Optimal. Leaf size=271 \[ \frac{b (a+b x) (A b-a B)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^3}+\frac{(a+b x) (A b-a B)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^2}-\frac{(a+b x) (B d-A e)}{3 e \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)}+\frac{b^2 (a+b x) (A b-a B) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{b^2 (a+b x) (A b-a B) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4} \]
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Rubi [A] time = 0.177455, antiderivative size = 271, normalized size of antiderivative = 1., 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} \[ \frac{b (a+b x) (A b-a B)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^3}+\frac{(a+b x) (A b-a B)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^2}-\frac{(a+b x) (B d-A e)}{3 e \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)}+\frac{b^2 (a+b x) (A b-a B) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{b^2 (a+b x) (A b-a B) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4} \]
Antiderivative was successfully verified.
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Rule 770
Rule 77
Rubi steps
\begin{align*} \int \frac{A+B x}{(d+e x)^4 \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)^4} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (a b+b^2 x\right ) \int \left (\frac{b^2 (A b-a B)}{(b d-a e)^4 (a+b x)}+\frac{B d-A e}{b (b d-a e) (d+e x)^4}+\frac{(-A b+a B) e}{b (b d-a e)^2 (d+e x)^3}+\frac{(-A b+a B) e}{(b d-a e)^3 (d+e x)^2}-\frac{b (A b-a B) e}{(-b d+a e)^4 (d+e x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{(B d-A e) (a+b x)}{3 e (b d-a e) (d+e x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) (a+b x)}{2 (b d-a e)^2 (d+e x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b (A b-a B) (a+b x)}{(b d-a e)^3 (d+e x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2 (A b-a B) (a+b x) \log (a+b x)}{(b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b^2 (A b-a B) (a+b x) \log (d+e x)}{(b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.18202, size = 168, normalized size = 0.62 \[ \frac{(a+b x) \left (6 b^2 e (d+e x)^3 (A b-a B) \log (a+b x)-6 b^2 e (d+e x)^3 (A b-a B) \log (d+e x)+3 e (d+e x) (A b-a B) (b d-a e)^2+6 b e (d+e x)^2 (A b-a B) (b d-a e)-2 (b d-a e)^3 (B d-A e)\right )}{6 e \sqrt{(a+b x)^2} (d+e x)^3 (b d-a e)^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.019, size = 544, normalized size = 2. \begin{align*} -{\frac{ \left ( bx+a \right ) \left ( 6\,B{x}^{2}a{b}^{2}d{e}^{3}-15\,Ax{b}^{3}{d}^{2}{e}^{2}-3\,Ax{a}^{2}b{e}^{4}+6\,A\ln \left ( ex+d \right ){b}^{3}{d}^{3}e-6\,B{x}^{2}{a}^{2}b{e}^{4}+6\,A{x}^{2}a{b}^{2}{e}^{4}-6\,A{x}^{2}{b}^{3}d{e}^{3}-9\,Ad{e}^{3}{a}^{2}b+2\,A{a}^{3}{e}^{4}+2\,B{b}^{3}{d}^{4}+3\,Ba{b}^{2}{d}^{3}e-6\,B{a}^{2}b{d}^{2}{e}^{2}+18\,Aa{b}^{2}{d}^{2}{e}^{2}-18\,B\ln \left ( ex+d \right ){x}^{2}a{b}^{2}d{e}^{3}-18\,B\ln \left ( ex+d \right ) xa{b}^{2}{d}^{2}{e}^{2}+18\,A\ln \left ( ex+d \right ){x}^{2}{b}^{3}d{e}^{3}-18\,A\ln \left ( bx+a \right ){x}^{2}{b}^{3}d{e}^{3}+18\,A\ln \left ( ex+d \right ) x{b}^{3}{d}^{2}{e}^{2}-18\,A\ln \left ( bx+a \right ) x{b}^{3}{d}^{2}{e}^{2}+6\,B\ln \left ( bx+a \right ) a{b}^{2}{d}^{3}e+6\,B\ln \left ( bx+a \right ){x}^{3}a{b}^{2}{e}^{4}-6\,B\ln \left ( ex+d \right ){x}^{3}a{b}^{2}{e}^{4}-18\,Bx{a}^{2}bd{e}^{3}+15\,Bxa{b}^{2}{d}^{2}{e}^{2}+18\,Axa{b}^{2}d{e}^{3}-6\,B\ln \left ( ex+d \right ) a{b}^{2}{d}^{3}e+3\,Bx{a}^{3}{e}^{4}+Bd{e}^{3}{a}^{3}-11\,A{b}^{3}{d}^{3}e-6\,A\ln \left ( bx+a \right ){x}^{3}{b}^{3}{e}^{4}-6\,A\ln \left ( bx+a \right ){b}^{3}{d}^{3}e+6\,A\ln \left ( ex+d \right ){x}^{3}{b}^{3}{e}^{4}+18\,B\ln \left ( bx+a \right ){x}^{2}a{b}^{2}d{e}^{3}+18\,B\ln \left ( bx+a \right ) xa{b}^{2}{d}^{2}{e}^{2} \right ) }{6\, \left ( ae-bd \right ) ^{4}e \left ( ex+d \right ) ^{3}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.40367, size = 1206, normalized size = 4.45 \begin{align*} -\frac{2 \, B b^{3} d^{4} + 2 \, A a^{3} e^{4} +{\left (3 \, B a b^{2} - 11 \, A b^{3}\right )} d^{3} e - 6 \,{\left (B a^{2} b - 3 \, A a b^{2}\right )} d^{2} e^{2} +{\left (B a^{3} - 9 \, A a^{2} b\right )} d e^{3} + 6 \,{\left ({\left (B a b^{2} - A b^{3}\right )} d e^{3} -{\left (B a^{2} b - A a b^{2}\right )} e^{4}\right )} x^{2} + 3 \,{\left (5 \,{\left (B a b^{2} - A b^{3}\right )} d^{2} e^{2} - 6 \,{\left (B a^{2} b - A a b^{2}\right )} d e^{3} +{\left (B a^{3} - A a^{2} b\right )} e^{4}\right )} x + 6 \,{\left ({\left (B a b^{2} - A b^{3}\right )} e^{4} x^{3} + 3 \,{\left (B a b^{2} - A b^{3}\right )} d e^{3} x^{2} + 3 \,{\left (B a b^{2} - A b^{3}\right )} d^{2} e^{2} x +{\left (B a b^{2} - A b^{3}\right )} d^{3} e\right )} \log \left (b x + a\right ) - 6 \,{\left ({\left (B a b^{2} - A b^{3}\right )} e^{4} x^{3} + 3 \,{\left (B a b^{2} - A b^{3}\right )} d e^{3} x^{2} + 3 \,{\left (B a b^{2} - A b^{3}\right )} d^{2} e^{2} x +{\left (B a b^{2} - A b^{3}\right )} d^{3} e\right )} \log \left (e x + d\right )}{6 \,{\left (b^{4} d^{7} e - 4 \, a b^{3} d^{6} e^{2} + 6 \, a^{2} b^{2} d^{5} e^{3} - 4 \, a^{3} b d^{4} e^{4} + a^{4} d^{3} e^{5} +{\left (b^{4} d^{4} e^{4} - 4 \, a b^{3} d^{3} e^{5} + 6 \, a^{2} b^{2} d^{2} e^{6} - 4 \, a^{3} b d e^{7} + a^{4} e^{8}\right )} x^{3} + 3 \,{\left (b^{4} d^{5} e^{3} - 4 \, a b^{3} d^{4} e^{4} + 6 \, a^{2} b^{2} d^{3} e^{5} - 4 \, a^{3} b d^{2} e^{6} + a^{4} d e^{7}\right )} x^{2} + 3 \,{\left (b^{4} d^{6} e^{2} - 4 \, a b^{3} d^{5} e^{3} + 6 \, a^{2} b^{2} d^{4} e^{4} - 4 \, a^{3} b d^{3} e^{5} + a^{4} d^{2} e^{6}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 4.07225, size = 818, normalized size = 3.02 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15431, size = 667, normalized size = 2.46 \begin{align*} -\frac{{\left (B a b^{3} \mathrm{sgn}\left (b x + a\right ) - A b^{4} \mathrm{sgn}\left (b x + a\right )\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}} + \frac{{\left (B a b^{2} e \mathrm{sgn}\left (b x + a\right ) - A b^{3} e \mathrm{sgn}\left (b x + a\right )\right )} \log \left ({\left | x e + d \right |}\right )}{b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}} - \frac{{\left (2 \, B b^{3} d^{4} \mathrm{sgn}\left (b x + a\right ) + 3 \, B a b^{2} d^{3} e \mathrm{sgn}\left (b x + a\right ) - 11 \, A b^{3} d^{3} e \mathrm{sgn}\left (b x + a\right ) - 6 \, B a^{2} b d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 18 \, A a b^{2} d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + B a^{3} d e^{3} \mathrm{sgn}\left (b x + a\right ) - 9 \, A a^{2} b d e^{3} \mathrm{sgn}\left (b x + a\right ) + 2 \, A a^{3} e^{4} \mathrm{sgn}\left (b x + a\right ) + 6 \,{\left (B a b^{2} d e^{3} \mathrm{sgn}\left (b x + a\right ) - A b^{3} d e^{3} \mathrm{sgn}\left (b x + a\right ) - B a^{2} b e^{4} \mathrm{sgn}\left (b x + a\right ) + A a b^{2} e^{4} \mathrm{sgn}\left (b x + a\right )\right )} x^{2} + 3 \,{\left (5 \, B a b^{2} d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) - 5 \, A b^{3} d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) - 6 \, B a^{2} b d e^{3} \mathrm{sgn}\left (b x + a\right ) + 6 \, A a b^{2} d e^{3} \mathrm{sgn}\left (b x + a\right ) + B a^{3} e^{4} \mathrm{sgn}\left (b x + a\right ) - A a^{2} b e^{4} \mathrm{sgn}\left (b x + a\right )\right )} x\right )} e^{\left (-1\right )}}{6 \,{\left (b d - a e\right )}^{4}{\left (x e + d\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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