Optimal. Leaf size=151 \[ -\frac {7 e}{5 (b d-a e)^2 (d+e x)^{5/2}}-\frac {1}{(b d-a e) (a+b x) (d+e x)^{5/2}}-\frac {7 b e}{3 (b d-a e)^3 (d+e x)^{3/2}}-\frac {7 b^2 e}{(b d-a e)^4 \sqrt {d+e x}}+\frac {7 b^{5/2} e \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{9/2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 44, 53, 65,
214} \begin {gather*} \frac {7 b^{5/2} e \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{9/2}}-\frac {7 b^2 e}{\sqrt {d+e x} (b d-a e)^4}-\frac {7 b e}{3 (d+e x)^{3/2} (b d-a e)^3}-\frac {1}{(a+b x) (d+e x)^{5/2} (b d-a e)}-\frac {7 e}{5 (d+e x)^{5/2} (b d-a e)^2} \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 {1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac {1}{(a+b x)^2 (d+e x)^{7/2}} \, dx\\ &=-\frac {1}{(b d-a e) (a+b x) (d+e x)^{5/2}}-\frac {(7 e) \int \frac {1}{(a+b x) (d+e x)^{7/2}} \, dx}{2 (b d-a e)}\\ &=-\frac {7 e}{5 (b d-a e)^2 (d+e x)^{5/2}}-\frac {1}{(b d-a e) (a+b x) (d+e x)^{5/2}}-\frac {(7 b e) \int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx}{2 (b d-a e)^2}\\ &=-\frac {7 e}{5 (b d-a e)^2 (d+e x)^{5/2}}-\frac {1}{(b d-a e) (a+b x) (d+e x)^{5/2}}-\frac {7 b e}{3 (b d-a e)^3 (d+e x)^{3/2}}-\frac {\left (7 b^2 e\right ) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{2 (b d-a e)^3}\\ &=-\frac {7 e}{5 (b d-a e)^2 (d+e x)^{5/2}}-\frac {1}{(b d-a e) (a+b x) (d+e x)^{5/2}}-\frac {7 b e}{3 (b d-a e)^3 (d+e x)^{3/2}}-\frac {7 b^2 e}{(b d-a e)^4 \sqrt {d+e x}}-\frac {\left (7 b^3 e\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{2 (b d-a e)^4}\\ &=-\frac {7 e}{5 (b d-a e)^2 (d+e x)^{5/2}}-\frac {1}{(b d-a e) (a+b x) (d+e x)^{5/2}}-\frac {7 b e}{3 (b d-a e)^3 (d+e x)^{3/2}}-\frac {7 b^2 e}{(b d-a e)^4 \sqrt {d+e x}}-\frac {\left (7 b^3\right ) \text {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)^4}\\ &=-\frac {7 e}{5 (b d-a e)^2 (d+e x)^{5/2}}-\frac {1}{(b d-a e) (a+b x) (d+e x)^{5/2}}-\frac {7 b e}{3 (b d-a e)^3 (d+e x)^{3/2}}-\frac {7 b^2 e}{(b d-a e)^4 \sqrt {d+e x}}+\frac {7 b^{5/2} e \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.33, size = 166, normalized size = 1.10 \begin {gather*} \frac {-6 a^3 e^3+2 a^2 b e^2 (16 d+7 e x)-2 a b^2 e \left (58 d^2+84 d e x+35 e^2 x^2\right )-b^3 \left (15 d^3+161 d^2 e x+245 d e^2 x^2+105 e^3 x^3\right )}{15 (b d-a e)^4 (a+b x) (d+e x)^{5/2}}-\frac {7 b^{5/2} e \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.67, size = 144, normalized size = 0.95
method | result | size |
derivativedivides | \(2 e \left (-\frac {1}{5 \left (a e -b d \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}-\frac {3 b^{2}}{\left (a e -b d \right )^{4} \sqrt {e x +d}}+\frac {2 b}{3 \left (a e -b d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}-\frac {b^{3} \left (\frac {\sqrt {e x +d}}{2 \left (e x +d \right ) b +2 a e -2 b d}+\frac {7 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{2 \sqrt {b \left (a e -b d \right )}}\right )}{\left (a e -b d \right )^{4}}\right )\) | \(144\) |
default | \(2 e \left (-\frac {1}{5 \left (a e -b d \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}-\frac {3 b^{2}}{\left (a e -b d \right )^{4} \sqrt {e x +d}}+\frac {2 b}{3 \left (a e -b d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}-\frac {b^{3} \left (\frac {\sqrt {e x +d}}{2 \left (e x +d \right ) b +2 a e -2 b d}+\frac {7 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{2 \sqrt {b \left (a e -b d \right )}}\right )}{\left (a e -b d \right )^{4}}\right )\) | \(144\) |
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. 592 vs.
\(2 (144) = 288\).
time = 2.59, size = 1195, normalized size = 7.91 \begin {gather*} \left [\frac {105 \, {\left ({\left (b^{3} x^{4} + a b^{2} x^{3}\right )} e^{4} + 3 \, {\left (b^{3} d x^{3} + a b^{2} d x^{2}\right )} e^{3} + 3 \, {\left (b^{3} d^{2} x^{2} + a b^{2} d^{2} x\right )} e^{2} + {\left (b^{3} d^{3} x + a b^{2} d^{3}\right )} e\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 (15 \, b^{3} d^{3} + {\left (105 \, b^{3} x^{3} + 70 \, a b^{2} x^{2} - 14 \, a^{2} b x + 6 \, a^{3}\right )} e^{3} + {\left (245 \, b^{3} d x^{2} + 168 \, a b^{2} d x - 32 \, a^{2} b d\right )} e^{2} + {\left (161 \, b^{3} d^{2} x + 116 \, a b^{2} d^{2}\right )} e\right )} \sqrt {x e + d}}{30 \, {\left (b^{5} d^{7} x + a b^{4} d^{7} + {\left (a^{4} b x^{4} + a^{5} x^{3}\right )} e^{7} - {\left (4 \, a^{3} b^{2} d x^{4} + a^{4} b d x^{3} - 3 \, a^{5} d x^{2}\right )} e^{6} + 3 \, {\left (2 \, a^{2} b^{3} d^{2} x^{4} - 2 \, a^{3} b^{2} d^{2} x^{3} - 3 \, a^{4} b d^{2} x^{2} + a^{5} d^{2} x\right )} e^{5} - {\left (4 \, a b^{4} d^{3} x^{4} - 14 \, a^{2} b^{3} d^{3} x^{3} - 6 \, a^{3} b^{2} d^{3} x^{2} + 11 \, a^{4} b d^{3} x - a^{5} d^{3}\right )} e^{4} + {\left (b^{5} d^{4} x^{4} - 11 \, a b^{4} d^{4} x^{3} + 6 \, a^{2} b^{3} d^{4} x^{2} + 14 \, a^{3} b^{2} d^{4} x - 4 \, a^{4} b d^{4}\right )} e^{3} + 3 \, {\left (b^{5} d^{5} x^{3} - 3 \, a b^{4} d^{5} x^{2} - 2 \, a^{2} b^{3} d^{5} x + 2 \, a^{3} b^{2} d^{5}\right )} e^{2} + {\left (3 \, b^{5} d^{6} x^{2} - a b^{4} d^{6} x - 4 \, a^{2} b^{3} d^{6}\right )} e\right )}}, \frac {105 \, {\left ({\left (b^{3} x^{4} + a b^{2} x^{3}\right )} e^{4} + 3 \, {\left (b^{3} d x^{3} + a b^{2} d x^{2}\right )} e^{3} + 3 \, {\left (b^{3} d^{2} x^{2} + a b^{2} d^{2} x\right )} e^{2} + {\left (b^{3} d^{3} x + a b^{2} d^{3}\right )} e\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 (15 \, b^{3} d^{3} + {\left (105 \, b^{3} x^{3} + 70 \, a b^{2} x^{2} - 14 \, a^{2} b x + 6 \, a^{3}\right )} e^{3} + {\left (245 \, b^{3} d x^{2} + 168 \, a b^{2} d x - 32 \, a^{2} b d\right )} e^{2} + {\left (161 \, b^{3} d^{2} x + 116 \, a b^{2} d^{2}\right )} e\right )} \sqrt {x e + d}}{15 \, {\left (b^{5} d^{7} x + a b^{4} d^{7} + {\left (a^{4} b x^{4} + a^{5} x^{3}\right )} e^{7} - {\left (4 \, a^{3} b^{2} d x^{4} + a^{4} b d x^{3} - 3 \, a^{5} d x^{2}\right )} e^{6} + 3 \, {\left (2 \, a^{2} b^{3} d^{2} x^{4} - 2 \, a^{3} b^{2} d^{2} x^{3} - 3 \, a^{4} b d^{2} x^{2} + a^{5} d^{2} x\right )} e^{5} - {\left (4 \, a b^{4} d^{3} x^{4} - 14 \, a^{2} b^{3} d^{3} x^{3} - 6 \, a^{3} b^{2} d^{3} x^{2} + 11 \, a^{4} b d^{3} x - a^{5} d^{3}\right )} e^{4} + {\left (b^{5} d^{4} x^{4} - 11 \, a b^{4} d^{4} x^{3} + 6 \, a^{2} b^{3} d^{4} x^{2} + 14 \, a^{3} b^{2} d^{4} x - 4 \, a^{4} b d^{4}\right )} e^{3} + 3 \, {\left (b^{5} d^{5} x^{3} - 3 \, a b^{4} d^{5} x^{2} - 2 \, a^{2} b^{3} d^{5} x + 2 \, a^{3} b^{2} d^{5}\right )} e^{2} + {\left (3 \, b^{5} d^{6} x^{2} - a b^{4} d^{6} x - 4 \, a^{2} b^{3} d^{6}\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 )^{2} \left (d + e x\right )^{\frac {7}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 304 vs.
\(2 (144) = 288\).
time = 0.96, size = 304, normalized size = 2.01 \begin {gather*} -\frac {7 \, b^{3} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e}{{\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 {\sqrt {x e + d} b^{3} e}{{\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 )}} - \frac {2 \, {\left (45 \, {\left (x e + d\right )}^{2} b^{2} e + 10 \, {\left (x e + d\right )} b^{2} d e + 3 \, b^{2} d^{2} e - 10 \, {\left (x e + d\right )} a b e^{2} - 6 \, a b d e^{2} + 3 \, a^{2} e^{3}\right )}}{15 \, {\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 {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.72, size = 198, normalized size = 1.31 \begin {gather*} -\frac {\frac {2\,e}{5\,\left (a\,e-b\,d\right )}-\frac {14\,b\,e\,\left (d+e\,x\right )}{15\,{\left (a\,e-b\,d\right )}^2}+\frac {14\,b^2\,e\,{\left (d+e\,x\right )}^2}{3\,{\left (a\,e-b\,d\right )}^3}+\frac {7\,b^3\,e\,{\left (d+e\,x\right )}^3}{{\left (a\,e-b\,d\right )}^4}}{b\,{\left (d+e\,x\right )}^{7/2}+\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{5/2}}-\frac {7\,b^{5/2}\,e\,\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 )}{{\left (a\,e-b\,d\right )}^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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