Optimal. Leaf size=107 \[ \frac {(a+b x) (A b-a B) \log (a+b x)}{b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac {(a+b x) (B d-A e) \log (d+e x)}{e \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \]
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Rubi [A] time = 0.08, antiderivative size = 107, 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, 72} \begin {gather*} \frac {(a+b x) (A b-a B) \log (a+b x)}{b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac {(a+b x) (B d-A e) \log (d+e x)}{e \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 72
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}{\left (a b+b^2 x\right ) (d+e x)} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int \left (\frac {A b-a B}{b (b d-a e) (a+b x)}+\frac {B d-A e}{b (b d-a e) (d+e x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {(A b-a B) (a+b x) \log (a+b x)}{b (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(B d-A e) (a+b x) \log (d+e x)}{e (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 66, normalized size = 0.62 \begin {gather*} \frac {(a+b x) (e (A b-a B) \log (a+b x)+b (B d-A e) \log (d+e x))}{b e \sqrt {(a+b x)^2} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.81, size = 352, normalized size = 3.29 \begin {gather*} -\frac {\left (\sqrt {b^2}+b\right ) \left (A \sqrt {b^2}-a B\right ) \log \left (\sqrt {a^2+2 a b x+b^2 x^2}-a-\sqrt {b^2} x\right )}{2 b \sqrt {b^2} (b d-a e)}+\frac {\left (b-\sqrt {b^2}\right ) \left (a B+A \sqrt {b^2}\right ) \log \left (\sqrt {a^2+2 a b x+b^2 x^2}+a-\sqrt {b^2} x\right )}{2 b \sqrt {b^2} (b d-a e)}+\frac {\left (A \sqrt {b^2} e+A b e-\sqrt {b^2} B d-b B d\right ) \log \left (-e \sqrt {a^2+2 a b x+b^2 x^2}-a e+\sqrt {b^2} e x+2 b d\right )}{2 b e (b d-a e)}+\frac {\left (A \sqrt {b^2} e-A b e-\sqrt {b^2} B d+b B d\right ) \log \left (e \sqrt {a^2+2 a b x+b^2 x^2}-a e-\sqrt {b^2} e x+2 b d\right )}{2 b e (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 53, normalized size = 0.50 \begin {gather*} -\frac {{\left (B a - A b\right )} e \log \left (b x + a\right ) - {\left (B b d - A b e\right )} \log \left (e x + d\right )}{b^{2} d e - a b e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 87, normalized size = 0.81 \begin {gather*} -\frac {{\left (B a \mathrm {sgn}\left (b x + a\right ) - A b \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | b x + a \right |}\right )}{b^{2} d - a b e} + \frac {{\left (B d \mathrm {sgn}\left (b x + a\right ) - A e \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | x e + d \right |}\right )}{b d e - a e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 76, normalized size = 0.71 \begin {gather*} -\frac {\left (b x +a \right ) \left (A b e \ln \left (b x +a \right )-A b e \ln \left (e x +d \right )-B a e \ln \left (b x +a \right )+B b d \ln \left (e x +d \right )\right )}{\sqrt {\left (b x +a \right )^{2}}\, \left (a e -b d \right ) b e} \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 {A+B\,x}{\sqrt {{\left (a+b\,x\right )}^2}\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.45, size = 226, normalized size = 2.11 \begin {gather*} - \frac {\left (- A e + B d\right ) \log {\left (x + \frac {- A a e - A b d + 2 B a d - \frac {a^{2} e \left (- A e + B d\right )}{a e - b d} + \frac {2 a b d \left (- A e + B d\right )}{a e - b d} - \frac {b^{2} d^{2} \left (- A e + B d\right )}{e \left (a e - b d\right )}}{- 2 A b e + B a e + B b d} \right )}}{e \left (a e - b d\right )} + \frac {\left (- A b + B a\right ) \log {\left (x + \frac {- A a e - A b d + 2 B a d + \frac {a^{2} e^{2} \left (- A b + B a\right )}{b \left (a e - b d\right )} - \frac {2 a d e \left (- A b + B a\right )}{a e - b d} + \frac {b d^{2} \left (- A b + B a\right )}{a e - b d}}{- 2 A b e + B a e + B b d} \right )}}{b \left (a e - b d\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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