Optimal. Leaf size=93 \[ -\frac {2 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{5/2}}+\frac {2 b}{\sqrt {d+e x} (b d-a e)^2}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)} \]
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Rubi [A] time = 0.05, antiderivative size = 93, 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 {2 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{5/2}}+\frac {2 b}{\sqrt {d+e x} (b d-a e)^2}+\frac {2}{3 (d+e x)^{3/2} (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}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx\\ &=\frac {2}{3 (b d-a e) (d+e x)^{3/2}}+\frac {b \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{b d-a e}\\ &=\frac {2}{3 (b d-a e) (d+e x)^{3/2}}+\frac {2 b}{(b d-a e)^2 \sqrt {d+e x}}+\frac {b^2 \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{(b d-a e)^2}\\ &=\frac {2}{3 (b d-a e) (d+e x)^{3/2}}+\frac {2 b}{(b d-a e)^2 \sqrt {d+e x}}+\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{e (b d-a e)^2}\\ &=\frac {2}{3 (b d-a e) (d+e x)^{3/2}}+\frac {2 b}{(b d-a e)^2 \sqrt {d+e x}}-\frac {2 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 48, normalized size = 0.52 \begin {gather*} \frac {2 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {b (d+e x)}{b d-a e}\right )}{3 (d+e x)^{3/2} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.15, size = 97, normalized size = 1.04 \begin {gather*} \frac {2 (-a e+3 b (d+e x)+b d)}{3 (d+e x)^{3/2} (b d-a e)^2}-\frac {2 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{(a e-b d)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 398, normalized size = 4.28 \begin {gather*} \left [\frac {3 \, {\left (b e^{2} x^{2} + 2 \, b d e x + b d^{2}\right )} \sqrt {\frac {b}{b d - a e}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, {\left (b d - a e\right )} \sqrt {e x + d} \sqrt {\frac {b}{b d - a e}}}{b x + a}\right ) + 2 \, {\left (3 \, b e x + 4 \, b d - a e\right )} \sqrt {e x + d}}{3 \, {\left (b^{2} d^{4} - 2 \, a b d^{3} e + a^{2} d^{2} e^{2} + {\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} x^{2} + 2 \, {\left (b^{2} d^{3} e - 2 \, a b d^{2} e^{2} + a^{2} d e^{3}\right )} x\right )}}, -\frac {2 \, {\left (3 \, {\left (b e^{2} x^{2} + 2 \, b d e x + b d^{2}\right )} \sqrt {-\frac {b}{b d - a e}} \arctan \left (-\frac {{\left (b d - a e\right )} \sqrt {e x + d} \sqrt {-\frac {b}{b d - a e}}}{b e x + b d}\right ) - {\left (3 \, b e x + 4 \, b d - a e\right )} \sqrt {e x + d}\right )}}{3 \, {\left (b^{2} d^{4} - 2 \, a b d^{3} e + a^{2} d^{2} e^{2} + {\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} x^{2} + 2 \, {\left (b^{2} d^{3} e - 2 \, a b d^{2} e^{2} + a^{2} d 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 = 0.18, size = 119, normalized size = 1.28 \begin {gather*} \frac {2 \, b^{2} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \sqrt {-b^{2} d + a b e}} + \frac {2 \, {\left (3 \, {\left (x e + d\right )} b + b d - a e\right )}}{3 \, {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} {\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 90, normalized size = 0.97 \begin {gather*} \frac {2 b^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{2} \sqrt {\left (a e -b d \right ) b}}+\frac {2 b}{\left (a e -b d \right )^{2} \sqrt {e x +d}}-\frac {2}{3 \left (a e -b d \right ) \left (e x +d \right )^{\frac {3}{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 = 2.09, size = 100, normalized size = 1.08 \begin {gather*} \frac {2\,b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}{{\left (a\,e-b\,d\right )}^{5/2}}\right )}{{\left (a\,e-b\,d\right )}^{5/2}}-\frac {\frac {2}{3\,\left (a\,e-b\,d\right )}-\frac {2\,b\,\left (d+e\,x\right )}{{\left (a\,e-b\,d\right )}^2}}{{\left (d+e\,x\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 136.91, size = 83, normalized size = 0.89 \begin {gather*} \frac {2 b}{\sqrt {d + e x} \left (a e - b d\right )^{2}} + \frac {2 b \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e - b d}{b}}} \right )}}{\sqrt {\frac {a e - b d}{b}} \left (a e - b d\right )^{2}} - \frac {2}{3 \left (d + e x\right )^{\frac {3}{2}} \left (a e - b d\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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