Optimal. Leaf size=140 \[ -\frac {(b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^3 (a+b x) (d+e x)^2}+\frac {2 b (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x) (d+e x)}+\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^3 (a+b x)} \]
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Rubi [A]
time = 0.06, antiderivative size = 140, 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, 45}
\begin {gather*} \frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^3 (a+b x) (d+e x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^3 (a+b x) (d+e x)^2}+\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^3 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 45
Rule 784
Rubi steps
\begin {align*} \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^3} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x) \left (a b+b^2 x\right )}{(d+e x)^3} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^2}{(d+e x)^3} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(-b d+a e)^2}{e^2 (d+e x)^3}-\frac {2 b (b d-a e)}{e^2 (d+e x)^2}+\frac {b^2}{e^2 (d+e x)}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {(b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^3 (a+b x) (d+e x)^2}+\frac {2 b (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x) (d+e x)}+\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^3 (a+b x)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 73, normalized size = 0.52 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left ((b d-a e) (3 b d+a e+4 b e x)+2 b^2 (d+e x)^2 \log (d+e x)\right )}{2 e^3 (a+b x) (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.07, size = 118, normalized size = 0.84
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {2 b \left (a e -b d \right ) x}{e^{2}}-\frac {a^{2} e^{2}+2 a b d e -3 b^{2} d^{2}}{2 e^{3}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{2}}+\frac {b^{2} \ln \left (e x +d \right ) \sqrt {\left (b x +a \right )^{2}}}{e^{3} \left (b x +a \right )}\) | \(98\) |
default | \(\frac {\mathrm {csgn}\left (b x +a \right ) \left (2 \ln \left (-b e x -b d \right ) b^{2} e^{2} x^{2}+4 \ln \left (-b e x -b d \right ) b^{2} d e x +2 \ln \left (-b e x -b d \right ) b^{2} d^{2}-4 a b \,e^{2} x +4 b^{2} d e x -a^{2} e^{2}-2 a b d e +3 b^{2} d^{2}\right )}{2 e^{3} \left (e x +d \right )^{2}}\) | \(118\) |
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 [A]
time = 1.97, size = 97, normalized size = 0.69 \begin {gather*} \frac {3 \, b^{2} d^{2} - {\left (4 \, a b x + a^{2}\right )} e^{2} + 2 \, {\left (2 \, b^{2} d x - a b d\right )} e + 2 \, {\left (b^{2} x^{2} e^{2} + 2 \, b^{2} d x e + b^{2} d^{2}\right )} \log \left (x e + d\right )}{2 \, {\left (x^{2} e^{5} + 2 \, d x e^{4} + d^{2} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.27, size = 80, normalized size = 0.57 \begin {gather*} \frac {b^{2} \log {\left (d + e x \right )}}{e^{3}} + \frac {- a^{2} e^{2} - 2 a b d e + 3 b^{2} d^{2} + x \left (- 4 a b e^{2} + 4 b^{2} d e\right )}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.78, size = 105, normalized size = 0.75 \begin {gather*} b^{2} e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (4 \, {\left (b^{2} d \mathrm {sgn}\left (b x + a\right ) - a b e \mathrm {sgn}\left (b x + a\right )\right )} x + {\left (3 \, b^{2} d^{2} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b d e \mathrm {sgn}\left (b x + a\right ) - a^{2} e^{2} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-1\right )}\right )} e^{\left (-2\right )}}{2 \, {\left (x e + d\right )}^{2}} \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 = -0.01 \begin {gather*} \int \frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (a+b\,x\right )}{{\left (d+e\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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