Optimal. Leaf size=126 \[ -\frac {5 e^2 \sqrt {d+e x}}{8 b^3 (a+b x)}-\frac {5 e (d+e x)^{3/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{5/2}}{3 b (a+b x)^3}-\frac {5 e^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{7/2} \sqrt {b d-a e}} \]
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Rubi [A]
time = 0.04, 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, 43, 65, 214}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rule 65
Rule 214
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 ) \text {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 = 0.53, size = 119, normalized size = 0.94 \begin {gather*} -\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}+\frac {5 e^3 \tan ^{-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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.66, size = 130, normalized size = 1.03
method | result | size |
derivativedivides | \(2 e^{3} \left (\frac {-\frac {11 \left (e x +d \right )^{\frac {5}{2}}}{16 b}-\frac {5 \left (a e -b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{6 b^{2}}-\frac {5 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {e x +d}}{16 b^{3}}}{\left (\left (e x +d \right ) b +a e -b d \right )^{3}}+\frac {5 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{16 b^{3} \sqrt {b \left (a e -b d \right )}}\right )\) | \(130\) |
default | \(2 e^{3} \left (\frac {-\frac {11 \left (e x +d \right )^{\frac {5}{2}}}{16 b}-\frac {5 \left (a e -b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{6 b^{2}}-\frac {5 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {e x +d}}{16 b^{3}}}{\left (\left (e x +d \right ) b +a e -b d \right )^{3}}+\frac {5 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{16 b^{3} \sqrt {b \left (a e -b d \right )}}\right )\) | \(130\) |
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. 262 vs.
\(2 (107) = 214\).
time = 3.37, size = 538, normalized size = 4.27 \begin {gather*} \left [\frac {15 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \sqrt {b^{2} d - a b e} e^{3} \log \left (\frac {2 \, b d + {\left (b x - a\right )} e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {x e + d}}{b x + a}\right ) - 2 \, {\left (8 \, b^{4} d^{3} - {\left (33 \, a b^{3} x^{2} + 40 \, a^{2} b^{2} x + 15 \, a^{3} b\right )} e^{3} + {\left (33 \, b^{4} d x^{2} + 14 \, a b^{3} d x + 5 \, a^{2} b^{2} d\right )} e^{2} + 2 \, {\left (13 \, b^{4} d^{2} x + a b^{3} d^{2}\right )} e\right )} \sqrt {x e + d}}{48 \, {\left (b^{8} d x^{3} + 3 \, a b^{7} d x^{2} + 3 \, a^{2} b^{6} d x + a^{3} b^{5} d - {\left (a b^{7} x^{3} + 3 \, a^{2} b^{6} x^{2} + 3 \, a^{3} b^{5} x + a^{4} b^{4}\right )} e\right )}}, \frac {15 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {x e + d}}{b x e + b d}\right ) e^{3} - {\left (8 \, b^{4} d^{3} - {\left (33 \, a b^{3} x^{2} + 40 \, a^{2} b^{2} x + 15 \, a^{3} b\right )} e^{3} + {\left (33 \, b^{4} d x^{2} + 14 \, a b^{3} d x + 5 \, a^{2} b^{2} d\right )} e^{2} + 2 \, {\left (13 \, b^{4} d^{2} x + a b^{3} d^{2}\right )} e\right )} \sqrt {x e + d}}{24 \, {\left (b^{8} d x^{3} + 3 \, a b^{7} d x^{2} + 3 \, a^{2} b^{6} d x + a^{3} b^{5} d - {\left (a b^{7} x^{3} + 3 \, a^{2} b^{6} x^{2} + 3 \, a^{3} b^{5} x + a^{4} b^{4}\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 = 0.93, size = 165, normalized size = 1.31 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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