Optimal. Leaf size=126 \[ -\frac {5 e^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{7/2} \sqrt {b d-a e}}-\frac {5 e^2 \sqrt {d+e x}}{8 b^3 (a+b x)}-\frac {5 e (d+e x)^{3/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{5/2}}{3 b (a+b x)^3} \]
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Rubi [A] time = 0.06, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {27, 47, 63, 208} \[ -\frac {5 e^2 \sqrt {d+e x}}{8 b^3 (a+b x)}-\frac {5 e^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{7/2} \sqrt {b d-a e}}-\frac {5 e (d+e x)^{3/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{5/2}}{3 b (a+b x)^3} \]
Antiderivative was successfully verified.
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Rule 27
Rule 47
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {(d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {(d+e x)^{5/2}}{(a+b x)^4} \, dx\\ &=-\frac {(d+e x)^{5/2}}{3 b (a+b x)^3}+\frac {(5 e) \int \frac {(d+e x)^{3/2}}{(a+b x)^3} \, dx}{6 b}\\ &=-\frac {5 e (d+e x)^{3/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{5/2}}{3 b (a+b x)^3}+\frac {\left (5 e^2\right ) \int \frac {\sqrt {d+e x}}{(a+b x)^2} \, dx}{8 b^2}\\ &=-\frac {5 e^2 \sqrt {d+e x}}{8 b^3 (a+b x)}-\frac {5 e (d+e x)^{3/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{5/2}}{3 b (a+b x)^3}+\frac {\left (5 e^3\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{16 b^3}\\ &=-\frac {5 e^2 \sqrt {d+e x}}{8 b^3 (a+b x)}-\frac {5 e (d+e x)^{3/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{5/2}}{3 b (a+b x)^3}+\frac {\left (5 e^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 )}{8 b^3}\\ &=-\frac {5 e^2 \sqrt {d+e x}}{8 b^3 (a+b x)}-\frac {5 e (d+e x)^{3/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{5/2}}{3 b (a+b x)^3}-\frac {5 e^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{7/2} \sqrt {b d-a e}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 119, normalized size = 0.94 \[ \frac {5 e^3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {a e-b d}}\right )}{8 b^{7/2} \sqrt {a e-b d}}-\frac {\sqrt {d+e x} \left (15 a^2 e^2+10 a b e (d+4 e x)+b^2 \left (8 d^2+26 d e x+33 e^2 x^2\right )\right )}{24 b^3 (a+b x)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.06, size = 563, normalized size = 4.47 \[ \left [\frac {15 \, {\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \sqrt {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) - 2 \, {\left (8 \, b^{4} d^{3} + 2 \, a b^{3} d^{2} e + 5 \, a^{2} b^{2} d e^{2} - 15 \, a^{3} b e^{3} + 33 \, {\left (b^{4} d e^{2} - a b^{3} e^{3}\right )} x^{2} + 2 \, {\left (13 \, b^{4} d^{2} e + 7 \, a b^{3} d e^{2} - 20 \, a^{2} b^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{48 \, {\left (a^{3} b^{5} d - a^{4} b^{4} e + {\left (b^{8} d - a b^{7} e\right )} x^{3} + 3 \, {\left (a b^{7} d - a^{2} b^{6} e\right )} x^{2} + 3 \, {\left (a^{2} b^{6} d - a^{3} b^{5} e\right )} x\right )}}, \frac {15 \, {\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) - {\left (8 \, b^{4} d^{3} + 2 \, a b^{3} d^{2} e + 5 \, a^{2} b^{2} d e^{2} - 15 \, a^{3} b e^{3} + 33 \, {\left (b^{4} d e^{2} - a b^{3} e^{3}\right )} x^{2} + 2 \, {\left (13 \, b^{4} d^{2} e + 7 \, a b^{3} d e^{2} - 20 \, a^{2} b^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{24 \, {\left (a^{3} b^{5} d - a^{4} b^{4} e + {\left (b^{8} d - a b^{7} e\right )} x^{3} + 3 \, {\left (a b^{7} d - a^{2} b^{6} e\right )} x^{2} + 3 \, {\left (a^{2} b^{6} d - a^{3} b^{5} e\right )} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 165, normalized size = 1.31 \[ \frac {5 \, \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{3}}{8 \, \sqrt {-b^{2} d + a b e} b^{3}} - \frac {33 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{2} e^{3} - 40 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} d e^{3} + 15 \, \sqrt {x e + d} b^{2} d^{2} e^{3} + 40 \, {\left (x e + d\right )}^{\frac {3}{2}} a b e^{4} - 30 \, \sqrt {x e + d} a b d e^{4} + 15 \, \sqrt {x e + d} a^{2} e^{5}}{24 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{3} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 204, normalized size = 1.62 \[ -\frac {5 \sqrt {e x +d}\, a^{2} e^{5}}{8 \left (b e x +a e \right )^{3} b^{3}}+\frac {5 \sqrt {e x +d}\, a d \,e^{4}}{4 \left (b e x +a e \right )^{3} b^{2}}-\frac {5 \sqrt {e x +d}\, d^{2} e^{3}}{8 \left (b e x +a e \right )^{3} b}-\frac {5 \left (e x +d \right )^{\frac {3}{2}} a \,e^{4}}{3 \left (b e x +a e \right )^{3} b^{2}}+\frac {5 \left (e x +d \right )^{\frac {3}{2}} d \,e^{3}}{3 \left (b e x +a e \right )^{3} b}-\frac {11 \left (e x +d \right )^{\frac {5}{2}} e^{3}}{8 \left (b e x +a e \right )^{3} b}+\frac {5 e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}\, b^{3}} \]
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 = 0.17, size = 222, normalized size = 1.76 \[ \frac {5\,e^3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{8\,b^{7/2}\,\sqrt {a\,e-b\,d}}-\frac {\frac {11\,e^3\,{\left (d+e\,x\right )}^{5/2}}{8\,b}+\frac {5\,e^3\,\sqrt {d+e\,x}\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}{8\,b^3}+\frac {5\,e^3\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,b^2}}{\left (d+e\,x\right )\,\left (3\,a^2\,b\,e^2-6\,a\,b^2\,d\,e+3\,b^3\,d^2\right )+b^3\,{\left (d+e\,x\right )}^3-\left (3\,b^3\,d-3\,a\,b^2\,e\right )\,{\left (d+e\,x\right )}^2+a^3\,e^3-b^3\,d^3+3\,a\,b^2\,d^2\,e-3\,a^2\,b\,d\,e^2} \]
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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