Optimal. Leaf size=134 \[ \frac {e (a+b x) (A+B x)^2}{2 b B \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (A b-a B) (b d-a e) \log (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B x (a+b x) (b d-a e)}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.08, antiderivative size = 134, 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 x) (A b-a B) (b d-a e) \log (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e (a+b x) (A+B x)^2}{2 b B \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B x (a+b x) (b d-a e)}{b^2 \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)}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {(A+B x) (d+e x)}{a b+b^2 x} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int \left (\frac {B (b d-a e)}{b^3}+\frac {(A b-a B) (b d-a e)}{b^3 (a+b x)}+\frac {e (A+B x)}{b^2}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {B (b d-a e) x (a+b x)}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e (a+b x) (A+B x)^2}{2 b B \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) (b d-a 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 = 0.04, size = 72, normalized size = 0.54 \begin {gather*} \frac {(a+b x) (b x (b (2 A e+2 B d+B e x)-2 a B e)+2 (A b-a B) (b d-a e) \log (a+b x))}{2 b^3 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.72, size = 344, normalized size = 2.57 \begin {gather*} \frac {\sqrt {a^2+2 a b x+b^2 x^2} (-3 a B e+2 A b e+2 b B d+b B e x)}{4 b^3}+\frac {\log \left (\sqrt {a^2+2 a b x+b^2 x^2}-a-\sqrt {b^2} x\right ) \left (-a^2 \sqrt {b^2} B e-a^2 b B e+a A b^2 e+a A \sqrt {b^2} b e+a b^2 B d+a \sqrt {b^2} b B d-A b^3 d-A \left (b^2\right )^{3/2} d\right )}{2 b^3 \sqrt {b^2}}+\frac {\log \left (\sqrt {a^2+2 a b x+b^2 x^2}+a-\sqrt {b^2} x\right ) \left (a^2 \sqrt {b^2} B e-a^2 b B e+a A b^2 e-a A \sqrt {b^2} b e+a b^2 B d-a \sqrt {b^2} b B d-A b^3 d+A \left (b^2\right )^{3/2} d\right )}{2 b^3 \sqrt {b^2}}+\frac {2 a B e x-2 A b e x-2 b B d x-b B e x^2}{4 b \sqrt {b^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 75, normalized size = 0.56 \begin {gather*} \frac {B b^{2} e x^{2} + 2 \, {\left (B b^{2} d - {\left (B a b - A b^{2}\right )} e\right )} x - 2 \, {\left ({\left (B a b - A b^{2}\right )} d - {\left (B a^{2} - A a b\right )} e\right )} \log \left (b x + a\right )}{2 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 122, normalized size = 0.91 \begin {gather*} \frac {B b x^{2} e \mathrm {sgn}\left (b x + a\right ) + 2 \, B b d x \mathrm {sgn}\left (b x + a\right ) - 2 \, B a x e \mathrm {sgn}\left (b x + a\right ) + 2 \, A b x e \mathrm {sgn}\left (b x + a\right )}{2 \, b^{2}} - \frac {{\left (B a b d \mathrm {sgn}\left (b x + a\right ) - A b^{2} d \mathrm {sgn}\left (b x + a\right ) - B a^{2} e \mathrm {sgn}\left (b x + a\right ) + A a b e \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | b x + a \right |}\right )}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 104, normalized size = 0.78 \begin {gather*} -\frac {\left (b x +a \right ) \left (-B \,b^{2} e \,x^{2}+2 A a b e \ln \left (b x +a \right )-2 A \,b^{2} d \ln \left (b x +a \right )-2 A \,b^{2} e x -2 B \,a^{2} e \ln \left (b x +a \right )+2 B a b d \ln \left (b x +a \right )+2 B a b e x -2 B \,b^{2} d x \right )}{2 \sqrt {\left (b x +a \right )^{2}}\, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 101, normalized size = 0.75 \begin {gather*} \frac {B e x^{2}}{2 \, b} - \frac {B a e x}{b^{2}} + \frac {A d \log \left (x + \frac {a}{b}\right )}{b} + \frac {B a^{2} e \log \left (x + \frac {a}{b}\right )}{b^{3}} - \frac {{\left (B d + A e\right )} a \log \left (x + \frac {a}{b}\right )}{b^{2}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} {\left (B d + A e\right )}}{b^{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 )}{\sqrt {{\left (a+b\,x\right )}^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.29, size = 53, normalized size = 0.40 \begin {gather*} \frac {B e x^{2}}{2 b} + x \left (\frac {A e}{b} - \frac {B a e}{b^{2}} + \frac {B d}{b}\right ) + \frac {\left (- A b + B a\right ) \left (a e - b d\right ) \log {\left (a + b x \right )}}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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