Optimal. Leaf size=132 \[ \frac {b^2 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^2 (a+b x)}-\frac {(b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x) (d+e x)}-\frac {2 b (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^3 (a+b x)} \]
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Rubi [A]
time = 0.06, antiderivative size = 132, 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 {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^3 (a+b x) (d+e x)}-\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^3 (a+b x)}+\frac {b^2 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^2 (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)^2} \, 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)^2} \, 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)^2} \, 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^2}{e^2}+\frac {(-b d+a e)^2}{e^2 (d+e x)^2}-\frac {2 b (b d-a e)}{e^2 (d+e x)}\right ) \, dx}{a b+b^2 x}\\ &=\frac {b^2 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^2 (a+b x)}-\frac {(b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x) (d+e x)}-\frac {2 b (b d-a e) \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 = 0.04, size = 87, normalized size = 0.66 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (2 a b d e-a^2 e^2+b^2 \left (-d^2+d e x+e^2 x^2\right )-2 b (b d-a e) (d+e x) \log (d+e x)\right )}{e^3 (a+b x) (d+e x)} \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 = 0.06, size = 139, normalized size = 1.05
method | result | size |
risch | \(\frac {b^{2} x \sqrt {\left (b x +a \right )^{2}}}{e^{2} \left (b x +a \right )}-\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{\left (b x +a \right ) e^{3} \left (e x +d \right )}+\frac {2 \sqrt {\left (b x +a \right )^{2}}\, b \left (a e -b d \right ) \ln \left (e x +d \right )}{\left (b x +a \right ) e^{3}}\) | \(111\) |
default | \(\frac {\mathrm {csgn}\left (b x +a \right ) \left (2 \ln \left (-b e x -b d \right ) a b \,e^{2} x -2 \ln \left (-b e x -b d \right ) b^{2} d e x +b^{2} e^{2} x^{2}+2 \ln \left (-b e x -b d \right ) a b d e -2 \ln \left (-b e x -b d \right ) b^{2} d^{2}+a b \,e^{2} x +b^{2} d e x -a^{2} e^{2}+3 a b d e -b^{2} d^{2}\right )}{e^{3} \left (e x +d \right )}\) | \(139\) |
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 = 1.95, size = 93, normalized size = 0.70 \begin {gather*} -\frac {b^{2} d^{2} - {\left (b^{2} x^{2} - a^{2}\right )} e^{2} - {\left (b^{2} d x + 2 \, a b d\right )} e + 2 \, {\left (b^{2} d^{2} - a b x e^{2} + {\left (b^{2} d x - a b d\right )} e\right )} \log \left (x e + d\right )}{x e^{4} + d e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 60, normalized size = 0.45 \begin {gather*} \frac {b^{2} x}{e^{2}} + \frac {2 b \left (a e - b d\right ) \log {\left (d + e x \right )}}{e^{3}} + \frac {- a^{2} e^{2} + 2 a b d e - b^{2} d^{2}}{d e^{3} + e^{4} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.79, size = 101, normalized size = 0.77 \begin {gather*} b^{2} x e^{\left (-2\right )} \mathrm {sgn}\left (b x + a\right ) - 2 \, {\left (b^{2} d \mathrm {sgn}\left (b x + a\right ) - a b e \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) - \frac {{\left (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 (-3\right )}}{x e + d} \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 )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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