Optimal. Leaf size=140 \[ -\frac {2 (b e-a f) (a d f-2 b c f+b d e)}{f^2 \sqrt {e+f x} (d e-c f)^2}+\frac {2 (b e-a f)^2}{3 f^2 (e+f x)^{3/2} (d e-c f)}-\frac {2 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{\sqrt {d} (d e-c f)^{5/2}} \]
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Rubi [A] time = 0.16, antiderivative size = 140, 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 e-a f) (a d f-2 b c f+b d e)}{f^2 \sqrt {e+f x} (d e-c f)^2}+\frac {2 (b e-a f)^2}{3 f^2 (e+f x)^{3/2} (d e-c f)}-\frac {2 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{\sqrt {d} (d e-c f)^{5/2}} \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)^{5/2}} \, dx &=\int \left (\frac {(-b e+a f)^2}{f (-d e+c f) (e+f x)^{5/2}}+\frac {(-b e+a f) (-b d e+2 b c f-a d f)}{f (-d e+c f)^2 (e+f x)^{3/2}}+\frac {(b c-a d)^2}{(d e-c f)^2 (c+d x) \sqrt {e+f x}}\right ) \, dx\\ &=\frac {2 (b e-a f)^2}{3 f^2 (d e-c f) (e+f x)^{3/2}}-\frac {2 (b e-a f) (b d e-2 b c f+a d f)}{f^2 (d e-c f)^2 \sqrt {e+f x}}+\frac {(b c-a d)^2 \int \frac {1}{(c+d x) \sqrt {e+f x}} \, dx}{(d e-c f)^2}\\ &=\frac {2 (b e-a f)^2}{3 f^2 (d e-c f) (e+f x)^{3/2}}-\frac {2 (b e-a f) (b d e-2 b c f+a d f)}{f^2 (d e-c f)^2 \sqrt {e+f x}}+\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 )}{f (d e-c f)^2}\\ &=\frac {2 (b e-a f)^2}{3 f^2 (d e-c f) (e+f x)^{3/2}}-\frac {2 (b e-a f) (b d e-2 b c f+a d f)}{f^2 (d e-c f)^2 \sqrt {e+f x}}-\frac {2 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{\sqrt {d} (d e-c f)^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.11, size = 103, normalized size = 0.74 \begin {gather*} \frac {2 b (d e-c f) (2 a d f+b (-c f+2 d e+3 d f x))-2 f^2 (b c-a d)^2 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {d (e+f x)}{d e-c f}\right )}{3 d^2 f^2 (e+f x)^{3/2} (c f-d e)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.22, size = 157, normalized size = 1.12 \begin {gather*} -\frac {2 (a f-b e) \left (a c f^2-3 a d f (e+f x)-a d e f+6 b c f (e+f x)-b c e f+b d e^2-3 b d e (e+f x)\right )}{3 f^2 (e+f x)^{3/2} (c f-d e)^2}-\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 )}{\sqrt {d} (c f-d e)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.83, size = 975, normalized size = 6.96 \begin {gather*} \left [\frac {3 \, {\left ({\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{4} x^{2} + 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} e f^{3} x + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} e^{2} 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^{4} - a^{2} c^{2} d f^{4} - {\left (7 \, b^{2} c d^{2} - 2 \, a b d^{3}\right )} e^{3} f + {\left (5 \, b^{2} c^{2} d + 2 \, a b c d^{2} - 4 \, a^{2} d^{3}\right )} e^{2} f^{2} - {\left (4 \, a b c^{2} d - 5 \, a^{2} c d^{2}\right )} e f^{3} + 3 \, {\left (b^{2} d^{3} e^{3} f - 3 \, b^{2} c d^{2} e^{2} f^{2} + {\left (2 \, b^{2} c^{2} d + 2 \, a b c d^{2} - a^{2} d^{3}\right )} e f^{3} - {\left (2 \, a b c^{2} d - a^{2} c d^{2}\right )} f^{4}\right )} x\right )} \sqrt {f x + e}}{3 \, {\left (d^{4} e^{5} f^{2} - 3 \, c d^{3} e^{4} f^{3} + 3 \, c^{2} d^{2} e^{3} f^{4} - c^{3} d e^{2} f^{5} + {\left (d^{4} e^{3} f^{4} - 3 \, c d^{3} e^{2} f^{5} + 3 \, c^{2} d^{2} e f^{6} - c^{3} d f^{7}\right )} x^{2} + 2 \, {\left (d^{4} e^{4} f^{3} - 3 \, c d^{3} e^{3} f^{4} + 3 \, c^{2} d^{2} e^{2} f^{5} - c^{3} d e f^{6}\right )} x\right )}}, \frac {2 \, {\left (3 \, {\left ({\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{4} x^{2} + 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} e f^{3} x + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} e^{2} 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^{4} - a^{2} c^{2} d f^{4} - {\left (7 \, b^{2} c d^{2} - 2 \, a b d^{3}\right )} e^{3} f + {\left (5 \, b^{2} c^{2} d + 2 \, a b c d^{2} - 4 \, a^{2} d^{3}\right )} e^{2} f^{2} - {\left (4 \, a b c^{2} d - 5 \, a^{2} c d^{2}\right )} e f^{3} + 3 \, {\left (b^{2} d^{3} e^{3} f - 3 \, b^{2} c d^{2} e^{2} f^{2} + {\left (2 \, b^{2} c^{2} d + 2 \, a b c d^{2} - a^{2} d^{3}\right )} e f^{3} - {\left (2 \, a b c^{2} d - a^{2} c d^{2}\right )} f^{4}\right )} x\right )} \sqrt {f x + e}\right )}}{3 \, {\left (d^{4} e^{5} f^{2} - 3 \, c d^{3} e^{4} f^{3} + 3 \, c^{2} d^{2} e^{3} f^{4} - c^{3} d e^{2} f^{5} + {\left (d^{4} e^{3} f^{4} - 3 \, c d^{3} e^{2} f^{5} + 3 \, c^{2} d^{2} e f^{6} - c^{3} d f^{7}\right )} x^{2} + 2 \, {\left (d^{4} e^{4} f^{3} - 3 \, c d^{3} e^{3} f^{4} + 3 \, c^{2} d^{2} e^{2} f^{5} - c^{3} d e f^{6}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.32, size = 236, normalized size = 1.69 \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^{2} f^{2} - 2 \, c d f e + d^{2} e^{2}\right )} \sqrt {c d f - d^{2} e}} - \frac {2 \, {\left (6 \, {\left (f x + e\right )} a b c f^{2} - 3 \, {\left (f x + e\right )} a^{2} d f^{2} + a^{2} c f^{3} - 6 \, {\left (f x + e\right )} b^{2} c f e - 2 \, a b c f^{2} e - a^{2} d f^{2} e + 3 \, {\left (f x + e\right )} b^{2} d e^{2} + b^{2} c f e^{2} + 2 \, a b d f e^{2} - b^{2} d e^{3}\right )}}{3 \, {\left (c^{2} f^{4} - 2 \, c d f^{3} e + d^{2} f^{2} e^{2}\right )} {\left (f x + e\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 332, normalized size = 2.37 \begin {gather*} \frac {2 a^{2} d^{2} \arctan \left (\frac {\sqrt {f x +e}\, d}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right )^{2} \sqrt {\left (c f -d e \right ) d}}-\frac {4 a b c d \arctan \left (\frac {\sqrt {f x +e}\, d}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right )^{2} \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 )^{2} \sqrt {\left (c f -d e \right ) d}}+\frac {2 a^{2} d}{\left (c f -d e \right )^{2} \sqrt {f x +e}}-\frac {4 a b c}{\left (c f -d e \right )^{2} \sqrt {f x +e}}+\frac {4 b^{2} c e}{\left (c f -d e \right )^{2} \sqrt {f x +e}\, f}-\frac {2 b^{2} d \,e^{2}}{\left (c f -d e \right )^{2} \sqrt {f x +e}\, f^{2}}-\frac {2 a^{2}}{3 \left (c f -d e \right ) \left (f x +e \right )^{\frac {3}{2}}}+\frac {4 a b e}{3 \left (c f -d e \right ) \left (f x +e \right )^{\frac {3}{2}} f}-\frac {2 b^{2} e^{2}}{3 \left (c f -d e \right ) \left (f x +e \right )^{\frac {3}{2}} 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 = 1.36, size = 203, normalized size = 1.45 \begin {gather*} \frac {2\,\mathrm {atan}\left (\frac {2\,\sqrt {d}\,\sqrt {e+f\,x}\,{\left (a\,d-b\,c\right )}^2\,\left (c^2\,f^2-2\,c\,d\,e\,f+d^2\,e^2\right )}{{\left (c\,f-d\,e\right )}^{5/2}\,\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}{\sqrt {d}\,{\left (c\,f-d\,e\right )}^{5/2}}-\frac {\frac {2\,\left (a^2\,f^2-2\,a\,b\,e\,f+b^2\,e^2\right )}{3\,\left (c\,f-d\,e\right )}-\frac {2\,\left (e+f\,x\right )\,\left (d\,a^2\,f^2-2\,c\,a\,b\,f^2-d\,b^2\,e^2+2\,c\,b^2\,e\,f\right )}{{\left (c\,f-d\,e\right )}^2}}{f^2\,{\left (e+f\,x\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 114.34, size = 129, normalized size = 0.92 \begin {gather*} \frac {2 \left (a f - b e\right ) \left (a d f - 2 b c f + b d e\right )}{f^{2} \sqrt {e + f x} \left (c f - d e\right )^{2}} - \frac {2 \left (a f - b e\right )^{2}}{3 f^{2} \left (e + f x\right )^{\frac {3}{2}} \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 \sqrt {\frac {c f - d e}{d}} \left (c f - d e\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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