\frac {2\,b^2\,\left (c\,f^3-d\,e\,f^2\right )}{d^2\,f^4}\right )\,\left (c\,f^3-d\,e\,f^2\right )}{d\,f^2}\right )-{\left (e+f\,x\right )}^{3/2}\,\left (\frac {4\,b^2\,e-4\,a\,b\,f}{3\,d\,f^2}+\frac {2\,b^2\,\left (c\,f^3-d\,e\,f^2\right )}{3\,d^2\,f^4}\right )+\frac {2\,b^2\,{\left (e+f\,x\right )}^{5/2}}{5\,d\,f^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}\,{\left (a\,d-b\,c\right )}^2\,\sqrt {d\,e-c\,f}\,1{}\mathrm {i}}{-f\,a^2\,c\,d^2+e\,a^2\,d^3+2\,f\,a\,b\,c^2\,d-2\,e\,a\,b\,c\,d^2-f\,b^2\,c^3+e\,b^2\,c^2\,d}\right )\,{\left (a\,d-b\,c\right )}^2\,\sqrt {d\,e-c\,f}\,2{}\mathrm {i}}{d^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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