Optimal. Leaf size=88 \[ \frac {2 (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{\sqrt {d} (d e-c f)^{3/2}}-\frac {2 (b e-a f)}{f \sqrt {e+f x} (d e-c f)} \]
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Rubi [A] time = 0.08, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {78, 63, 208} \begin {gather*} \frac {2 (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{\sqrt {d} (d e-c f)^{3/2}}-\frac {2 (b e-a f)}{f \sqrt {e+f x} (d e-c f)} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 78
Rule 208
Rubi steps
\begin {align*} \int \frac {a+b x}{(c+d x) (e+f x)^{3/2}} \, dx &=-\frac {2 (b e-a f)}{f (d e-c f) \sqrt {e+f x}}-\frac {(b c-a d) \int \frac {1}{(c+d x) \sqrt {e+f x}} \, dx}{d e-c f}\\ &=-\frac {2 (b e-a f)}{f (d e-c f) \sqrt {e+f x}}-\frac {(2 (b c-a d)) \operatorname {Subst}\left (\int \frac {1}{c-\frac {d e}{f}+\frac {d x^2}{f}} \, dx,x,\sqrt {e+f x}\right )}{f (d e-c f)}\\ &=-\frac {2 (b e-a f)}{f (d e-c f) \sqrt {e+f x}}+\frac {2 (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{\sqrt {d} (d e-c f)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 88, normalized size = 1.00 \begin {gather*} \frac {2 b e-2 a f}{f \sqrt {e+f x} (c f-d e)}+\frac {2 (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{\sqrt {d} (d e-c f)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 98, normalized size = 1.11 \begin {gather*} \frac {2 (a d-b c) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x} \sqrt {c f-d e}}{d e-c f}\right )}{\sqrt {d} (c f-d e)^{3/2}}-\frac {2 (a f-b e)}{f \sqrt {e+f x} (c f-d e)} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.42, size = 363, normalized size = 4.12 \begin {gather*} \left [\frac {{\left ({\left (b c - a d\right )} f^{2} x + {\left (b c - a d\right )} e f\right )} \sqrt {d^{2} e - c d f} \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 (b d^{2} e^{2} + a c d f^{2} - {\left (b c d + a d^{2}\right )} e f\right )} \sqrt {f x + e}}{d^{3} e^{3} f - 2 \, c d^{2} e^{2} f^{2} + c^{2} d e f^{3} + {\left (d^{3} e^{2} f^{2} - 2 \, c d^{2} e f^{3} + c^{2} d f^{4}\right )} x}, -\frac {2 \, {\left ({\left ({\left (b c - a d\right )} f^{2} x + {\left (b c - a d\right )} e f\right )} \sqrt {-d^{2} e + c d f} \arctan \left (\frac {\sqrt {-d^{2} e + c d f} \sqrt {f x + e}}{d f x + d e}\right ) + {\left (b d^{2} e^{2} + a c d f^{2} - {\left (b c d + a d^{2}\right )} e f\right )} \sqrt {f x + e}\right )}}{d^{3} e^{3} f - 2 \, c d^{2} e^{2} f^{2} + c^{2} d e f^{3} + {\left (d^{3} e^{2} f^{2} - 2 \, c d^{2} e f^{3} + c^{2} d f^{4}\right )} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.20, size = 94, normalized size = 1.07 \begin {gather*} \frac {2 \, {\left (b c - a d\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {c d f - d^{2} e}}\right )}{\sqrt {c d f - d^{2} e} {\left (c f - d e\right )}} - \frac {2 \, {\left (a f - b e\right )}}{{\left (c f^{2} - d f e\right )} \sqrt {f x + e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 142, normalized size = 1.61 \begin {gather*} -\frac {2 a d \arctan \left (\frac {\sqrt {f x +e}\, d}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right ) \sqrt {\left (c f -d e \right ) d}}+\frac {2 b c \arctan \left (\frac {\sqrt {f x +e}\, d}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right ) \sqrt {\left (c f -d e \right ) d}}-\frac {2 a}{\left (c f -d e \right ) \sqrt {f x +e}}+\frac {2 b e}{\left (c f -d e \right ) \sqrt {f x +e}\, f} \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 = 1.27, size = 96, normalized size = 1.09 \begin {gather*} -\frac {2\,\left (a\,f-b\,e\right )}{f\,\sqrt {e+f\,x}\,\left (c\,f-d\,e\right )}-\frac {2\,\mathrm {atan}\left (\frac {2\,\sqrt {d}\,\sqrt {e+f\,x}\,\left (a\,d-b\,c\right )}{\left (2\,a\,d-2\,b\,c\right )\,\sqrt {c\,f-d\,e}}\right )\,\left (a\,d-b\,c\right )}{\sqrt {d}\,{\left (c\,f-d\,e\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 30.55, size = 78, normalized size = 0.89 \begin {gather*} - \frac {2 \left (a f - b e\right )}{f \sqrt {e + f x} \left (c f - d e\right )} - \frac {2 \left (a d - b c\right ) \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{d \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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