Optimal. Leaf size=98 \[ -\frac {2 (b c-a d) \sqrt {e+f x}}{d^2}+\frac {2 b (e+f x)^{3/2}}{3 d f}+\frac {2 (b c-a d) \sqrt {d e-c f} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{5/2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {81, 52, 65, 214}
\begin {gather*} \frac {2 (b c-a d) \sqrt {d e-c f} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{5/2}}-\frac {2 \sqrt {e+f x} (b c-a d)}{d^2}+\frac {2 b (e+f x)^{3/2}}{3 d f} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 81
Rule 214
Rubi steps
\begin {align*} \int \frac {(a+b x) \sqrt {e+f x}}{c+d x} \, dx &=\frac {2 b (e+f x)^{3/2}}{3 d f}+\frac {\left (2 \left (-\frac {3}{2} b c f+\frac {3 a d f}{2}\right )\right ) \int \frac {\sqrt {e+f x}}{c+d x} \, dx}{3 d f}\\ &=-\frac {2 (b c-a d) \sqrt {e+f x}}{d^2}+\frac {2 b (e+f x)^{3/2}}{3 d f}-\frac {((b c-a d) (d e-c f)) \int \frac {1}{(c+d x) \sqrt {e+f x}} \, dx}{d^2}\\ &=-\frac {2 (b c-a d) \sqrt {e+f x}}{d^2}+\frac {2 b (e+f x)^{3/2}}{3 d f}-\frac {(2 (b c-a d) (d e-c f)) \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}\\ &=-\frac {2 (b c-a d) \sqrt {e+f x}}{d^2}+\frac {2 b (e+f x)^{3/2}}{3 d f}+\frac {2 (b c-a d) \sqrt {d e-c f} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 94, normalized size = 0.96 \begin {gather*} \frac {2 \sqrt {e+f x} (-3 b c f+3 a d f+b d (e+f x))}{3 d^2 f}+\frac {2 (b c-a d) \sqrt {-d e+c f} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 111, normalized size = 1.13
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (\frac {b \left (f x +e \right )^{\frac {3}{2}} d}{3}+a d f \sqrt {f x +e}-b c f \sqrt {f x +e}\right )}{d^{2}}-\frac {2 f \left (a c d f -a \,d^{2} e -b \,c^{2} f +b c d e \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{2} \sqrt {\left (c f -d e \right ) d}}}{f}\) | \(111\) |
default | \(\frac {\frac {2 \left (\frac {b \left (f x +e \right )^{\frac {3}{2}} d}{3}+a d f \sqrt {f x +e}-b c f \sqrt {f x +e}\right )}{d^{2}}-\frac {2 f \left (a c d f -a \,d^{2} e -b \,c^{2} f +b c d e \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{2} \sqrt {\left (c f -d e \right ) d}}}{f}\) | \(111\) |
risch | \(\frac {2 \left (b d f x +3 a d f -3 b c f +b d e \right ) \sqrt {f x +e}}{3 f \,d^{2}}-\frac {2 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a c f}{d \sqrt {\left (c f -d e \right ) d}}+\frac {2 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a e}{\sqrt {\left (c f -d e \right ) d}}+\frac {2 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) b \,c^{2} f}{d^{2} \sqrt {\left (c f -d e \right ) d}}-\frac {2 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) b c e}{d \sqrt {\left (c f -d e \right ) d}}\) | \(203\) |
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 = 1.12, size = 224, normalized size = 2.29 \begin {gather*} \left [-\frac {3 \, {\left (b c - a d\right )} f \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 (b d f x + b d e - 3 \, {\left (b c - a d\right )} f\right )} \sqrt {f x + e}}{3 \, d^{2} f}, -\frac {2 \, {\left (3 \, {\left (b c - a d\right )} f \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 (b d f x + b d e - 3 \, {\left (b c - a d\right )} f\right )} \sqrt {f x + e}\right )}}{3 \, d^{2} f}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 6.09, size = 94, normalized size = 0.96 \begin {gather*} \frac {2 \left (\frac {b \left (e + f x\right )^{\frac {3}{2}}}{3 d} + \frac {\sqrt {e + f x} \left (a d f - b c f\right )}{d^{2}} - \frac {f \left (a d - b c\right ) \left (c f - d e\right ) \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}}}\right )}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.92, size = 130, normalized size = 1.33 \begin {gather*} \frac {2 \, {\left (b c^{2} f - a c d f - b c d e + a d^{2} 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^{2}} + \frac {2 \, {\left ({\left (f x + e\right )}^{\frac {3}{2}} b d^{2} f^{2} - 3 \, \sqrt {f x + e} b c d f^{3} + 3 \, \sqrt {f x + e} a d^{2} f^{3}\right )}}{3 \, d^{3} f^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.13, size = 107, normalized size = 1.09 \begin {gather*} \sqrt {e+f\,x}\,\left (\frac {2\,a\,f-2\,b\,e}{d\,f}-\frac {2\,b\,\left (c\,f^2-d\,e\,f\right )}{d^2\,f^2}\right )+\frac {2\,b\,{\left (e+f\,x\right )}^{3/2}}{3\,d\,f}-\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}}{\sqrt {d\,e-c\,f}}\right )\,\left (a\,d-b\,c\right )\,\sqrt {d\,e-c\,f}}{d^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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