Optimal. Leaf size=127 \[ -\frac {(A b-a B) (b d-a e)}{2 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {-2 a B e+A b e+b B d}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B e (a+b x) \log (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.10, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {770, 77} \begin {gather*} -\frac {(A b-a B) (b d-a e)}{2 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {-2 a B e+A b e+b B d}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B e (a+b x) \log (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 770
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {(A+B x) (d+e x)}{\left (a b+b^2 x\right )^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \left (\frac {(A b-a B) (b d-a e)}{b^5 (a+b x)^3}+\frac {b B d+A b e-2 a B e}{b^5 (a+b x)^2}+\frac {B e}{b^5 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {b B d+A b e-2 a B e}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (b d-a e)}{2 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B e (a+b x) \log (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 86, normalized size = 0.68 \begin {gather*} \frac {B \left (3 a^2 e-a b d+4 a b e x-2 b^2 d x\right )-A b (a e+b d+2 b e x)+2 B e (a+b x)^2 \log (a+b x)}{2 b^3 (a+b x) \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 2.14, size = 1723, normalized size = 13.57
result too large to display
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 110, normalized size = 0.87 \begin {gather*} -\frac {{\left (B a b + A b^{2}\right )} d - {\left (3 \, B a^{2} - A a b\right )} e + 2 \, {\left (B b^{2} d - {\left (2 \, B a b - A b^{2}\right )} e\right )} x - 2 \, {\left (B b^{2} e x^{2} + 2 \, B a b e x + B a^{2} e\right )} \log \left (b x + a\right )}{2 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 109, normalized size = 0.86 \begin {gather*} -\frac {\left (-2 B \,b^{2} e \,x^{2} \ln \left (b x +a \right )-4 B a b e x \ln \left (b x +a \right )+2 A \,b^{2} e x -2 B \,a^{2} e \ln \left (b x +a \right )-4 B a b e x +2 B \,b^{2} d x +A a b e +A \,b^{2} d -3 B \,a^{2} e +B a b d \right ) \left (b x +a \right )}{2 \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 120, normalized size = 0.94 \begin {gather*} \frac {B e \log \left (x + \frac {a}{b}\right )}{b^{3}} - \frac {B d + A e}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} + \frac {2 \, B a e x}{b^{4} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {A d}{2 \, b^{3} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {3 \, B a^{2} e}{2 \, b^{5} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {{\left (B d + A e\right )} a}{2 \, b^{4} {\left (x + \frac {a}{b}\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 {\left (A+B\,x\right )\,\left (d+e\,x\right )}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \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 {\left (A + B x\right ) \left (d + e x\right )}{\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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