Optimal. Leaf size=124 \[ \frac {g^2 (a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 b}-\frac {B g^2 n (b c-a d)^3 \log (c+d x)}{3 b d^3}+\frac {B g^2 n x (b c-a d)^2}{3 d^2}-\frac {B g^2 n (a+b x)^2 (b c-a d)}{6 b d} \]
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Rubi [A] time = 0.09, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2525, 12, 43} \[ \frac {g^2 (a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 b}+\frac {B g^2 n x (b c-a d)^2}{3 d^2}-\frac {B g^2 n (b c-a d)^3 \log (c+d x)}{3 b d^3}-\frac {B g^2 n (a+b x)^2 (b c-a d)}{6 b d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2525
Rubi steps
\begin {align*} \int (a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx &=\frac {g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b}-\frac {(B n) \int \frac {(b c-a d) g^3 (a+b x)^2}{c+d x} \, dx}{3 b g}\\ &=\frac {g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b}-\frac {\left (B (b c-a d) g^2 n\right ) \int \frac {(a+b x)^2}{c+d x} \, dx}{3 b}\\ &=\frac {g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b}-\frac {\left (B (b c-a d) g^2 n\right ) \int \left (-\frac {b (b c-a d)}{d^2}+\frac {b (a+b x)}{d}+\frac {(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{3 b}\\ &=\frac {B (b c-a d)^2 g^2 n x}{3 d^2}-\frac {B (b c-a d) g^2 n (a+b x)^2}{6 b d}+\frac {g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b}-\frac {B (b c-a d)^3 g^2 n \log (c+d x)}{3 b d^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 103, normalized size = 0.83 \[ \frac {g^2 \left (\frac {B n (a d-b c) \left (d \left (a^2 d+4 a b d x+b^2 x (d x-2 c)\right )+2 (b c-a d)^2 \log (c+d x)\right )}{2 d^3}+(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )\right )}{3 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.77, size = 296, normalized size = 2.39 \[ \frac {2 \, A b^{3} d^{3} g^{2} x^{3} + 2 \, B a^{3} d^{3} g^{2} n \log \left (b x + a\right ) - 2 \, {\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2}\right )} g^{2} n \log \left (d x + c\right ) + {\left (6 \, A a b^{2} d^{3} g^{2} - {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} g^{2} n\right )} x^{2} + 2 \, {\left (3 \, A a^{2} b d^{3} g^{2} + {\left (B b^{3} c^{2} d - 3 \, B a b^{2} c d^{2} + 2 \, B a^{2} b d^{3}\right )} g^{2} n\right )} x + 2 \, {\left (B b^{3} d^{3} g^{2} x^{3} + 3 \, B a b^{2} d^{3} g^{2} x^{2} + 3 \, B a^{2} b d^{3} g^{2} x\right )} \log \relax (e) + 2 \, {\left (B b^{3} d^{3} g^{2} n x^{3} + 3 \, B a b^{2} d^{3} g^{2} n x^{2} + 3 \, B a^{2} b d^{3} g^{2} n x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{6 \, b d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.29, size = 1836, normalized size = 14.81 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.27, size = 0, normalized size = 0.00 \[ \int \left (b g x +a g \right )^{2} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.39, size = 309, normalized size = 2.49 \[ \frac {1}{3} \, B b^{2} g^{2} x^{3} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + \frac {1}{3} \, A b^{2} g^{2} x^{3} + B a b g^{2} x^{2} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + A a b g^{2} x^{2} + \frac {1}{6} \, B b^{2} g^{2} n {\left (\frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} - B a b g^{2} n {\left (\frac {a^{2} \log \left (b x + a\right )}{b^{2}} - \frac {c^{2} \log \left (d x + c\right )}{d^{2}} + \frac {{\left (b c - a d\right )} x}{b d}\right )} + B a^{2} g^{2} n {\left (\frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} + B a^{2} g^{2} x \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + A a^{2} g^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.28, size = 303, normalized size = 2.44 \[ \ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (B\,a^2\,g^2\,x+B\,a\,b\,g^2\,x^2+\frac {B\,b^2\,g^2\,x^3}{3}\right )-x\,\left (\frac {\left (3\,a\,d+3\,b\,c\right )\,\left (\frac {b\,g^2\,\left (9\,A\,a\,d+3\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{3\,d}-\frac {A\,b\,g^2\,\left (3\,a\,d+3\,b\,c\right )}{3\,d}\right )}{3\,b\,d}-\frac {a\,g^2\,\left (3\,A\,a\,d+3\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{d}+\frac {A\,a\,b\,c\,g^2}{d}\right )+x^2\,\left (\frac {b\,g^2\,\left (9\,A\,a\,d+3\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{6\,d}-\frac {A\,b\,g^2\,\left (3\,a\,d+3\,b\,c\right )}{6\,d}\right )-\frac {\ln \left (c+d\,x\right )\,\left (3\,B\,n\,a^2\,c\,d^2\,g^2-3\,B\,n\,a\,b\,c^2\,d\,g^2+B\,n\,b^2\,c^3\,g^2\right )}{3\,d^3}+\frac {A\,b^2\,g^2\,x^3}{3}+\frac {B\,a^3\,g^2\,n\,\ln \left (a+b\,x\right )}{3\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 60.49, size = 673, normalized size = 5.43 \[ \begin {cases} a^{2} g^{2} x \left (A + B \log {\left (e \left (\frac {a}{c}\right )^{n} \right )}\right ) & \text {for}\: b = 0 \wedge d = 0 \\A a^{2} g^{2} x + A a b g^{2} x^{2} + \frac {A b^{2} g^{2} x^{3}}{3} + \frac {B a^{3} g^{2} n \log {\left (\frac {a}{c} + \frac {b x}{c} \right )}}{3 b} + B a^{2} g^{2} n x \log {\left (\frac {a}{c} + \frac {b x}{c} \right )} - \frac {B a^{2} g^{2} n x}{3} + B a^{2} g^{2} x \log {\relax (e )} + B a b g^{2} n x^{2} \log {\left (\frac {a}{c} + \frac {b x}{c} \right )} - \frac {B a b g^{2} n x^{2}}{3} + B a b g^{2} x^{2} \log {\relax (e )} + \frac {B b^{2} g^{2} n x^{3} \log {\left (\frac {a}{c} + \frac {b x}{c} \right )}}{3} - \frac {B b^{2} g^{2} n x^{3}}{9} + \frac {B b^{2} g^{2} x^{3} \log {\relax (e )}}{3} & \text {for}\: d = 0 \\a^{2} g^{2} \left (A x - \frac {B c n \log {\left (c + d x \right )}}{d} + B n x \log {\relax (a )} - B n x \log {\left (c + d x \right )} + B n x + B x \log {\relax (e )}\right ) & \text {for}\: b = 0 \\A a^{2} g^{2} x + A a b g^{2} x^{2} + \frac {A b^{2} g^{2} x^{3}}{3} + \frac {B a^{3} g^{2} n \log {\left (\frac {a}{c + d x} + \frac {b x}{c + d x} \right )}}{3 b} + \frac {B a^{3} g^{2} n \log {\left (\frac {c}{d} + x \right )}}{3 b} - \frac {B a^{2} c g^{2} n \log {\left (\frac {c}{d} + x \right )}}{d} + B a^{2} g^{2} n x \log {\left (\frac {a}{c + d x} + \frac {b x}{c + d x} \right )} + \frac {2 B a^{2} g^{2} n x}{3} + B a^{2} g^{2} x \log {\relax (e )} + \frac {B a b c^{2} g^{2} n \log {\left (\frac {c}{d} + x \right )}}{d^{2}} - \frac {B a b c g^{2} n x}{d} + B a b g^{2} n x^{2} \log {\left (\frac {a}{c + d x} + \frac {b x}{c + d x} \right )} + \frac {B a b g^{2} n x^{2}}{6} + B a b g^{2} x^{2} \log {\relax (e )} - \frac {B b^{2} c^{3} g^{2} n \log {\left (\frac {c}{d} + x \right )}}{3 d^{3}} + \frac {B b^{2} c^{2} g^{2} n x}{3 d^{2}} - \frac {B b^{2} c g^{2} n x^{2}}{6 d} + \frac {B b^{2} g^{2} n x^{3} \log {\left (\frac {a}{c + d x} + \frac {b x}{c + d x} \right )}}{3} + \frac {B b^{2} g^{2} x^{3} \log {\relax (e )}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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