Optimal. Leaf size=230 \[ \frac {e^4 (a+b x)^3 (-6 a B e+A b e+5 b B d)}{3 b^7}+\frac {5 e^3 (a+b x)^2 (b d-a e) (-3 a B e+A b e+2 b B d)}{2 b^7}-\frac {(b d-a e)^4 (-6 a B e+5 A b e+b B d)}{b^7 (a+b x)}-\frac {(A b-a B) (b d-a e)^5}{2 b^7 (a+b x)^2}+\frac {5 e (b d-a e)^3 \log (a+b x) (-3 a B e+2 A b e+b B d)}{b^7}+\frac {10 e^2 x (b d-a e)^2 (-2 a B e+A b e+b B d)}{b^6}+\frac {B e^5 (a+b x)^4}{4 b^7} \]
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Rubi [A] time = 0.35, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \begin {gather*} \frac {e^4 (a+b x)^3 (-6 a B e+A b e+5 b B d)}{3 b^7}+\frac {5 e^3 (a+b x)^2 (b d-a e) (-3 a B e+A b e+2 b B d)}{2 b^7}+\frac {10 e^2 x (b d-a e)^2 (-2 a B e+A b e+b B d)}{b^6}-\frac {(b d-a e)^4 (-6 a B e+5 A b e+b B d)}{b^7 (a+b x)}-\frac {(A b-a B) (b d-a e)^5}{2 b^7 (a+b x)^2}+\frac {5 e (b d-a e)^3 \log (a+b x) (-3 a B e+2 A b e+b B d)}{b^7}+\frac {B e^5 (a+b x)^4}{4 b^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^5}{(a+b x)^3} \, dx &=\int \left (\frac {10 e^2 (b d-a e)^2 (b B d+A b e-2 a B e)}{b^6}+\frac {(A b-a B) (b d-a e)^5}{b^6 (a+b x)^3}+\frac {(b d-a e)^4 (b B d+5 A b e-6 a B e)}{b^6 (a+b x)^2}+\frac {5 e (b d-a e)^3 (b B d+2 A b e-3 a B e)}{b^6 (a+b x)}+\frac {5 e^3 (b d-a e) (2 b B d+A b e-3 a B e) (a+b x)}{b^6}+\frac {e^4 (5 b B d+A b e-6 a B e) (a+b x)^2}{b^6}+\frac {B e^5 (a+b x)^3}{b^6}\right ) \, dx\\ &=\frac {10 e^2 (b d-a e)^2 (b B d+A b e-2 a B e) x}{b^6}-\frac {(A b-a B) (b d-a e)^5}{2 b^7 (a+b x)^2}-\frac {(b d-a e)^4 (b B d+5 A b e-6 a B e)}{b^7 (a+b x)}+\frac {5 e^3 (b d-a e) (2 b B d+A b e-3 a B e) (a+b x)^2}{2 b^7}+\frac {e^4 (5 b B d+A b e-6 a B e) (a+b x)^3}{3 b^7}+\frac {B e^5 (a+b x)^4}{4 b^7}+\frac {5 e (b d-a e)^3 (b B d+2 A b e-3 a B e) \log (a+b x)}{b^7}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 254, normalized size = 1.10 \begin {gather*} \frac {6 b^2 e^3 x^2 \left (6 a^2 B e^2-3 a b e (A e+5 B d)+5 b^2 d (A e+2 B d)\right )+12 b e^2 x \left (-10 a^3 B e^3+6 a^2 b e^2 (A e+5 B d)-15 a b^2 d e (A e+2 B d)+10 b^3 d^2 (A e+B d)\right )+4 b^3 e^4 x^3 (-3 a B e+A b e+5 b B d)-\frac {12 (b d-a e)^4 (-6 a B e+5 A b e+b B d)}{a+b x}-\frac {6 (A b-a B) (b d-a e)^5}{(a+b x)^2}+60 e (b d-a e)^3 \log (a+b x) (-3 a B e+2 A b e+b B d)+3 b^4 B e^5 x^4}{12 b^7} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) (d+e x)^5}{(a+b x)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.15, size = 912, normalized size = 3.97 \begin {gather*} \frac {3 \, B b^{6} e^{5} x^{6} - 6 \, {\left (B a b^{5} + A b^{6}\right )} d^{5} + 30 \, {\left (3 \, B a^{2} b^{4} - A a b^{5}\right )} d^{4} e - 60 \, {\left (5 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} d^{3} e^{2} + 60 \, {\left (7 \, B a^{4} b^{2} - 5 \, A a^{3} b^{3}\right )} d^{2} e^{3} - 30 \, {\left (9 \, B a^{5} b - 7 \, A a^{4} b^{2}\right )} d e^{4} + 6 \, {\left (11 \, B a^{6} - 9 \, A a^{5} b\right )} e^{5} + 2 \, {\left (10 \, B b^{6} d e^{4} - {\left (3 \, B a b^{5} - 2 \, A b^{6}\right )} e^{5}\right )} x^{5} + 5 \, {\left (12 \, B b^{6} d^{2} e^{3} - 2 \, {\left (5 \, B a b^{5} - 3 \, A b^{6}\right )} d e^{4} + {\left (3 \, B a^{2} b^{4} - 2 \, A a b^{5}\right )} e^{5}\right )} x^{4} + 20 \, {\left (6 \, B b^{6} d^{3} e^{2} - 6 \, {\left (2 \, B a b^{5} - A b^{6}\right )} d^{2} e^{3} + 2 \, {\left (5 \, B a^{2} b^{4} - 3 \, A a b^{5}\right )} d e^{4} - {\left (3 \, B a^{3} b^{3} - 2 \, A a^{2} b^{4}\right )} e^{5}\right )} x^{3} + 6 \, {\left (40 \, B a b^{5} d^{3} e^{2} - 10 \, {\left (11 \, B a^{2} b^{4} - 4 \, A a b^{5}\right )} d^{2} e^{3} + 5 \, {\left (21 \, B a^{3} b^{3} - 11 \, A a^{2} b^{4}\right )} d e^{4} - {\left (34 \, B a^{4} b^{2} - 21 \, A a^{3} b^{3}\right )} e^{5}\right )} x^{2} - 12 \, {\left (B b^{6} d^{5} - 5 \, {\left (2 \, B a b^{5} - A b^{6}\right )} d^{4} e + 20 \, {\left (B a^{2} b^{4} - A a b^{5}\right )} d^{3} e^{2} - 10 \, {\left (B a^{3} b^{3} - 2 \, A a^{2} b^{4}\right )} d^{2} e^{3} - 5 \, {\left (B a^{4} b^{2} + A a^{3} b^{3}\right )} d e^{4} + {\left (4 \, B a^{5} b - A a^{4} b^{2}\right )} e^{5}\right )} x + 60 \, {\left (B a^{2} b^{4} d^{4} e - 2 \, {\left (3 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} d^{3} e^{2} + 6 \, {\left (2 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} d^{2} e^{3} - 2 \, {\left (5 \, B a^{5} b - 3 \, A a^{4} b^{2}\right )} d e^{4} + {\left (3 \, B a^{6} - 2 \, A a^{5} b\right )} e^{5} + {\left (B b^{6} d^{4} e - 2 \, {\left (3 \, B a b^{5} - A b^{6}\right )} d^{3} e^{2} + 6 \, {\left (2 \, B a^{2} b^{4} - A a b^{5}\right )} d^{2} e^{3} - 2 \, {\left (5 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} d e^{4} + {\left (3 \, B a^{4} b^{2} - 2 \, A a^{3} b^{3}\right )} e^{5}\right )} x^{2} + 2 \, {\left (B a b^{5} d^{4} e - 2 \, {\left (3 \, B a^{2} b^{4} - A a b^{5}\right )} d^{3} e^{2} + 6 \, {\left (2 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} d^{2} e^{3} - 2 \, {\left (5 \, B a^{4} b^{2} - 3 \, A a^{3} b^{3}\right )} d e^{4} + {\left (3 \, B a^{5} b - 2 \, A a^{4} b^{2}\right )} e^{5}\right )} x\right )} \log \left (b x + a\right )}{12 \, {\left (b^{9} x^{2} + 2 \, a b^{8} x + a^{2} b^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.21, size = 598, normalized size = 2.60 \begin {gather*} \frac {5 \, {\left (B b^{4} d^{4} e - 6 \, B a b^{3} d^{3} e^{2} + 2 \, A b^{4} d^{3} e^{2} + 12 \, B a^{2} b^{2} d^{2} e^{3} - 6 \, A a b^{3} d^{2} e^{3} - 10 \, B a^{3} b d e^{4} + 6 \, A a^{2} b^{2} d e^{4} + 3 \, B a^{4} e^{5} - 2 \, A a^{3} b e^{5}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7}} - \frac {B a b^{5} d^{5} + A b^{6} d^{5} - 15 \, B a^{2} b^{4} d^{4} e + 5 \, A a b^{5} d^{4} e + 50 \, B a^{3} b^{3} d^{3} e^{2} - 30 \, A a^{2} b^{4} d^{3} e^{2} - 70 \, B a^{4} b^{2} d^{2} e^{3} + 50 \, A a^{3} b^{3} d^{2} e^{3} + 45 \, B a^{5} b d e^{4} - 35 \, A a^{4} b^{2} d e^{4} - 11 \, B a^{6} e^{5} + 9 \, A a^{5} b e^{5} + 2 \, {\left (B b^{6} d^{5} - 10 \, B a b^{5} d^{4} e + 5 \, A b^{6} d^{4} e + 30 \, B a^{2} b^{4} d^{3} e^{2} - 20 \, A a b^{5} d^{3} e^{2} - 40 \, B a^{3} b^{3} d^{2} e^{3} + 30 \, A a^{2} b^{4} d^{2} e^{3} + 25 \, B a^{4} b^{2} d e^{4} - 20 \, A a^{3} b^{3} d e^{4} - 6 \, B a^{5} b e^{5} + 5 \, A a^{4} b^{2} e^{5}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{7}} + \frac {3 \, B b^{9} x^{4} e^{5} + 20 \, B b^{9} d x^{3} e^{4} + 60 \, B b^{9} d^{2} x^{2} e^{3} + 120 \, B b^{9} d^{3} x e^{2} - 12 \, B a b^{8} x^{3} e^{5} + 4 \, A b^{9} x^{3} e^{5} - 90 \, B a b^{8} d x^{2} e^{4} + 30 \, A b^{9} d x^{2} e^{4} - 360 \, B a b^{8} d^{2} x e^{3} + 120 \, A b^{9} d^{2} x e^{3} + 36 \, B a^{2} b^{7} x^{2} e^{5} - 18 \, A a b^{8} x^{2} e^{5} + 360 \, B a^{2} b^{7} d x e^{4} - 180 \, A a b^{8} d x e^{4} - 120 \, B a^{3} b^{6} x e^{5} + 72 \, A a^{2} b^{7} x e^{5}}{12 \, b^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 833, normalized size = 3.62 \begin {gather*} \frac {B \,e^{5} x^{4}}{4 b^{3}}+\frac {A \,e^{5} x^{3}}{3 b^{3}}-\frac {B a \,e^{5} x^{3}}{b^{4}}+\frac {5 B d \,e^{4} x^{3}}{3 b^{3}}+\frac {A \,a^{5} e^{5}}{2 \left (b x +a \right )^{2} b^{6}}-\frac {5 A \,a^{4} d \,e^{4}}{2 \left (b x +a \right )^{2} b^{5}}+\frac {5 A \,a^{3} d^{2} e^{3}}{\left (b x +a \right )^{2} b^{4}}-\frac {5 A \,a^{2} d^{3} e^{2}}{\left (b x +a \right )^{2} b^{3}}+\frac {5 A a \,d^{4} e}{2 \left (b x +a \right )^{2} b^{2}}-\frac {3 A a \,e^{5} x^{2}}{2 b^{4}}-\frac {A \,d^{5}}{2 \left (b x +a \right )^{2} b}+\frac {5 A d \,e^{4} x^{2}}{2 b^{3}}-\frac {B \,a^{6} e^{5}}{2 \left (b x +a \right )^{2} b^{7}}+\frac {5 B \,a^{5} d \,e^{4}}{2 \left (b x +a \right )^{2} b^{6}}-\frac {5 B \,a^{4} d^{2} e^{3}}{\left (b x +a \right )^{2} b^{5}}+\frac {5 B \,a^{3} d^{3} e^{2}}{\left (b x +a \right )^{2} b^{4}}-\frac {5 B \,a^{2} d^{4} e}{2 \left (b x +a \right )^{2} b^{3}}+\frac {3 B \,a^{2} e^{5} x^{2}}{b^{5}}+\frac {B a \,d^{5}}{2 \left (b x +a \right )^{2} b^{2}}-\frac {15 B a d \,e^{4} x^{2}}{2 b^{4}}+\frac {5 B \,d^{2} e^{3} x^{2}}{b^{3}}-\frac {5 A \,a^{4} e^{5}}{\left (b x +a \right ) b^{6}}+\frac {20 A \,a^{3} d \,e^{4}}{\left (b x +a \right ) b^{5}}-\frac {10 A \,a^{3} e^{5} \ln \left (b x +a \right )}{b^{6}}-\frac {30 A \,a^{2} d^{2} e^{3}}{\left (b x +a \right ) b^{4}}+\frac {30 A \,a^{2} d \,e^{4} \ln \left (b x +a \right )}{b^{5}}+\frac {6 A \,a^{2} e^{5} x}{b^{5}}+\frac {20 A a \,d^{3} e^{2}}{\left (b x +a \right ) b^{3}}-\frac {30 A a \,d^{2} e^{3} \ln \left (b x +a \right )}{b^{4}}-\frac {15 A a d \,e^{4} x}{b^{4}}-\frac {5 A \,d^{4} e}{\left (b x +a \right ) b^{2}}+\frac {10 A \,d^{3} e^{2} \ln \left (b x +a \right )}{b^{3}}+\frac {10 A \,d^{2} e^{3} x}{b^{3}}+\frac {6 B \,a^{5} e^{5}}{\left (b x +a \right ) b^{7}}-\frac {25 B \,a^{4} d \,e^{4}}{\left (b x +a \right ) b^{6}}+\frac {15 B \,a^{4} e^{5} \ln \left (b x +a \right )}{b^{7}}+\frac {40 B \,a^{3} d^{2} e^{3}}{\left (b x +a \right ) b^{5}}-\frac {50 B \,a^{3} d \,e^{4} \ln \left (b x +a \right )}{b^{6}}-\frac {10 B \,a^{3} e^{5} x}{b^{6}}-\frac {30 B \,a^{2} d^{3} e^{2}}{\left (b x +a \right ) b^{4}}+\frac {60 B \,a^{2} d^{2} e^{3} \ln \left (b x +a \right )}{b^{5}}+\frac {30 B \,a^{2} d \,e^{4} x}{b^{5}}+\frac {10 B a \,d^{4} e}{\left (b x +a \right ) b^{3}}-\frac {30 B a \,d^{3} e^{2} \ln \left (b x +a \right )}{b^{4}}-\frac {30 B a \,d^{2} e^{3} x}{b^{4}}-\frac {B \,d^{5}}{\left (b x +a \right ) b^{2}}+\frac {5 B \,d^{4} e \ln \left (b x +a \right )}{b^{3}}+\frac {10 B \,d^{3} e^{2} x}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.66, size = 592, normalized size = 2.57 \begin {gather*} -\frac {{\left (B a b^{5} + A b^{6}\right )} d^{5} - 5 \, {\left (3 \, B a^{2} b^{4} - A a b^{5}\right )} d^{4} e + 10 \, {\left (5 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} d^{3} e^{2} - 10 \, {\left (7 \, B a^{4} b^{2} - 5 \, A a^{3} b^{3}\right )} d^{2} e^{3} + 5 \, {\left (9 \, B a^{5} b - 7 \, A a^{4} b^{2}\right )} d e^{4} - {\left (11 \, B a^{6} - 9 \, A a^{5} b\right )} e^{5} + 2 \, {\left (B b^{6} d^{5} - 5 \, {\left (2 \, B a b^{5} - A b^{6}\right )} d^{4} e + 10 \, {\left (3 \, B a^{2} b^{4} - 2 \, A a b^{5}\right )} d^{3} e^{2} - 10 \, {\left (4 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} d^{2} e^{3} + 5 \, {\left (5 \, B a^{4} b^{2} - 4 \, A a^{3} b^{3}\right )} d e^{4} - {\left (6 \, B a^{5} b - 5 \, A a^{4} b^{2}\right )} e^{5}\right )} x}{2 \, {\left (b^{9} x^{2} + 2 \, a b^{8} x + a^{2} b^{7}\right )}} + \frac {3 \, B b^{3} e^{5} x^{4} + 4 \, {\left (5 \, B b^{3} d e^{4} - {\left (3 \, B a b^{2} - A b^{3}\right )} e^{5}\right )} x^{3} + 6 \, {\left (10 \, B b^{3} d^{2} e^{3} - 5 \, {\left (3 \, B a b^{2} - A b^{3}\right )} d e^{4} + 3 \, {\left (2 \, B a^{2} b - A a b^{2}\right )} e^{5}\right )} x^{2} + 12 \, {\left (10 \, B b^{3} d^{3} e^{2} - 10 \, {\left (3 \, B a b^{2} - A b^{3}\right )} d^{2} e^{3} + 15 \, {\left (2 \, B a^{2} b - A a b^{2}\right )} d e^{4} - 2 \, {\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} e^{5}\right )} x}{12 \, b^{6}} + \frac {5 \, {\left (B b^{4} d^{4} e - 2 \, {\left (3 \, B a b^{3} - A b^{4}\right )} d^{3} e^{2} + 6 \, {\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} d^{2} e^{3} - 2 \, {\left (5 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} d e^{4} + {\left (3 \, B a^{4} - 2 \, A a^{3} b\right )} e^{5}\right )} \log \left (b x + a\right )}{b^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 681, normalized size = 2.96 \begin {gather*} x^3\,\left (\frac {A\,e^5+5\,B\,d\,e^4}{3\,b^3}-\frac {B\,a\,e^5}{b^4}\right )-\frac {x\,\left (-6\,B\,a^5\,e^5+25\,B\,a^4\,b\,d\,e^4+5\,A\,a^4\,b\,e^5-40\,B\,a^3\,b^2\,d^2\,e^3-20\,A\,a^3\,b^2\,d\,e^4+30\,B\,a^2\,b^3\,d^3\,e^2+30\,A\,a^2\,b^3\,d^2\,e^3-10\,B\,a\,b^4\,d^4\,e-20\,A\,a\,b^4\,d^3\,e^2+B\,b^5\,d^5+5\,A\,b^5\,d^4\,e\right )+\frac {-11\,B\,a^6\,e^5+45\,B\,a^5\,b\,d\,e^4+9\,A\,a^5\,b\,e^5-70\,B\,a^4\,b^2\,d^2\,e^3-35\,A\,a^4\,b^2\,d\,e^4+50\,B\,a^3\,b^3\,d^3\,e^2+50\,A\,a^3\,b^3\,d^2\,e^3-15\,B\,a^2\,b^4\,d^4\,e-30\,A\,a^2\,b^4\,d^3\,e^2+B\,a\,b^5\,d^5+5\,A\,a\,b^5\,d^4\,e+A\,b^6\,d^5}{2\,b}}{a^2\,b^6+2\,a\,b^7\,x+b^8\,x^2}-x^2\,\left (\frac {3\,a\,\left (\frac {A\,e^5+5\,B\,d\,e^4}{b^3}-\frac {3\,B\,a\,e^5}{b^4}\right )}{2\,b}-\frac {5\,d\,e^3\,\left (A\,e+2\,B\,d\right )}{2\,b^3}+\frac {3\,B\,a^2\,e^5}{2\,b^5}\right )-x\,\left (\frac {3\,a^2\,\left (\frac {A\,e^5+5\,B\,d\,e^4}{b^3}-\frac {3\,B\,a\,e^5}{b^4}\right )}{b^2}-\frac {3\,a\,\left (\frac {3\,a\,\left (\frac {A\,e^5+5\,B\,d\,e^4}{b^3}-\frac {3\,B\,a\,e^5}{b^4}\right )}{b}-\frac {5\,d\,e^3\,\left (A\,e+2\,B\,d\right )}{b^3}+\frac {3\,B\,a^2\,e^5}{b^5}\right )}{b}-\frac {10\,d^2\,e^2\,\left (A\,e+B\,d\right )}{b^3}+\frac {B\,a^3\,e^5}{b^6}\right )+\frac {\ln \left (a+b\,x\right )\,\left (15\,B\,a^4\,e^5-50\,B\,a^3\,b\,d\,e^4-10\,A\,a^3\,b\,e^5+60\,B\,a^2\,b^2\,d^2\,e^3+30\,A\,a^2\,b^2\,d\,e^4-30\,B\,a\,b^3\,d^3\,e^2-30\,A\,a\,b^3\,d^2\,e^3+5\,B\,b^4\,d^4\,e+10\,A\,b^4\,d^3\,e^2\right )}{b^7}+\frac {B\,e^5\,x^4}{4\,b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 11.38, size = 615, normalized size = 2.67 \begin {gather*} \frac {B e^{5} x^{4}}{4 b^{3}} + x^{3} \left (\frac {A e^{5}}{3 b^{3}} - \frac {B a e^{5}}{b^{4}} + \frac {5 B d e^{4}}{3 b^{3}}\right ) + x^{2} \left (- \frac {3 A a e^{5}}{2 b^{4}} + \frac {5 A d e^{4}}{2 b^{3}} + \frac {3 B a^{2} e^{5}}{b^{5}} - \frac {15 B a d e^{4}}{2 b^{4}} + \frac {5 B d^{2} e^{3}}{b^{3}}\right ) + x \left (\frac {6 A a^{2} e^{5}}{b^{5}} - \frac {15 A a d e^{4}}{b^{4}} + \frac {10 A d^{2} e^{3}}{b^{3}} - \frac {10 B a^{3} e^{5}}{b^{6}} + \frac {30 B a^{2} d e^{4}}{b^{5}} - \frac {30 B a d^{2} e^{3}}{b^{4}} + \frac {10 B d^{3} e^{2}}{b^{3}}\right ) + \frac {- 9 A a^{5} b e^{5} + 35 A a^{4} b^{2} d e^{4} - 50 A a^{3} b^{3} d^{2} e^{3} + 30 A a^{2} b^{4} d^{3} e^{2} - 5 A a b^{5} d^{4} e - A b^{6} d^{5} + 11 B a^{6} e^{5} - 45 B a^{5} b d e^{4} + 70 B a^{4} b^{2} d^{2} e^{3} - 50 B a^{3} b^{3} d^{3} e^{2} + 15 B a^{2} b^{4} d^{4} e - B a b^{5} d^{5} + x \left (- 10 A a^{4} b^{2} e^{5} + 40 A a^{3} b^{3} d e^{4} - 60 A a^{2} b^{4} d^{2} e^{3} + 40 A a b^{5} d^{3} e^{2} - 10 A b^{6} d^{4} e + 12 B a^{5} b e^{5} - 50 B a^{4} b^{2} d e^{4} + 80 B a^{3} b^{3} d^{2} e^{3} - 60 B a^{2} b^{4} d^{3} e^{2} + 20 B a b^{5} d^{4} e - 2 B b^{6} d^{5}\right )}{2 a^{2} b^{7} + 4 a b^{8} x + 2 b^{9} x^{2}} + \frac {5 e \left (a e - b d\right )^{3} \left (- 2 A b e + 3 B a e - B b d\right ) \log {\left (a + b x \right )}}{b^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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