Optimal. Leaf size=144 \[ -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{e^3 (a+b x) (d+e x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x) (-a B e-A b e+2 b B d)}{e^3 (a+b x)}+\frac {b B x \sqrt {a^2+2 a b x+b^2 x^2}}{e^2 (a+b x)} \]
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Rubi [A] time = 0.10, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {770, 77} \begin {gather*} -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{e^3 (a+b x) (d+e x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x) (-a B e-A b e+2 b B d)}{e^3 (a+b x)}+\frac {b B 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 77
Rule 770
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 {\left (a b+b^2 x\right ) (A+B x)}{(d+e x)^2} \, dx}{a b+b^2 x}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^2 B}{e^2}-\frac {b (b d-a e) (-B d+A e)}{e^2 (d+e x)^2}+\frac {b (-2 b B d+A b e+a B e)}{e^2 (d+e x)}\right ) \, dx}{a b+b^2 x}\\ &=\frac {b B x \sqrt {a^2+2 a b x+b^2 x^2}}{e^2 (a+b x)}-\frac {(b d-a e) (B d-A e) \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 b e-a B 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 = 0.05, size = 96, normalized size = 0.67 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (-(d+e x) \log (d+e x) (-a B e-A b e+2 b B d)+a e (B d-A e)+b \left (A d e-B d^2+B d e x+B e^2 x^2\right )\right )}{e^3 (a+b x) (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 2.42, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.42, size = 102, normalized size = 0.71 \begin {gather*} \frac {B b e^{2} x^{2} + B b d e x - B b d^{2} - A a e^{2} + {\left (B a + A b\right )} d e - {\left (2 \, B b d^{2} - {\left (B a + A b\right )} d e + {\left (2 \, B b d e - {\left (B a + A b\right )} e^{2}\right )} x\right )} \log \left (e x + d\right )}{e^{4} x + d e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 123, normalized size = 0.85 \begin {gather*} B b x e^{\left (-2\right )} \mathrm {sgn}\left (b x + a\right ) - {\left (2 \, B b d \mathrm {sgn}\left (b x + a\right ) - B a e \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 b d^{2} \mathrm {sgn}\left (b x + a\right ) - B a d e \mathrm {sgn}\left (b x + a\right ) - A b d e \mathrm {sgn}\left (b x + a\right ) + A a 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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maple [C] time = 0.07, size = 158, normalized size = 1.10 \begin {gather*} \frac {\left (A b \,e^{2} x \ln \left (b e x +b d \right )+B a \,e^{2} x \ln \left (b e x +b d \right )-2 B b d e x \ln \left (b e x +b d \right )+B b \,e^{2} x^{2}+A b d e \ln \left (b e x +b d \right )+B a d e \ln \left (b e x +b d \right )+B a \,e^{2} x -2 B b \,d^{2} \ln \left (b e x +b d \right )+B b d e x -A a \,e^{2}+A b d e +2 B a d e -B b \,d^{2}\right ) \mathrm {csgn}\left (b x +a \right )}{\left (e x +d \right ) e^{3}} \end {gather*}
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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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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sympy [A] time = 0.46, size = 71, normalized size = 0.49 \begin {gather*} \frac {B b x}{e^{2}} + \frac {- A a e^{2} + A b d e + B a d e - B b d^{2}}{d e^{3} + e^{4} x} + \frac {\left (A b e + B a e - 2 B b d\right ) \log {\left (d + e x \right )}}{e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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