Optimal. Leaf size=100 \[ -\frac {3 e^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{5/2} \sqrt {b d-a e}}-\frac {3 e \sqrt {d+e x}}{4 b^2 (a+b x)}-\frac {(d+e x)^{3/2}}{2 b (a+b x)^2} \]
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Rubi [A] time = 0.05, antiderivative size = 100, 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, 47, 63, 208} \[ -\frac {3 e^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{5/2} \sqrt {b d-a e}}-\frac {3 e \sqrt {d+e x}}{4 b^2 (a+b x)}-\frac {(d+e x)^{3/2}}{2 b (a+b x)^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 47
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {(a+b x) (d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {(d+e x)^{3/2}}{(a+b x)^3} \, dx\\ &=-\frac {(d+e x)^{3/2}}{2 b (a+b x)^2}+\frac {(3 e) \int \frac {\sqrt {d+e x}}{(a+b x)^2} \, dx}{4 b}\\ &=-\frac {3 e \sqrt {d+e x}}{4 b^2 (a+b x)}-\frac {(d+e x)^{3/2}}{2 b (a+b x)^2}+\frac {\left (3 e^2\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{8 b^2}\\ &=-\frac {3 e \sqrt {d+e x}}{4 b^2 (a+b x)}-\frac {(d+e x)^{3/2}}{2 b (a+b x)^2}+\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^2}\\ &=-\frac {3 e \sqrt {d+e x}}{4 b^2 (a+b x)}-\frac {(d+e x)^{3/2}}{2 b (a+b x)^2}-\frac {3 e^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{5/2} \sqrt {b d-a e}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 90, normalized size = 0.90 \[ \frac {3 e^2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {a e-b d}}\right )}{4 b^{5/2} \sqrt {a e-b d}}-\frac {\sqrt {d+e x} (3 a e+2 b d+5 b e x)}{4 b^2 (a+b x)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.96, size = 383, normalized size = 3.83 \[ \left [\frac {3 \, {\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \sqrt {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) - 2 \, {\left (2 \, b^{3} d^{2} + a b^{2} d e - 3 \, a^{2} b e^{2} + 5 \, {\left (b^{3} d e - a b^{2} e^{2}\right )} x\right )} \sqrt {e x + d}}{8 \, {\left (a^{2} b^{4} d - a^{3} b^{3} e + {\left (b^{6} d - a b^{5} e\right )} x^{2} + 2 \, {\left (a b^{5} d - a^{2} b^{4} e\right )} x\right )}}, \frac {3 \, {\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) - {\left (2 \, b^{3} d^{2} + a b^{2} d e - 3 \, a^{2} b e^{2} + 5 \, {\left (b^{3} d e - a b^{2} e^{2}\right )} x\right )} \sqrt {e x + d}}{4 \, {\left (a^{2} b^{4} d - a^{3} b^{3} e + {\left (b^{6} d - a b^{5} e\right )} x^{2} + 2 \, {\left (a b^{5} d - a^{2} b^{4} e\right )} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 112, normalized size = 1.12 \[ \frac {3 \, \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{2}}{4 \, \sqrt {-b^{2} d + a b e} b^{2}} - \frac {5 \, {\left (x e + d\right )}^{\frac {3}{2}} b e^{2} - 3 \, \sqrt {x e + d} b d e^{2} + 3 \, \sqrt {x e + d} a e^{3}}{4 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 121, normalized size = 1.21 \[ -\frac {3 \sqrt {e x +d}\, a \,e^{3}}{4 \left (b e x +a e \right )^{2} b^{2}}+\frac {3 \sqrt {e x +d}\, d \,e^{2}}{4 \left (b e x +a e \right )^{2} b}-\frac {5 \left (e x +d \right )^{\frac {3}{2}} e^{2}}{4 \left (b e x +a e \right )^{2} b}+\frac {3 e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{4 \sqrt {\left (a e -b d \right ) b}\, b^{2}} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 135, normalized size = 1.35 \[ \frac {3\,e^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{4\,b^{5/2}\,\sqrt {a\,e-b\,d}}-\frac {\frac {5\,e^2\,{\left (d+e\,x\right )}^{3/2}}{4\,b}+\frac {3\,e^2\,\left (a\,e-b\,d\right )\,\sqrt {d+e\,x}}{4\,b^2}}{b^2\,{\left (d+e\,x\right )}^2-\left (2\,b^2\,d-2\,a\,b\,e\right )\,\left (d+e\,x\right )+a^2\,e^2+b^2\,d^2-2\,a\,b\,d\,e} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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