Optimal. Leaf size=112 \[ -\frac {2 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{3/2} (d e-c f)^{3/2}}+\frac {2 (b e-a f)^2}{f^2 \sqrt {e+f x} (d e-c f)}+\frac {2 b^2 \sqrt {e+f x}}{d f^2} \]
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Rubi [A] time = 0.14, antiderivative size = 112, 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 = {87, 63, 208} \begin {gather*} -\frac {2 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{3/2} (d e-c f)^{3/2}}+\frac {2 (b e-a f)^2}{f^2 \sqrt {e+f x} (d e-c f)}+\frac {2 b^2 \sqrt {e+f x}}{d f^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 87
Rule 208
Rubi steps
\begin {align*} \int \frac {(a+b x)^2}{(c+d x) (e+f x)^{3/2}} \, dx &=\int \left (\frac {(-b e+a f)^2}{f (-d e+c f) (e+f x)^{3/2}}+\frac {b^2}{d f \sqrt {e+f x}}+\frac {(-b c+a d)^2}{d (d e-c f) (c+d x) \sqrt {e+f x}}\right ) \, dx\\ &=\frac {2 (b e-a f)^2}{f^2 (d e-c f) \sqrt {e+f x}}+\frac {2 b^2 \sqrt {e+f x}}{d f^2}+\frac {(b c-a d)^2 \int \frac {1}{(c+d x) \sqrt {e+f x}} \, dx}{d (d e-c f)}\\ &=\frac {2 (b e-a f)^2}{f^2 (d e-c f) \sqrt {e+f x}}+\frac {2 b^2 \sqrt {e+f x}}{d f^2}+\frac {\left (2 (b c-a d)^2\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 f (d e-c f)}\\ &=\frac {2 (b e-a f)^2}{f^2 (d e-c f) \sqrt {e+f x}}+\frac {2 b^2 \sqrt {e+f x}}{d f^2}-\frac {2 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{3/2} (d e-c f)^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.11, size = 98, normalized size = 0.88 \begin {gather*} \frac {-2 f^2 (b c-a d)^2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {d (e+f x)}{d e-c f}\right )-2 b (d e-c f) (b (c f+2 d e+d f x)-2 a d f)}{d^2 f^2 \sqrt {e+f x} (c f-d e)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.16, size = 142, normalized size = 1.27 \begin {gather*} \frac {2 \left (a^2 d f^2-2 a b d e f-b^2 c f (e+f x)+b^2 d e^2+b^2 d e (e+f x)\right )}{d f^2 \sqrt {e+f x} (d e-c f)}+\frac {2 (a d-b c)^2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x} \sqrt {c f-d e}}{d e-c f}\right )}{d^{3/2} (c f-d e)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.74, size = 600, normalized size = 5.36 \begin {gather*} \left [-\frac {{\left ({\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{3} x + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} e f^{2}\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 (2 \, b^{2} d^{3} e^{3} - a^{2} c d^{2} f^{3} - {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} e^{2} f + {\left (b^{2} c^{2} d + 2 \, a b c d^{2} + a^{2} d^{3}\right )} e f^{2} + {\left (b^{2} d^{3} e^{2} f - 2 \, b^{2} c d^{2} e f^{2} + b^{2} c^{2} d f^{3}\right )} x\right )} \sqrt {f x + e}}{d^{4} e^{3} f^{2} - 2 \, c d^{3} e^{2} f^{3} + c^{2} d^{2} e f^{4} + {\left (d^{4} e^{2} f^{3} - 2 \, c d^{3} e f^{4} + c^{2} d^{2} f^{5}\right )} x}, \frac {2 \, {\left ({\left ({\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{3} x + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} e f^{2}\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 (2 \, b^{2} d^{3} e^{3} - a^{2} c d^{2} f^{3} - {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} e^{2} f + {\left (b^{2} c^{2} d + 2 \, a b c d^{2} + a^{2} d^{3}\right )} e f^{2} + {\left (b^{2} d^{3} e^{2} f - 2 \, b^{2} c d^{2} e f^{2} + b^{2} c^{2} d f^{3}\right )} x\right )} \sqrt {f x + e}\right )}}{d^{4} e^{3} f^{2} - 2 \, c d^{3} e^{2} f^{3} + c^{2} d^{2} e f^{4} + {\left (d^{4} e^{2} f^{3} - 2 \, c d^{3} e f^{4} + c^{2} d^{2} f^{5}\right )} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.33, size = 129, normalized size = 1.15 \begin {gather*} -\frac {2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {c d f - d^{2} e}}\right )}{{\left (c d f - d^{2} e\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (a^{2} f^{2} - 2 \, a b f e + b^{2} e^{2}\right )}}{{\left (c f^{3} - d f^{2} e\right )} \sqrt {f x + e}} + \frac {2 \, \sqrt {f x + e} b^{2}}{d f^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 249, normalized size = 2.22 \begin {gather*} -\frac {2 a^{2} 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 {4 a 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 b^{2} c^{2} \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}\, d}-\frac {2 a^{2}}{\left (c f -d e \right ) \sqrt {f x +e}}+\frac {4 a b e}{\left (c f -d e \right ) \sqrt {f x +e}\, f}-\frac {2 b^{2} e^{2}}{\left (c f -d e \right ) \sqrt {f x +e}\, f^{2}}+\frac {2 \sqrt {f x +e}\, b^{2}}{d \,f^{2}} \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 = 1.30, size = 162, normalized size = 1.45 \begin {gather*} \frac {2\,b^2\,\sqrt {e+f\,x}}{d\,f^2}+\frac {2\,\mathrm {atan}\left (\frac {2\,\sqrt {e+f\,x}\,\left (d^2\,e-c\,d\,f\right )\,{\left (a\,d-b\,c\right )}^2}{\sqrt {d}\,{\left (c\,f-d\,e\right )}^{3/2}\,\left (2\,a^2\,d^2-4\,a\,b\,c\,d+2\,b^2\,c^2\right )}\right )\,{\left (a\,d-b\,c\right )}^2}{d^{3/2}\,{\left (c\,f-d\,e\right )}^{3/2}}-\frac {2\,\left (d\,a^2\,f^2-2\,d\,a\,b\,e\,f+d\,b^2\,e^2\right )}{d\,f^2\,\sqrt {e+f\,x}\,\left (c\,f-d\,e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 53.45, size = 100, normalized size = 0.89 \begin {gather*} \frac {2 b^{2} \sqrt {e + f x}}{d f^{2}} - \frac {2 \left (a f - b e\right )^{2}}{f^{2} \sqrt {e + f x} \left (c f - d e\right )} - \frac {2 \left (a d - b c\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{d^{2} \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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