Optimal. Leaf size=167 \[ \frac {35 e^2}{12 (b d-a e)^3 (d+e x)^{3/2}}-\frac {1}{2 (b d-a e) (a+b x)^2 (d+e x)^{3/2}}+\frac {7 e}{4 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}+\frac {35 b e^2}{4 (b d-a e)^4 \sqrt {d+e x}}-\frac {35 b^{3/2} e^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 (b d-a e)^{9/2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {27, 44, 53, 65,
214} \begin {gather*} -\frac {35 b^{3/2} e^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 (b d-a e)^{9/2}}+\frac {35 b e^2}{4 \sqrt {d+e x} (b d-a e)^4}+\frac {35 e^2}{12 (d+e x)^{3/2} (b d-a e)^3}+\frac {7 e}{4 (a+b x) (d+e x)^{3/2} (b d-a e)^2}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 44
Rule 53
Rule 65
Rule 214
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 )^2} \, dx &=\int \frac {1}{(a+b x)^3 (d+e x)^{5/2}} \, dx\\ &=-\frac {1}{2 (b d-a e) (a+b x)^2 (d+e x)^{3/2}}-\frac {(7 e) \int \frac {1}{(a+b x)^2 (d+e x)^{5/2}} \, dx}{4 (b d-a e)}\\ &=-\frac {1}{2 (b d-a e) (a+b x)^2 (d+e x)^{3/2}}+\frac {7 e}{4 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}+\frac {\left (35 e^2\right ) \int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx}{8 (b d-a e)^2}\\ &=\frac {35 e^2}{12 (b d-a e)^3 (d+e x)^{3/2}}-\frac {1}{2 (b d-a e) (a+b x)^2 (d+e x)^{3/2}}+\frac {7 e}{4 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}+\frac {\left (35 b e^2\right ) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{8 (b d-a e)^3}\\ &=\frac {35 e^2}{12 (b d-a e)^3 (d+e x)^{3/2}}-\frac {1}{2 (b d-a e) (a+b x)^2 (d+e x)^{3/2}}+\frac {7 e}{4 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}+\frac {35 b e^2}{4 (b d-a e)^4 \sqrt {d+e x}}+\frac {\left (35 b^2 e^2\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{8 (b d-a e)^4}\\ &=\frac {35 e^2}{12 (b d-a e)^3 (d+e x)^{3/2}}-\frac {1}{2 (b d-a e) (a+b x)^2 (d+e x)^{3/2}}+\frac {7 e}{4 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}+\frac {35 b e^2}{4 (b d-a e)^4 \sqrt {d+e x}}+\frac {\left (35 b^2 e\right ) \text {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)^4}\\ &=\frac {35 e^2}{12 (b d-a e)^3 (d+e x)^{3/2}}-\frac {1}{2 (b d-a e) (a+b x)^2 (d+e x)^{3/2}}+\frac {7 e}{4 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}+\frac {35 b e^2}{4 (b d-a e)^4 \sqrt {d+e x}}-\frac {35 b^{3/2} e^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 (b d-a e)^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 168, normalized size = 1.01 \begin {gather*} \frac {-8 a^3 e^3+8 a^2 b e^2 (10 d+7 e x)+a b^2 e \left (39 d^2+238 d e x+175 e^2 x^2\right )+b^3 \left (-6 d^3+21 d^2 e x+140 d e^2 x^2+105 e^3 x^3\right )}{12 (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}+\frac {35 b^{3/2} e^2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{4 (-b d+a e)^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 143, normalized size = 0.86
method | result | size |
derivativedivides | \(2 e^{2} \left (\frac {b^{2} \left (\frac {\frac {11 b \left (e x +d \right )^{\frac {3}{2}}}{8}+\left (\frac {13 a e}{8}-\frac {13 b d}{8}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{2}}+\frac {35 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{4}}-\frac {1}{3 \left (a e -b d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}+\frac {3 b}{\left (a e -b d \right )^{4} \sqrt {e x +d}}\right )\) | \(143\) |
default | \(2 e^{2} \left (\frac {b^{2} \left (\frac {\frac {11 b \left (e x +d \right )^{\frac {3}{2}}}{8}+\left (\frac {13 a e}{8}-\frac {13 b d}{8}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{2}}+\frac {35 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{4}}-\frac {1}{3 \left (a e -b d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}+\frac {3 b}{\left (a e -b d \right )^{4} \sqrt {e x +d}}\right )\) | \(143\) |
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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 590 vs.
\(2 (148) = 296\).
time = 2.49, size = 1192, normalized size = 7.14 \begin {gather*} \left [\frac {105 \, {\left ({\left (b^{3} x^{4} + 2 \, a b^{2} x^{3} + a^{2} b x^{2}\right )} e^{4} + 2 \, {\left (b^{3} d x^{3} + 2 \, a b^{2} d x^{2} + a^{2} b d x\right )} e^{3} + {\left (b^{3} d^{2} x^{2} + 2 \, a b^{2} d^{2} x + a^{2} b d^{2}\right )} e^{2}\right )} \sqrt {\frac {b}{b d - a e}} \log \left (\frac {2 \, b d - 2 \, {\left (b d - a e\right )} \sqrt {x e + d} \sqrt {\frac {b}{b d - a e}} + {\left (b x - a\right )} e}{b x + a}\right ) - 2 \, {\left (6 \, b^{3} d^{3} - {\left (105 \, b^{3} x^{3} + 175 \, a b^{2} x^{2} + 56 \, a^{2} b x - 8 \, a^{3}\right )} e^{3} - 2 \, {\left (70 \, b^{3} d x^{2} + 119 \, a b^{2} d x + 40 \, a^{2} b d\right )} e^{2} - 3 \, {\left (7 \, b^{3} d^{2} x + 13 \, a b^{2} d^{2}\right )} e\right )} \sqrt {x e + d}}{24 \, {\left (b^{6} d^{6} x^{2} + 2 \, a b^{5} d^{6} x + a^{2} b^{4} d^{6} + {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )} e^{6} - 2 \, {\left (2 \, a^{3} b^{3} d x^{4} + 3 \, a^{4} b^{2} d x^{3} - a^{6} d x\right )} e^{5} + {\left (6 \, a^{2} b^{4} d^{2} x^{4} + 4 \, a^{3} b^{3} d^{2} x^{3} - 9 \, a^{4} b^{2} d^{2} x^{2} - 6 \, a^{5} b d^{2} x + a^{6} d^{2}\right )} e^{4} - 4 \, {\left (a b^{5} d^{3} x^{4} - a^{2} b^{4} d^{3} x^{3} - 4 \, a^{3} b^{3} d^{3} x^{2} - a^{4} b^{2} d^{3} x + a^{5} b d^{3}\right )} e^{3} + {\left (b^{6} d^{4} x^{4} - 6 \, a b^{5} d^{4} x^{3} - 9 \, a^{2} b^{4} d^{4} x^{2} + 4 \, a^{3} b^{3} d^{4} x + 6 \, a^{4} b^{2} d^{4}\right )} e^{2} + 2 \, {\left (b^{6} d^{5} x^{3} - 3 \, a^{2} b^{4} d^{5} x - 2 \, a^{3} b^{3} d^{5}\right )} e\right )}}, -\frac {105 \, {\left ({\left (b^{3} x^{4} + 2 \, a b^{2} x^{3} + a^{2} b x^{2}\right )} e^{4} + 2 \, {\left (b^{3} d x^{3} + 2 \, a b^{2} d x^{2} + a^{2} b d x\right )} e^{3} + {\left (b^{3} d^{2} x^{2} + 2 \, a b^{2} d^{2} x + a^{2} b d^{2}\right )} e^{2}\right )} \sqrt {-\frac {b}{b d - a e}} \arctan \left (-\frac {{\left (b d - a e\right )} \sqrt {x e + d} \sqrt {-\frac {b}{b d - a e}}}{b x e + b d}\right ) + {\left (6 \, b^{3} d^{3} - {\left (105 \, b^{3} x^{3} + 175 \, a b^{2} x^{2} + 56 \, a^{2} b x - 8 \, a^{3}\right )} e^{3} - 2 \, {\left (70 \, b^{3} d x^{2} + 119 \, a b^{2} d x + 40 \, a^{2} b d\right )} e^{2} - 3 \, {\left (7 \, b^{3} d^{2} x + 13 \, a b^{2} d^{2}\right )} e\right )} \sqrt {x e + d}}{12 \, {\left (b^{6} d^{6} x^{2} + 2 \, a b^{5} d^{6} x + a^{2} b^{4} d^{6} + {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )} e^{6} - 2 \, {\left (2 \, a^{3} b^{3} d x^{4} + 3 \, a^{4} b^{2} d x^{3} - a^{6} d x\right )} e^{5} + {\left (6 \, a^{2} b^{4} d^{2} x^{4} + 4 \, a^{3} b^{3} d^{2} x^{3} - 9 \, a^{4} b^{2} d^{2} x^{2} - 6 \, a^{5} b d^{2} x + a^{6} d^{2}\right )} e^{4} - 4 \, {\left (a b^{5} d^{3} x^{4} - a^{2} b^{4} d^{3} x^{3} - 4 \, a^{3} b^{3} d^{3} x^{2} - a^{4} b^{2} d^{3} x + a^{5} b d^{3}\right )} e^{3} + {\left (b^{6} d^{4} x^{4} - 6 \, a b^{5} d^{4} x^{3} - 9 \, a^{2} b^{4} d^{4} x^{2} + 4 \, a^{3} b^{3} d^{4} x + 6 \, a^{4} b^{2} d^{4}\right )} e^{2} + 2 \, {\left (b^{6} d^{5} x^{3} - 3 \, a^{2} b^{4} d^{5} x - 2 \, a^{3} b^{3} d^{5}\right )} e\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [A]
time = 1.67, size = 295, normalized size = 1.77 \begin {gather*} \frac {35 \, b^{2} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{2}}{4 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt {-b^{2} d + a b e}} + \frac {2 \, {\left (9 \, {\left (x e + d\right )} b e^{2} + b d e^{2} - a e^{3}\right )}}{3 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left (x e + d\right )}^{\frac {3}{2}}} + \frac {11 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} e^{2} - 13 \, \sqrt {x e + d} b^{3} d e^{2} + 13 \, \sqrt {x e + d} a b^{2} e^{3}}{4 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\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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Mupad [B]
time = 2.54, size = 243, normalized size = 1.46 \begin {gather*} \frac {\frac {175\,b^2\,e^2\,{\left (d+e\,x\right )}^2}{12\,{\left (a\,e-b\,d\right )}^3}-\frac {2\,e^2}{3\,\left (a\,e-b\,d\right )}+\frac {35\,b^3\,e^2\,{\left (d+e\,x\right )}^3}{4\,{\left (a\,e-b\,d\right )}^4}+\frac {14\,b\,e^2\,\left (d+e\,x\right )}{3\,{\left (a\,e-b\,d\right )}^2}}{b^2\,{\left (d+e\,x\right )}^{7/2}-\left (2\,b^2\,d-2\,a\,b\,e\right )\,{\left (d+e\,x\right )}^{5/2}+{\left (d+e\,x\right )}^{3/2}\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}+\frac {35\,b^{3/2}\,e^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{{\left (a\,e-b\,d\right )}^{9/2}}\right )}{4\,{\left (a\,e-b\,d\right )}^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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