Optimal. Leaf size=260 \[ -\frac {3 b e^2}{(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b}{3 (b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b e}{(b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^3 (a+b x)}{(b d-a e)^4 (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {4 b e^3 (a+b x) \log (a+b x)}{(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {4 b e^3 (a+b x) \log (d+e x)}{(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A]
time = 0.13, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {784, 21, 46}
\begin {gather*} -\frac {e^3 (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^4}-\frac {4 b e^3 (a+b x) \log (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac {4 b e^3 (a+b x) \log (d+e x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac {3 b e^2}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac {b e}{(a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac {b}{3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 46
Rule 784
Rubi steps
\begin {align*} \int \frac {a+b x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {a+b x}{\left (a b+b^2 x\right )^5 (d+e x)^2} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int \frac {1}{(a+b x)^4 (d+e x)^2} \, dx}{b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int \left (\frac {b^2}{(b d-a e)^2 (a+b x)^4}-\frac {2 b^2 e}{(b d-a e)^3 (a+b x)^3}+\frac {3 b^2 e^2}{(b d-a e)^4 (a+b x)^2}-\frac {4 b^2 e^3}{(b d-a e)^5 (a+b x)}+\frac {e^4}{(b d-a e)^4 (d+e x)^2}+\frac {4 b e^4}{(b d-a e)^5 (d+e x)}\right ) \, dx}{b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {3 b e^2}{(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b}{3 (b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b e}{(b d-a e)^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^3 (a+b x)}{(b d-a e)^4 (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {4 b e^3 (a+b x) \log (a+b x)}{(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {4 b e^3 (a+b x) \log (d+e x)}{(b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 144, normalized size = 0.55 \begin {gather*} \frac {-b (b d-a e)^3+3 b e (b d-a e)^2 (a+b x)-9 b e^2 (b d-a e) (a+b x)^2+\frac {3 e^3 (-b d+a e) (a+b x)^3}{d+e x}-12 b e^3 (a+b x)^3 \log (a+b x)+12 b e^3 (a+b x)^3 \log (d+e x)}{3 (b d-a e)^5 \left ((a+b x)^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(482\) vs.
\(2(192)=384\).
time = 0.13, size = 483, normalized size = 1.86
method | result | size |
default | \(\frac {\left (-36 \ln \left (e x +d \right ) a^{2} b^{2} d \,e^{3} x -36 \ln \left (e x +d \right ) a \,b^{3} d \,e^{3} x^{2}+36 \ln \left (b x +a \right ) a^{2} b^{2} d \,e^{3} x +36 \ln \left (b x +a \right ) a \,b^{3} d \,e^{3} x^{2}+12 \ln \left (b x +a \right ) b^{4} d \,e^{3} x^{3}+36 \ln \left (b x +a \right ) a^{2} b^{2} e^{4} x^{2}+36 \ln \left (b x +a \right ) a \,b^{3} e^{4} x^{3}+12 \ln \left (b x +a \right ) a^{3} b d \,e^{3}+12 \ln \left (b x +a \right ) a^{3} b \,e^{4} x -36 \ln \left (e x +d \right ) a \,b^{3} e^{4} x^{3}-12 \ln \left (e x +d \right ) b^{4} d \,e^{3} x^{3}-36 \ln \left (e x +d \right ) a^{2} b^{2} e^{4} x^{2}-12 \ln \left (e x +d \right ) a^{3} b \,e^{4} x +18 a \,b^{3} d^{2} e^{2} x -12 a \,b^{3} e^{4} x^{3}+12 b^{4} d \,e^{3} x^{3}-30 a^{2} b^{2} e^{4} x^{2}+6 b^{4} d^{2} e^{2} x^{2}-22 a^{3} b \,e^{4} x -2 b^{4} d^{3} e x +12 \ln \left (b x +a \right ) b^{4} e^{4} x^{4}+b^{4} d^{4}-3 a^{4} e^{4}-12 \ln \left (e x +d \right ) b^{4} e^{4} x^{4}-6 a \,b^{3} d^{3} e +18 a^{2} b^{2} d^{2} e^{2}-10 a^{3} b d \,e^{3}+24 a \,b^{3} d \,e^{3} x^{2}+6 a^{2} b^{2} d \,e^{3} x -12 \ln \left (e x +d \right ) a^{3} b d \,e^{3}\right ) \left (b x +a \right )^{2}}{3 \left (e x +d \right ) \left (a e -b d \right )^{5} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(483\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {4 e^{3} b^{3} x^{3}}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}-\frac {2 b^{2} \left (5 a e +b d \right ) e^{2} x^{2}}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}-\frac {2 \left (11 a^{2} e^{2}+8 a b d e -b^{2} d^{2}\right ) b e x}{3 \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}-\frac {3 a^{3} e^{3}+13 a^{2} b d \,e^{2}-5 a \,b^{2} d^{2} e +b^{3} d^{3}}{3 \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}\right )}{\left (b x +a \right )^{4} \left (e x +d \right )}+\frac {4 \sqrt {\left (b x +a \right )^{2}}\, e^{3} b \ln \left (-b x -a \right )}{\left (b x +a \right ) \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}-\frac {4 \sqrt {\left (b x +a \right )^{2}}\, e^{3} b \ln \left (e x +d \right )}{\left (b x +a \right ) \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}\) | \(518\) |
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. 708 vs.
\(2 (197) = 394\).
time = 2.38, size = 708, normalized size = 2.72 \begin {gather*} -\frac {b^{4} d^{4} - {\left (12 \, a b^{3} x^{3} + 30 \, a^{2} b^{2} x^{2} + 22 \, a^{3} b x + 3 \, a^{4}\right )} e^{4} + 2 \, {\left (6 \, b^{4} d x^{3} + 12 \, a b^{3} d x^{2} + 3 \, a^{2} b^{2} d x - 5 \, a^{3} b d\right )} e^{3} + 6 \, {\left (b^{4} d^{2} x^{2} + 3 \, a b^{3} d^{2} x + 3 \, a^{2} b^{2} d^{2}\right )} e^{2} - 2 \, {\left (b^{4} d^{3} x + 3 \, a b^{3} d^{3}\right )} e + 12 \, {\left ({\left (b^{4} x^{4} + 3 \, a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} + a^{3} b x\right )} e^{4} + {\left (b^{4} d x^{3} + 3 \, a b^{3} d x^{2} + 3 \, a^{2} b^{2} d x + a^{3} b d\right )} e^{3}\right )} \log \left (b x + a\right ) - 12 \, {\left ({\left (b^{4} x^{4} + 3 \, a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} + a^{3} b x\right )} e^{4} + {\left (b^{4} d x^{3} + 3 \, a b^{3} d x^{2} + 3 \, a^{2} b^{2} d x + a^{3} b d\right )} e^{3}\right )} \log \left (x e + d\right )}{3 \, {\left (b^{8} d^{6} x^{3} + 3 \, a b^{7} d^{6} x^{2} + 3 \, a^{2} b^{6} d^{6} x + a^{3} b^{5} d^{6} - {\left (a^{5} b^{3} x^{4} + 3 \, a^{6} b^{2} x^{3} + 3 \, a^{7} b x^{2} + a^{8} x\right )} e^{6} + {\left (5 \, a^{4} b^{4} d x^{4} + 14 \, a^{5} b^{3} d x^{3} + 12 \, a^{6} b^{2} d x^{2} + 2 \, a^{7} b d x - a^{8} d\right )} e^{5} - 5 \, {\left (2 \, a^{3} b^{5} d^{2} x^{4} + 5 \, a^{4} b^{4} d^{2} x^{3} + 3 \, a^{5} b^{3} d^{2} x^{2} - a^{6} b^{2} d^{2} x - a^{7} b d^{2}\right )} e^{4} + 10 \, {\left (a^{2} b^{6} d^{3} x^{4} + 2 \, a^{3} b^{5} d^{3} x^{3} - 2 \, a^{5} b^{3} d^{3} x - a^{6} b^{2} d^{3}\right )} e^{3} - 5 \, {\left (a b^{7} d^{4} x^{4} + a^{2} b^{6} d^{4} x^{3} - 3 \, a^{3} b^{5} d^{4} x^{2} - 5 \, a^{4} b^{4} d^{4} x - 2 \, a^{5} b^{3} d^{4}\right )} e^{2} + {\left (b^{8} d^{5} x^{4} - 2 \, a b^{7} d^{5} x^{3} - 12 \, a^{2} b^{6} d^{5} x^{2} - 14 \, a^{3} b^{5} d^{5} x - 5 \, a^{4} b^{4} d^{5}\right )} e\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 413 vs.
\(2 (197) = 394\).
time = 1.60, size = 413, normalized size = 1.59 \begin {gather*} -\frac {4 \, b^{2} e^{3} \log \left ({\left | b x + a \right |}\right )}{b^{6} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{5} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{4} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{3} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{5} b e^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {4 \, b e^{4} \log \left ({\left | x e + d \right |}\right )}{b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b d e^{5} \mathrm {sgn}\left (b x + a\right ) - a^{5} e^{6} \mathrm {sgn}\left (b x + a\right )} - \frac {b^{4} d^{4} - 6 \, a b^{3} d^{3} e + 18 \, a^{2} b^{2} d^{2} e^{2} - 10 \, a^{3} b d e^{3} - 3 \, a^{4} e^{4} + 12 \, {\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \, {\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} - 5 \, a^{2} b^{2} e^{4}\right )} x^{2} - 2 \, {\left (b^{4} d^{3} e - 9 \, a b^{3} d^{2} e^{2} - 3 \, a^{2} b^{2} d e^{3} + 11 \, a^{3} b e^{4}\right )} x}{3 \, {\left (b d - a e\right )}^{5} {\left (b x + a\right )}^{3} {\left (x e + d\right )} \mathrm {sgn}\left (b x + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {a+b\,x}{{\left (d+e\,x\right )}^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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