Optimal. Leaf size=114 \[ -\frac {3 e^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 \sqrt {b} (b d-a e)^{5/2}}+\frac {3 e \sqrt {d+e x}}{4 (a+b x) (b d-a e)^2}-\frac {\sqrt {d+e x}}{2 (a+b x)^2 (b d-a e)} \]
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Rubi [A] time = 0.05, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {27, 51, 63, 208} \begin {gather*} -\frac {3 e^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 \sqrt {b} (b d-a e)^{5/2}}+\frac {3 e \sqrt {d+e x}}{4 (a+b x) (b d-a e)^2}-\frac {\sqrt {d+e x}}{2 (a+b x)^2 (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 51
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {a+b x}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {1}{(a+b x)^3 \sqrt {d+e x}} \, dx\\ &=-\frac {\sqrt {d+e x}}{2 (b d-a e) (a+b x)^2}-\frac {(3 e) \int \frac {1}{(a+b x)^2 \sqrt {d+e x}} \, dx}{4 (b d-a e)}\\ &=-\frac {\sqrt {d+e x}}{2 (b d-a e) (a+b x)^2}+\frac {3 e \sqrt {d+e x}}{4 (b d-a e)^2 (a+b x)}+\frac {\left (3 e^2\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{8 (b d-a e)^2}\\ &=-\frac {\sqrt {d+e x}}{2 (b d-a e) (a+b x)^2}+\frac {3 e \sqrt {d+e x}}{4 (b d-a e)^2 (a+b x)}+\frac {(3 e) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{4 (b d-a e)^2}\\ &=-\frac {\sqrt {d+e x}}{2 (b d-a e) (a+b x)^2}+\frac {3 e \sqrt {d+e x}}{4 (b d-a e)^2 (a+b x)}-\frac {3 e^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 \sqrt {b} (b d-a e)^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 50, normalized size = 0.44 \begin {gather*} \frac {2 e^2 \sqrt {d+e x} \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};-\frac {b (d+e x)}{a e-b d}\right )}{(a e-b d)^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.25, size = 124, normalized size = 1.09 \begin {gather*} \frac {e^2 \sqrt {d+e x} (5 a e+3 b (d+e x)-5 b d)}{4 (b d-a e)^2 (-a e-b (d+e x)+b d)^2}-\frac {3 e^2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{4 \sqrt {b} (a e-b d)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 549, normalized size = 4.82 \begin {gather*} \left [\frac {3 \, {\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \sqrt {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) - 2 \, {\left (2 \, b^{3} d^{2} - 7 \, a b^{2} d e + 5 \, a^{2} b e^{2} - 3 \, {\left (b^{3} d e - a b^{2} e^{2}\right )} x\right )} \sqrt {e x + d}}{8 \, {\left (a^{2} b^{4} d^{3} - 3 \, a^{3} b^{3} d^{2} e + 3 \, a^{4} b^{2} d e^{2} - a^{5} b e^{3} + {\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} x^{2} + 2 \, {\left (a b^{5} d^{3} - 3 \, a^{2} b^{4} d^{2} e + 3 \, a^{3} b^{3} d e^{2} - a^{4} b^{2} e^{3}\right )} x\right )}}, \frac {3 \, {\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) - {\left (2 \, b^{3} d^{2} - 7 \, a b^{2} d e + 5 \, a^{2} b e^{2} - 3 \, {\left (b^{3} d e - a b^{2} e^{2}\right )} x\right )} \sqrt {e x + d}}{4 \, {\left (a^{2} b^{4} d^{3} - 3 \, a^{3} b^{3} d^{2} e + 3 \, a^{4} b^{2} d e^{2} - a^{5} b e^{3} + {\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} x^{2} + 2 \, {\left (a b^{5} d^{3} - 3 \, a^{2} b^{4} d^{2} e + 3 \, a^{3} b^{3} d e^{2} - a^{4} b^{2} e^{3}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 152, normalized size = 1.33 \begin {gather*} \frac {3 \, \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{2}}{4 \, {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \sqrt {-b^{2} d + a b e}} + \frac {3 \, {\left (x e + d\right )}^{\frac {3}{2}} b e^{2} - 5 \, \sqrt {x e + d} b d e^{2} + 5 \, \sqrt {x e + d} a e^{3}}{4 \, {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 115, normalized size = 1.01 \begin {gather*} \frac {3 e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{4 \left (a e -b d \right )^{2} \sqrt {\left (a e -b d \right ) b}}+\frac {\sqrt {e x +d}\, e^{2}}{2 \left (a e -b d \right ) \left (b e x +a e \right )^{2}}+\frac {3 \sqrt {e x +d}\, e^{2}}{4 \left (a e -b d \right )^{2} \left (b e x +a e \right )} \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 = 2.15, size = 142, normalized size = 1.25 \begin {gather*} \frac {\frac {5\,e^2\,\sqrt {d+e\,x}}{4\,\left (a\,e-b\,d\right )}+\frac {3\,b\,e^2\,{\left (d+e\,x\right )}^{3/2}}{4\,{\left (a\,e-b\,d\right )}^2}}{b^2\,{\left (d+e\,x\right )}^2-\left (2\,b^2\,d-2\,a\,b\,e\right )\,\left (d+e\,x\right )+a^2\,e^2+b^2\,d^2-2\,a\,b\,d\,e}+\frac {3\,e^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{4\,\sqrt {b}\,{\left (a\,e-b\,d\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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