Optimal. Leaf size=124 \[ \frac {5 b^{3/2} e \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{7/2}}-\frac {5 b e}{\sqrt {d+e x} (b d-a e)^3}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}-\frac {5 e}{3 (d+e x)^{3/2} (b d-a e)^2} \]
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Rubi [A] time = 0.06, antiderivative size = 124, 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, 51, 63, 208} \[ \frac {5 b^{3/2} e \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{7/2}}-\frac {5 b e}{\sqrt {d+e x} (b d-a e)^3}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}-\frac {5 e}{3 (d+e x)^{3/2} (b d-a e)^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 51
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac {1}{(a+b x)^2 (d+e x)^{5/2}} \, dx\\ &=-\frac {1}{(b d-a e) (a+b x) (d+e x)^{3/2}}-\frac {(5 e) \int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx}{2 (b d-a e)}\\ &=-\frac {5 e}{3 (b d-a e)^2 (d+e x)^{3/2}}-\frac {1}{(b d-a e) (a+b x) (d+e x)^{3/2}}-\frac {(5 b e) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{2 (b d-a e)^2}\\ &=-\frac {5 e}{3 (b d-a e)^2 (d+e x)^{3/2}}-\frac {1}{(b d-a e) (a+b x) (d+e x)^{3/2}}-\frac {5 b e}{(b d-a e)^3 \sqrt {d+e x}}-\frac {\left (5 b^2 e\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{2 (b d-a e)^3}\\ &=-\frac {5 e}{3 (b d-a e)^2 (d+e x)^{3/2}}-\frac {1}{(b d-a e) (a+b x) (d+e x)^{3/2}}-\frac {5 b e}{(b d-a e)^3 \sqrt {d+e x}}-\frac {\left (5 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 )}{(b d-a e)^3}\\ &=-\frac {5 e}{3 (b d-a e)^2 (d+e x)^{3/2}}-\frac {1}{(b d-a e) (a+b x) (d+e x)^{3/2}}-\frac {5 b e}{(b d-a e)^3 \sqrt {d+e x}}+\frac {5 b^{3/2} e \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 50, normalized size = 0.40 \[ -\frac {2 e \, _2F_1\left (-\frac {3}{2},2;-\frac {1}{2};-\frac {b (d+e x)}{a e-b d}\right )}{3 (d+e x)^{3/2} (a e-b d)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.97, size = 782, normalized size = 6.31 \[ \left [-\frac {15 \, {\left (b^{2} e^{3} x^{3} + a b d^{2} e + {\left (2 \, b^{2} d e^{2} + a b e^{3}\right )} x^{2} + {\left (b^{2} d^{2} e + 2 \, a b d e^{2}\right )} x\right )} \sqrt {\frac {b}{b d - a e}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, {\left (b d - a e\right )} \sqrt {e x + d} \sqrt {\frac {b}{b d - a e}}}{b x + a}\right ) + 2 \, {\left (15 \, b^{2} e^{2} x^{2} + 3 \, b^{2} d^{2} + 14 \, a b d e - 2 \, a^{2} e^{2} + 10 \, {\left (2 \, b^{2} d e + a b e^{2}\right )} x\right )} \sqrt {e x + d}}{6 \, {\left (a b^{3} d^{5} - 3 \, a^{2} b^{2} d^{4} e + 3 \, a^{3} b d^{3} e^{2} - a^{4} d^{2} e^{3} + {\left (b^{4} d^{3} e^{2} - 3 \, a b^{3} d^{2} e^{3} + 3 \, a^{2} b^{2} d e^{4} - a^{3} b e^{5}\right )} x^{3} + {\left (2 \, b^{4} d^{4} e - 5 \, a b^{3} d^{3} e^{2} + 3 \, a^{2} b^{2} d^{2} e^{3} + a^{3} b d e^{4} - a^{4} e^{5}\right )} x^{2} + {\left (b^{4} d^{5} - a b^{3} d^{4} e - 3 \, a^{2} b^{2} d^{3} e^{2} + 5 \, a^{3} b d^{2} e^{3} - 2 \, a^{4} d e^{4}\right )} x\right )}}, \frac {15 \, {\left (b^{2} e^{3} x^{3} + a b d^{2} e + {\left (2 \, b^{2} d e^{2} + a b e^{3}\right )} x^{2} + {\left (b^{2} d^{2} e + 2 \, a b d e^{2}\right )} x\right )} \sqrt {-\frac {b}{b d - a e}} \arctan \left (-\frac {{\left (b d - a e\right )} \sqrt {e x + d} \sqrt {-\frac {b}{b d - a e}}}{b e x + b d}\right ) - {\left (15 \, b^{2} e^{2} x^{2} + 3 \, b^{2} d^{2} + 14 \, a b d e - 2 \, a^{2} e^{2} + 10 \, {\left (2 \, b^{2} d e + a b e^{2}\right )} x\right )} \sqrt {e x + d}}{3 \, {\left (a b^{3} d^{5} - 3 \, a^{2} b^{2} d^{4} e + 3 \, a^{3} b d^{3} e^{2} - a^{4} d^{2} e^{3} + {\left (b^{4} d^{3} e^{2} - 3 \, a b^{3} d^{2} e^{3} + 3 \, a^{2} b^{2} d e^{4} - a^{3} b e^{5}\right )} x^{3} + {\left (2 \, b^{4} d^{4} e - 5 \, a b^{3} d^{3} e^{2} + 3 \, a^{2} b^{2} d^{2} e^{3} + a^{3} b d e^{4} - a^{4} e^{5}\right )} x^{2} + {\left (b^{4} d^{5} - a b^{3} d^{4} e - 3 \, a^{2} b^{2} d^{3} e^{2} + 5 \, a^{3} b d^{2} e^{3} - 2 \, a^{4} d e^{4}\right )} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 224, normalized size = 1.81 \[ -\frac {5 \, b^{2} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt {-b^{2} d + a b e}} - \frac {\sqrt {x e + d} b^{2} e}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}} - \frac {2 \, {\left (6 \, {\left (x e + d\right )} b e + b d e - a e^{2}\right )}}{3 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} {\left (x e + d\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 125, normalized size = 1.01 \[ \frac {5 b^{2} e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{3} \sqrt {\left (a e -b d \right ) b}}+\frac {\sqrt {e x +d}\, b^{2} e}{\left (a e -b d \right )^{3} \left (b e x +a e \right )}+\frac {4 b e}{\left (a e -b d \right )^{3} \sqrt {e x +d}}-\frac {2 e}{3 \left (a e -b d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}} \]
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 = 0.67, size = 161, normalized size = 1.30 \[ \frac {\frac {10\,b\,e\,\left (d+e\,x\right )}{3\,{\left (a\,e-b\,d\right )}^2}-\frac {2\,e}{3\,\left (a\,e-b\,d\right )}+\frac {5\,b^2\,e\,{\left (d+e\,x\right )}^2}{{\left (a\,e-b\,d\right )}^3}}{b\,{\left (d+e\,x\right )}^{5/2}+\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{3/2}}+\frac {5\,b^{3/2}\,e\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{{\left (a\,e-b\,d\right )}^{7/2}}\right )}{{\left (a\,e-b\,d\right )}^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b x\right )^{2} \left (d + e x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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