Optimal. Leaf size=541 \[ \frac {A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )}{3 e^8 (d+e x)^3}+\frac {3 c \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{e^8 (d+e x)}+\frac {3 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{4 e^8 (d+e x)^4}+\frac {B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^8 (d+e x)^2}+\frac {\left (a e^2-b d e+c d^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right )}{5 e^8 (d+e x)^5}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{6 e^8 (d+e x)^6}-\frac {c^2 \log (d+e x) (-A c e-3 b B e+7 B c d)}{e^8}+\frac {B c^3 x}{e^7} \]
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Rubi [A] time = 0.81, antiderivative size = 539, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {771} \begin {gather*} \frac {A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )}{3 e^8 (d+e x)^3}+\frac {3 c \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{e^8 (d+e x)}+\frac {B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^8 (d+e x)^2}+\frac {3 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{4 e^8 (d+e x)^4}-\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{5 e^8 (d+e x)^5}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{6 e^8 (d+e x)^6}-\frac {c^2 \log (d+e x) (-A c e-3 b B e+7 B c d)}{e^8}+\frac {B c^3 x}{e^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx &=\int \left (\frac {B c^3}{e^7}+\frac {(-B d+A e) \left (c d^2-b d e+a e^2\right )^3}{e^7 (d+e x)^7}+\frac {\left (c d^2-b d e+a e^2\right )^2 \left (7 B c d^2-B e (4 b d-a e)-3 A e (2 c d-b e)\right )}{e^7 (d+e x)^6}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (-B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )+A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right )}{e^7 (d+e x)^5}+\frac {-A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )+B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )}{e^7 (d+e x)^4}+\frac {-B \left (35 c^3 d^3-b^3 e^3+3 b c e^2 (5 b d-2 a e)-15 c^2 d e (3 b d-a e)\right )+3 A c e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^7 (d+e x)^3}+\frac {3 c \left (-A c e (2 c d-b e)+B \left (7 c^2 d^2+b^2 e^2-c e (6 b d-a e)\right )\right )}{e^7 (d+e x)^2}+\frac {c^2 (-7 B c d+3 b B e+A c e)}{e^7 (d+e x)}\right ) \, dx\\ &=\frac {B c^3 x}{e^7}+\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^3}{6 e^8 (d+e x)^6}-\frac {\left (c d^2-b d e+a e^2\right )^2 \left (7 B c d^2-B e (4 b d-a e)-3 A e (2 c d-b e)\right )}{5 e^8 (d+e x)^5}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )-A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right )}{4 e^8 (d+e x)^4}+\frac {A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )-B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )}{3 e^8 (d+e x)^3}+\frac {B \left (35 c^3 d^3-b^3 e^3+3 b c e^2 (5 b d-2 a e)-15 c^2 d e (3 b d-a e)\right )-3 A c e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{2 e^8 (d+e x)^2}+\frac {3 c \left (A c e (2 c d-b e)-B \left (7 c^2 d^2+b^2 e^2-c e (6 b d-a e)\right )\right )}{e^8 (d+e x)}-\frac {c^2 (7 B c d-3 b B e-A c e) \log (d+e x)}{e^8}\\ \end {align*}
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Mathematica [A] time = 0.65, size = 868, normalized size = 1.60 \begin {gather*} -\frac {60 c^2 (7 B c d-3 b B e-A c e) \log (d+e x) (d+e x)^6+A e \left (-d \left (147 d^5+822 e x d^4+1875 e^2 x^2 d^3+2200 e^3 x^3 d^2+1350 e^4 x^4 d+360 e^5 x^5\right ) c^3+6 e \left (a e \left (d^4+6 e x d^3+15 e^2 x^2 d^2+20 e^3 x^3 d+15 e^4 x^4\right )+5 b \left (d^5+6 e x d^4+15 e^2 x^2 d^3+20 e^3 x^3 d^2+15 e^4 x^4 d+6 e^5 x^5\right )\right ) c^2+3 e^2 \left (2 \left (d^4+6 e x d^3+15 e^2 x^2 d^2+20 e^3 x^3 d+15 e^4 x^4\right ) b^2+2 a e \left (d^3+6 e x d^2+15 e^2 x^2 d+20 e^3 x^3\right ) b+a^2 e^2 \left (d^2+6 e x d+15 e^2 x^2\right )\right ) c+e^3 \left (\left (d^3+6 e x d^2+15 e^2 x^2 d+20 e^3 x^3\right ) b^3+3 a e \left (d^2+6 e x d+15 e^2 x^2\right ) b^2+6 a^2 e^2 (d+6 e x) b+10 a^3 e^3\right )\right )+B \left (\left (669 d^7+3594 e x d^6+7725 e^2 x^2 d^5+8200 e^3 x^3 d^4+4050 e^4 x^4 d^3+360 e^5 x^5 d^2-360 e^6 x^6 d-60 e^7 x^7\right ) c^3+3 e \left (10 a e \left (d^5+6 e x d^4+15 e^2 x^2 d^3+20 e^3 x^3 d^2+15 e^4 x^4 d+6 e^5 x^5\right )-b d \left (147 d^5+822 e x d^4+1875 e^2 x^2 d^3+2200 e^3 x^3 d^2+1350 e^4 x^4 d+360 e^5 x^5\right )\right ) c^2+3 e^2 \left (10 \left (d^5+6 e x d^4+15 e^2 x^2 d^3+20 e^3 x^3 d^2+15 e^4 x^4 d+6 e^5 x^5\right ) b^2+4 a e \left (d^4+6 e x d^3+15 e^2 x^2 d^2+20 e^3 x^3 d+15 e^4 x^4\right ) b+a^2 e^2 \left (d^3+6 e x d^2+15 e^2 x^2 d+20 e^3 x^3\right )\right ) c+e^3 \left (2 \left (d^4+6 e x d^3+15 e^2 x^2 d^2+20 e^3 x^3 d+15 e^4 x^4\right ) b^3+3 a e \left (d^3+6 e x d^2+15 e^2 x^2 d+20 e^3 x^3\right ) b^2+3 a^2 e^2 \left (d^2+6 e x d+15 e^2 x^2\right ) b+2 a^3 e^3 (d+6 e x)\right )\right )}{60 e^8 (d+e x)^6} \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) \left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.39, size = 1145, normalized size = 2.12
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 989, normalized size = 1.83 \begin {gather*} B c^{3} x e^{\left (-7\right )} - {\left (7 \, B c^{3} d - 3 \, B b c^{2} e - A c^{3} e\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) - \frac {{\left (669 \, B c^{3} d^{7} - 441 \, B b c^{2} d^{6} e - 147 \, A c^{3} d^{6} e + 30 \, B b^{2} c d^{5} e^{2} + 30 \, B a c^{2} d^{5} e^{2} + 30 \, A b c^{2} d^{5} e^{2} + 2 \, B b^{3} d^{4} e^{3} + 12 \, B a b c d^{4} e^{3} + 6 \, A b^{2} c d^{4} e^{3} + 6 \, A a c^{2} d^{4} e^{3} + 3 \, B a b^{2} d^{3} e^{4} + A b^{3} d^{3} e^{4} + 3 \, B a^{2} c d^{3} e^{4} + 6 \, A a b c d^{3} e^{4} + 3 \, B a^{2} b d^{2} e^{5} + 3 \, A a b^{2} d^{2} e^{5} + 3 \, A a^{2} c d^{2} e^{5} + 180 \, {\left (7 \, B c^{3} d^{2} e^{5} - 6 \, B b c^{2} d e^{6} - 2 \, A c^{3} d e^{6} + B b^{2} c e^{7} + B a c^{2} e^{7} + A b c^{2} e^{7}\right )} x^{5} + 2 \, B a^{3} d e^{6} + 6 \, A a^{2} b d e^{6} + 30 \, {\left (175 \, B c^{3} d^{3} e^{4} - 135 \, B b c^{2} d^{2} e^{5} - 45 \, A c^{3} d^{2} e^{5} + 15 \, B b^{2} c d e^{6} + 15 \, B a c^{2} d e^{6} + 15 \, A b c^{2} d e^{6} + B b^{3} e^{7} + 6 \, B a b c e^{7} + 3 \, A b^{2} c e^{7} + 3 \, A a c^{2} e^{7}\right )} x^{4} + 10 \, A a^{3} e^{7} + 20 \, {\left (455 \, B c^{3} d^{4} e^{3} - 330 \, B b c^{2} d^{3} e^{4} - 110 \, A c^{3} d^{3} e^{4} + 30 \, B b^{2} c d^{2} e^{5} + 30 \, B a c^{2} d^{2} e^{5} + 30 \, A b c^{2} d^{2} e^{5} + 2 \, B b^{3} d e^{6} + 12 \, B a b c d e^{6} + 6 \, A b^{2} c d e^{6} + 6 \, A a c^{2} d e^{6} + 3 \, B a b^{2} e^{7} + A b^{3} e^{7} + 3 \, B a^{2} c e^{7} + 6 \, A a b c e^{7}\right )} x^{3} + 15 \, {\left (539 \, B c^{3} d^{5} e^{2} - 375 \, B b c^{2} d^{4} e^{3} - 125 \, A c^{3} d^{4} e^{3} + 30 \, B b^{2} c d^{3} e^{4} + 30 \, B a c^{2} d^{3} e^{4} + 30 \, A b c^{2} d^{3} e^{4} + 2 \, B b^{3} d^{2} e^{5} + 12 \, B a b c d^{2} e^{5} + 6 \, A b^{2} c d^{2} e^{5} + 6 \, A a c^{2} d^{2} e^{5} + 3 \, B a b^{2} d e^{6} + A b^{3} d e^{6} + 3 \, B a^{2} c d e^{6} + 6 \, A a b c d e^{6} + 3 \, B a^{2} b e^{7} + 3 \, A a b^{2} e^{7} + 3 \, A a^{2} c e^{7}\right )} x^{2} + 6 \, {\left (609 \, B c^{3} d^{6} e - 411 \, B b c^{2} d^{5} e^{2} - 137 \, A c^{3} d^{5} e^{2} + 30 \, B b^{2} c d^{4} e^{3} + 30 \, B a c^{2} d^{4} e^{3} + 30 \, A b c^{2} d^{4} e^{3} + 2 \, B b^{3} d^{3} e^{4} + 12 \, B a b c d^{3} e^{4} + 6 \, A b^{2} c d^{3} e^{4} + 6 \, A a c^{2} d^{3} e^{4} + 3 \, B a b^{2} d^{2} e^{5} + A b^{3} d^{2} e^{5} + 3 \, B a^{2} c d^{2} e^{5} + 6 \, A a b c d^{2} e^{5} + 3 \, B a^{2} b d e^{6} + 3 \, A a b^{2} d e^{6} + 3 \, A a^{2} c d e^{6} + 2 \, B a^{3} e^{7} + 6 \, A a^{2} b e^{7}\right )} x\right )} e^{\left (-8\right )}}{60 \, {\left (x e + d\right )}^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 1656, normalized size = 3.06
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.71, size = 905, normalized size = 1.67 \begin {gather*} -\frac {669 \, B c^{3} d^{7} + 10 \, A a^{3} e^{7} - 147 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 30 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{5} e^{2} + 2 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{4} e^{3} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{3} e^{4} + 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{2} e^{5} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{6} + 180 \, {\left (7 \, B c^{3} d^{2} e^{5} - 2 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{6} + {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} e^{7}\right )} x^{5} + 30 \, {\left (175 \, B c^{3} d^{3} e^{4} - 45 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{5} + 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d e^{6} + {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} e^{7}\right )} x^{4} + 20 \, {\left (455 \, B c^{3} d^{4} e^{3} - 110 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{4} + 30 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} e^{5} + 2 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d e^{6} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} e^{7}\right )} x^{3} + 15 \, {\left (539 \, B c^{3} d^{5} e^{2} - 125 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} + 30 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} e^{4} + 2 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{2} e^{5} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d e^{6} + 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e^{7}\right )} x^{2} + 6 \, {\left (609 \, B c^{3} d^{6} e - 137 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 30 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} e^{3} + 2 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{3} e^{4} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{2} e^{5} + 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d e^{6} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{7}\right )} x}{60 \, {\left (e^{14} x^{6} + 6 \, d e^{13} x^{5} + 15 \, d^{2} e^{12} x^{4} + 20 \, d^{3} e^{11} x^{3} + 15 \, d^{4} e^{10} x^{2} + 6 \, d^{5} e^{9} x + d^{6} e^{8}\right )}} + \frac {B c^{3} x}{e^{7}} - \frac {{\left (7 \, B c^{3} d - {\left (3 \, B b c^{2} + A c^{3}\right )} e\right )} \log \left (e x + d\right )}{e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.54, size = 1598, normalized size = 2.95
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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