Optimal. Leaf size=147 \[ -\frac {\sqrt {d+e x}}{3 (b d-a e) (a+b x)^3}+\frac {5 e \sqrt {d+e x}}{12 (b d-a e)^2 (a+b x)^2}-\frac {5 e^2 \sqrt {d+e x}}{8 (b d-a e)^3 (a+b x)}+\frac {5 e^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 \sqrt {b} (b d-a e)^{7/2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 147, 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, 44, 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 \sqrt {b} (b d-a e)^{7/2}}-\frac {5 e^2 \sqrt {d+e x}}{8 (a+b x) (b d-a e)^3}+\frac {5 e \sqrt {d+e x}}{12 (a+b x)^2 (b d-a e)^2}-\frac {\sqrt {d+e x}}{3 (a+b x)^3 (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 44
Rule 65
Rule 214
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {1}{(a+b x)^4 \sqrt {d+e x}} \, dx\\ &=-\frac {\sqrt {d+e x}}{3 (b d-a e) (a+b x)^3}-\frac {(5 e) \int \frac {1}{(a+b x)^3 \sqrt {d+e x}} \, dx}{6 (b d-a e)}\\ &=-\frac {\sqrt {d+e x}}{3 (b d-a e) (a+b x)^3}+\frac {5 e \sqrt {d+e x}}{12 (b d-a e)^2 (a+b x)^2}+\frac {\left (5 e^2\right ) \int \frac {1}{(a+b x)^2 \sqrt {d+e x}} \, dx}{8 (b d-a e)^2}\\ &=-\frac {\sqrt {d+e x}}{3 (b d-a e) (a+b x)^3}+\frac {5 e \sqrt {d+e x}}{12 (b d-a e)^2 (a+b x)^2}-\frac {5 e^2 \sqrt {d+e x}}{8 (b d-a e)^3 (a+b x)}-\frac {\left (5 e^3\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{16 (b d-a e)^3}\\ &=-\frac {\sqrt {d+e x}}{3 (b d-a e) (a+b x)^3}+\frac {5 e \sqrt {d+e x}}{12 (b d-a e)^2 (a+b x)^2}-\frac {5 e^2 \sqrt {d+e x}}{8 (b d-a e)^3 (a+b x)}-\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 d-a e)^3}\\ &=-\frac {\sqrt {d+e x}}{3 (b d-a e) (a+b x)^3}+\frac {5 e \sqrt {d+e x}}{12 (b d-a e)^2 (a+b x)^2}-\frac {5 e^2 \sqrt {d+e x}}{8 (b d-a e)^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 \sqrt {b} (b d-a e)^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 128, normalized size = 0.87 \begin {gather*} \frac {\sqrt {d+e x} \left (33 a^2 e^2+2 a b e (-13 d+20 e x)+b^2 \left (8 d^2-10 d e x+15 e^2 x^2\right )\right )}{24 (-b d+a e)^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 \sqrt {b} (-b d+a e)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.75, size = 187, normalized size = 1.27
method | result | size |
derivativedivides | \(2 e^{3} \left (\frac {\sqrt {e x +d}}{6 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )^{3}}+\frac {\frac {5 \sqrt {e x +d}}{24 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )^{2}}+\frac {5 \left (\frac {3 \sqrt {e x +d}}{8 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )}+\frac {3 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{8 \left (a e -b d \right ) \sqrt {b \left (a e -b d \right )}}\right )}{6 \left (a e -b d \right )}}{a e -b d}\right )\) | \(187\) |
default | \(2 e^{3} \left (\frac {\sqrt {e x +d}}{6 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )^{3}}+\frac {\frac {5 \sqrt {e x +d}}{24 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )^{2}}+\frac {5 \left (\frac {3 \sqrt {e x +d}}{8 \left (a e -b d \right ) \left (\left (e x +d \right ) b +a e -b d \right )}+\frac {3 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{8 \left (a e -b d \right ) \sqrt {b \left (a e -b d \right )}}\right )}{6 \left (a e -b d \right )}}{a e -b d}\right )\) | \(187\) |
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. 421 vs.
\(2 (131) = 262\).
time = 3.48, size = 857, normalized size = 5.83 \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 (15 \, a b^{3} x^{2} + 40 \, a^{2} b^{2} x + 33 \, a^{3} b\right )} e^{3} + {\left (15 \, b^{4} d x^{2} + 50 \, a b^{3} d x + 59 \, a^{2} b^{2} d\right )} e^{2} - 2 \, {\left (5 \, b^{4} d^{2} x + 17 \, a b^{3} d^{2}\right )} e\right )} \sqrt {x e + d}}{48 \, {\left (b^{8} d^{4} x^{3} + 3 \, a b^{7} d^{4} x^{2} + 3 \, a^{2} b^{6} d^{4} x + a^{3} b^{5} d^{4} + {\left (a^{4} b^{4} x^{3} + 3 \, a^{5} b^{3} x^{2} + 3 \, a^{6} b^{2} x + a^{7} b\right )} e^{4} - 4 \, {\left (a^{3} b^{5} d x^{3} + 3 \, a^{4} b^{4} d x^{2} + 3 \, a^{5} b^{3} d x + a^{6} b^{2} d\right )} e^{3} + 6 \, {\left (a^{2} b^{6} d^{2} x^{3} + 3 \, a^{3} b^{5} d^{2} x^{2} + 3 \, a^{4} b^{4} d^{2} x + a^{5} b^{3} d^{2}\right )} e^{2} - 4 \, {\left (a b^{7} d^{3} x^{3} + 3 \, a^{2} b^{6} d^{3} x^{2} + 3 \, a^{3} b^{5} d^{3} x + a^{4} b^{4} d^{3}\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 (15 \, a b^{3} x^{2} + 40 \, a^{2} b^{2} x + 33 \, a^{3} b\right )} e^{3} + {\left (15 \, b^{4} d x^{2} + 50 \, a b^{3} d x + 59 \, a^{2} b^{2} d\right )} e^{2} - 2 \, {\left (5 \, b^{4} d^{2} x + 17 \, a b^{3} d^{2}\right )} e\right )} \sqrt {x e + d}}{24 \, {\left (b^{8} d^{4} x^{3} + 3 \, a b^{7} d^{4} x^{2} + 3 \, a^{2} b^{6} d^{4} x + a^{3} b^{5} d^{4} + {\left (a^{4} b^{4} x^{3} + 3 \, a^{5} b^{3} x^{2} + 3 \, a^{6} b^{2} x + a^{7} b\right )} e^{4} - 4 \, {\left (a^{3} b^{5} d x^{3} + 3 \, a^{4} b^{4} d x^{2} + 3 \, a^{5} b^{3} d x + a^{6} b^{2} d\right )} e^{3} + 6 \, {\left (a^{2} b^{6} d^{2} x^{3} + 3 \, a^{3} b^{5} d^{2} x^{2} + 3 \, a^{4} b^{4} d^{2} x + a^{5} b^{3} d^{2}\right )} e^{2} - 4 \, {\left (a b^{7} d^{3} x^{3} + 3 \, a^{2} b^{6} d^{3} x^{2} + 3 \, a^{3} b^{5} d^{3} x + a^{4} b^{4} d^{3}\right )} e\right )}}\right ] \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (a + b x\right )^{4} \sqrt {d + e x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.97, size = 233, normalized size = 1.59 \begin {gather*} -\frac {5 \, \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{3}}{8 \, {\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 {15 \, {\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} + 33 \, \sqrt {x e + d} b^{2} d^{2} e^{3} + 40 \, {\left (x e + d\right )}^{\frac {3}{2}} a b e^{4} - 66 \, \sqrt {x e + d} a b d e^{4} + 33 \, \sqrt {x e + d} a^{2} e^{5}}{24 \, {\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 )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.65, size = 218, normalized size = 1.48 \begin {gather*} \frac {\frac {11\,e^3\,\sqrt {d+e\,x}}{8\,\left (a\,e-b\,d\right )}+\frac {5\,b^2\,e^3\,{\left (d+e\,x\right )}^{5/2}}{8\,{\left (a\,e-b\,d\right )}^3}+\frac {5\,b\,e^3\,{\left (d+e\,x\right )}^{3/2}}{3\,{\left (a\,e-b\,d\right )}^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}+\frac {5\,e^3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{8\,\sqrt {b}\,{\left (a\,e-b\,d\right )}^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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