Optimal. Leaf size=302 \[ \frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^2}{6 e^6 (d+e x)^6}+\frac {\left (c d^2-b d e+a e^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{5 e^6 (d+e x)^5}+\frac {B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )-A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )}{4 e^6 (d+e x)^4}+\frac {2 A c e (2 c d-b e)-B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )}{3 e^6 (d+e x)^3}+\frac {c (5 B c d-2 b B e-A c e)}{2 e^6 (d+e x)^2}-\frac {B c^2}{e^6 (d+e x)} \]
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Rubi [A]
time = 0.21, antiderivative size = 300, normalized size of antiderivative = 0.99, 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 = {785}
\begin {gather*} \frac {2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^6 (d+e x)^3}+\frac {B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{4 e^6 (d+e x)^4}-\frac {\left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{5 e^6 (d+e x)^5}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{6 e^6 (d+e x)^6}+\frac {c (-A c e-2 b B e+5 B c d)}{2 e^6 (d+e x)^2}-\frac {B c^2}{e^6 (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 785
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^7} \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2-b d e+a e^2\right )^2}{e^5 (d+e x)^7}+\frac {\left (c d^2-b d e+a e^2\right ) \left (5 B c d^2-B e (3 b d-a e)-2 A e (2 c d-b e)\right )}{e^5 (d+e x)^6}+\frac {-B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )+A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )}{e^5 (d+e x)^5}+\frac {-2 A c e (2 c d-b e)+B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )}{e^5 (d+e x)^4}+\frac {c (-5 B c d+2 b B e+A c e)}{e^5 (d+e x)^3}+\frac {B c^2}{e^5 (d+e x)^2}\right ) \, dx\\ &=\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^2}{6 e^6 (d+e x)^6}-\frac {\left (c d^2-b d e+a e^2\right ) \left (5 B c d^2-B e (3 b d-a e)-2 A e (2 c d-b e)\right )}{5 e^6 (d+e x)^5}+\frac {B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )-A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )}{4 e^6 (d+e x)^4}+\frac {2 A c e (2 c d-b e)-B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )}{3 e^6 (d+e x)^3}+\frac {c (5 B c d-2 b B e-A c e)}{2 e^6 (d+e x)^2}-\frac {B c^2}{e^6 (d+e x)}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 372, normalized size = 1.23 \begin {gather*} -\frac {A e \left (2 c^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+e^2 \left (10 a^2 e^2+4 a b e (d+6 e x)+b^2 \left (d^2+6 d e x+15 e^2 x^2\right )\right )+2 c e \left (a e \left (d^2+6 d e x+15 e^2 x^2\right )+b \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )\right )+B \left (10 c^2 \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )+e^2 \left (2 a^2 e^2 (d+6 e x)+2 a b e \left (d^2+6 d e x+15 e^2 x^2\right )+b^2 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )+2 c e \left (a e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+2 b \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )\right )}{60 e^6 (d+e x)^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 453, normalized size = 1.50
method | result | size |
risch | \(\frac {-\frac {B \,c^{2} x^{5}}{e}-\frac {c \left (A c e +2 b B e +5 B c d \right ) x^{4}}{2 e^{2}}-\frac {\left (2 A b c \,e^{2}+2 A \,c^{2} d e +2 B a c \,e^{2}+B \,b^{2} e^{2}+4 B b c d e +10 B \,c^{2} d^{2}\right ) x^{3}}{3 e^{3}}-\frac {\left (2 A a c \,e^{3}+A \,b^{2} e^{3}+2 A b c d \,e^{2}+2 A \,c^{2} d^{2} e +2 B \,e^{3} a b +2 B a c d \,e^{2}+B \,b^{2} d \,e^{2}+4 B b c \,d^{2} e +10 B \,c^{2} d^{3}\right ) x^{2}}{4 e^{4}}-\frac {\left (4 A a b \,e^{4}+2 A a c d \,e^{3}+A \,b^{2} d \,e^{3}+2 A b c \,d^{2} e^{2}+2 A \,c^{2} d^{3} e +2 B \,e^{4} a^{2}+2 B a b d \,e^{3}+2 B a c \,d^{2} e^{2}+B \,b^{2} d^{2} e^{2}+4 B b c \,d^{3} e +10 B \,c^{2} d^{4}\right ) x}{10 e^{5}}-\frac {10 A \,a^{2} e^{5}+4 A a b d \,e^{4}+2 A a c \,d^{2} e^{3}+A \,b^{2} d^{2} e^{3}+2 A b c \,d^{3} e^{2}+2 A \,c^{2} d^{4} e +2 B \,a^{2} d \,e^{4}+2 B a b \,d^{2} e^{3}+2 B a c \,d^{3} e^{2}+B \,b^{2} d^{3} e^{2}+4 B b c \,d^{4} e +10 B \,c^{2} d^{5}}{60 e^{6}}}{\left (e x +d \right )^{6}}\) | \(431\) |
norman | \(\frac {-\frac {B \,c^{2} x^{5}}{e}-\frac {\left (A \,c^{2} e +2 B e b c +5 B \,c^{2} d \right ) x^{4}}{2 e^{2}}-\frac {\left (2 A b c \,e^{2}+2 A \,c^{2} d e +2 B a c \,e^{2}+B \,b^{2} e^{2}+4 B b c d e +10 B \,c^{2} d^{2}\right ) x^{3}}{3 e^{3}}-\frac {\left (2 A a c \,e^{3}+A \,b^{2} e^{3}+2 A b c d \,e^{2}+2 A \,c^{2} d^{2} e +2 B \,e^{3} a b +2 B a c d \,e^{2}+B \,b^{2} d \,e^{2}+4 B b c \,d^{2} e +10 B \,c^{2} d^{3}\right ) x^{2}}{4 e^{4}}-\frac {\left (4 A a b \,e^{4}+2 A a c d \,e^{3}+A \,b^{2} d \,e^{3}+2 A b c \,d^{2} e^{2}+2 A \,c^{2} d^{3} e +2 B \,e^{4} a^{2}+2 B a b d \,e^{3}+2 B a c \,d^{2} e^{2}+B \,b^{2} d^{2} e^{2}+4 B b c \,d^{3} e +10 B \,c^{2} d^{4}\right ) x}{10 e^{5}}-\frac {10 A \,a^{2} e^{5}+4 A a b d \,e^{4}+2 A a c \,d^{2} e^{3}+A \,b^{2} d^{2} e^{3}+2 A b c \,d^{3} e^{2}+2 A \,c^{2} d^{4} e +2 B \,a^{2} d \,e^{4}+2 B a b \,d^{2} e^{3}+2 B a c \,d^{3} e^{2}+B \,b^{2} d^{3} e^{2}+4 B b c \,d^{4} e +10 B \,c^{2} d^{5}}{60 e^{6}}}{\left (e x +d \right )^{6}}\) | \(435\) |
default | \(-\frac {A \,a^{2} e^{5}-2 A a b d \,e^{4}+2 A a c \,d^{2} e^{3}+A \,b^{2} d^{2} e^{3}-2 A b c \,d^{3} e^{2}+A \,c^{2} d^{4} e -B \,a^{2} d \,e^{4}+2 B a b \,d^{2} e^{3}-2 B a c \,d^{3} e^{2}-B \,b^{2} d^{3} e^{2}+2 B b c \,d^{4} e -B \,c^{2} d^{5}}{6 e^{6} \left (e x +d \right )^{6}}-\frac {c \left (A c e +2 b B e -5 B c d \right )}{2 e^{6} \left (e x +d \right )^{2}}-\frac {B \,c^{2}}{e^{6} \left (e x +d \right )}-\frac {2 A b c \,e^{2}-4 A \,c^{2} d e +2 B a c \,e^{2}+B \,b^{2} e^{2}-8 B b c d e +10 B \,c^{2} d^{2}}{3 e^{6} \left (e x +d \right )^{3}}-\frac {2 A a b \,e^{4}-4 A a c d \,e^{3}-2 A \,b^{2} d \,e^{3}+6 A b c \,d^{2} e^{2}-4 A \,c^{2} d^{3} e +B \,e^{4} a^{2}-4 B a b d \,e^{3}+6 B a c \,d^{2} e^{2}+3 B \,b^{2} d^{2} e^{2}-8 B b c \,d^{3} e +5 B \,c^{2} d^{4}}{5 e^{6} \left (e x +d \right )^{5}}-\frac {2 A a c \,e^{3}+A \,b^{2} e^{3}-6 A b c d \,e^{2}+6 A \,c^{2} d^{2} e +2 B \,e^{3} a b -6 B a c d \,e^{2}-3 B \,b^{2} d \,e^{2}+12 B b c \,d^{2} e -10 B \,c^{2} d^{3}}{4 e^{6} \left (e x +d \right )^{4}}\) | \(453\) |
gosper | \(-\frac {60 B \,c^{2} x^{5} e^{5}+30 A \,c^{2} e^{5} x^{4}+60 B b c \,e^{5} x^{4}+150 B \,c^{2} d \,e^{4} x^{4}+40 A b c \,e^{5} x^{3}+40 A \,c^{2} d \,e^{4} x^{3}+40 B a c \,e^{5} x^{3}+20 B \,b^{2} e^{5} x^{3}+80 B b c d \,e^{4} x^{3}+200 B \,c^{2} d^{2} e^{3} x^{3}+30 A a c \,e^{5} x^{2}+15 A \,b^{2} e^{5} x^{2}+30 A b c d \,e^{4} x^{2}+30 A \,c^{2} d^{2} e^{3} x^{2}+30 B a b \,e^{5} x^{2}+30 B a c d \,e^{4} x^{2}+15 B \,b^{2} d \,e^{4} x^{2}+60 B b c \,d^{2} e^{3} x^{2}+150 B \,c^{2} d^{3} e^{2} x^{2}+24 A a b \,e^{5} x +12 A a c d \,e^{4} x +6 A \,b^{2} d \,e^{4} x +12 A b c \,d^{2} e^{3} x +12 A \,c^{2} d^{3} e^{2} x +12 B \,a^{2} e^{5} x +12 B a b d \,e^{4} x +12 B a c \,d^{2} e^{3} x +6 B \,b^{2} d^{2} e^{3} x +24 B b c \,d^{3} e^{2} x +60 B \,c^{2} d^{4} e x +10 A \,a^{2} e^{5}+4 A a b d \,e^{4}+2 A a c \,d^{2} e^{3}+A \,b^{2} d^{2} e^{3}+2 A b c \,d^{3} e^{2}+2 A \,c^{2} d^{4} e +2 B \,a^{2} d \,e^{4}+2 B a b \,d^{2} e^{3}+2 B a c \,d^{3} e^{2}+B \,b^{2} d^{3} e^{2}+4 B b c \,d^{4} e +10 B \,c^{2} d^{5}}{60 \left (e x +d \right )^{6} e^{6}}\) | \(496\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 449, normalized size = 1.49 \begin {gather*} -\frac {60 \, B c^{2} x^{5} e^{5} + 10 \, B c^{2} d^{5} + 2 \, {\left (2 \, B b c e + A c^{2} e\right )} d^{4} + 30 \, {\left (5 \, B c^{2} d e^{4} + 2 \, B b c e^{5} + A c^{2} e^{5}\right )} x^{4} + {\left (B b^{2} e^{2} + 2 \, {\left (B a e^{2} + A b e^{2}\right )} c\right )} d^{3} + 20 \, {\left (10 \, B c^{2} d^{2} e^{3} + B b^{2} e^{5} + 2 \, {\left (B a e^{5} + A b e^{5}\right )} c + 2 \, {\left (2 \, B b c e^{4} + A c^{2} e^{4}\right )} d\right )} x^{3} + 10 \, A a^{2} e^{5} + {\left (2 \, B a b e^{3} + A b^{2} e^{3} + 2 \, A a c e^{3}\right )} d^{2} + 15 \, {\left (10 \, B c^{2} d^{3} e^{2} + 2 \, B a b e^{5} + A b^{2} e^{5} + 2 \, A a c e^{5} + 2 \, {\left (2 \, B b c e^{3} + A c^{2} e^{3}\right )} d^{2} + {\left (B b^{2} e^{4} + 2 \, {\left (B a e^{4} + A b e^{4}\right )} c\right )} d\right )} x^{2} + 2 \, {\left (B a^{2} e^{4} + 2 \, A a b e^{4}\right )} d + 6 \, {\left (10 \, B c^{2} d^{4} e + 2 \, {\left (2 \, B b c e^{2} + A c^{2} e^{2}\right )} d^{3} + 2 \, B a^{2} e^{5} + 4 \, A a b e^{5} + {\left (B b^{2} e^{3} + 2 \, {\left (B a e^{3} + A b e^{3}\right )} c\right )} d^{2} + {\left (2 \, B a b e^{4} + A b^{2} e^{4} + 2 \, A a c e^{4}\right )} d\right )} x}{60 \, {\left (x^{6} e^{12} + 6 \, d x^{5} e^{11} + 15 \, d^{2} x^{4} e^{10} + 20 \, d^{3} x^{3} e^{9} + 15 \, d^{4} x^{2} e^{8} + 6 \, d^{5} x e^{7} + d^{6} e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.46, size = 411, normalized size = 1.36 \begin {gather*} -\frac {10 \, B c^{2} d^{5} + {\left (60 \, B c^{2} x^{5} + 30 \, {\left (2 \, B b c + A c^{2}\right )} x^{4} + 20 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{3} + 10 \, A a^{2} + 15 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 12 \, {\left (B a^{2} + 2 \, A a b\right )} x\right )} e^{5} + {\left (150 \, B c^{2} d x^{4} + 40 \, {\left (2 \, B b c + A c^{2}\right )} d x^{3} + 15 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d x^{2} + 6 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d x + 2 \, {\left (B a^{2} + 2 \, A a b\right )} d\right )} e^{4} + {\left (200 \, B c^{2} d^{2} x^{3} + 30 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} x^{2} + 6 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} x + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2}\right )} e^{3} + {\left (150 \, B c^{2} d^{3} x^{2} + 12 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} x + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3}\right )} e^{2} + 2 \, {\left (30 \, B c^{2} d^{4} x + {\left (2 \, B b c + A c^{2}\right )} d^{4}\right )} e}{60 \, {\left (x^{6} e^{12} + 6 \, d x^{5} e^{11} + 15 \, d^{2} x^{4} e^{10} + 20 \, d^{3} x^{3} e^{9} + 15 \, d^{4} x^{2} e^{8} + 6 \, d^{5} x e^{7} + d^{6} e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [A]
time = 2.55, size = 460, normalized size = 1.52 \begin {gather*} -\frac {{\left (60 \, B c^{2} x^{5} e^{5} + 150 \, B c^{2} d x^{4} e^{4} + 200 \, B c^{2} d^{2} x^{3} e^{3} + 150 \, B c^{2} d^{3} x^{2} e^{2} + 60 \, B c^{2} d^{4} x e + 10 \, B c^{2} d^{5} + 60 \, B b c x^{4} e^{5} + 30 \, A c^{2} x^{4} e^{5} + 80 \, B b c d x^{3} e^{4} + 40 \, A c^{2} d x^{3} e^{4} + 60 \, B b c d^{2} x^{2} e^{3} + 30 \, A c^{2} d^{2} x^{2} e^{3} + 24 \, B b c d^{3} x e^{2} + 12 \, A c^{2} d^{3} x e^{2} + 4 \, B b c d^{4} e + 2 \, A c^{2} d^{4} e + 20 \, B b^{2} x^{3} e^{5} + 40 \, B a c x^{3} e^{5} + 40 \, A b c x^{3} e^{5} + 15 \, B b^{2} d x^{2} e^{4} + 30 \, B a c d x^{2} e^{4} + 30 \, A b c d x^{2} e^{4} + 6 \, B b^{2} d^{2} x e^{3} + 12 \, B a c d^{2} x e^{3} + 12 \, A b c d^{2} x e^{3} + B b^{2} d^{3} e^{2} + 2 \, B a c d^{3} e^{2} + 2 \, A b c d^{3} e^{2} + 30 \, B a b x^{2} e^{5} + 15 \, A b^{2} x^{2} e^{5} + 30 \, A a c x^{2} e^{5} + 12 \, B a b d x e^{4} + 6 \, A b^{2} d x e^{4} + 12 \, A a c d x e^{4} + 2 \, B a b d^{2} e^{3} + A b^{2} d^{2} e^{3} + 2 \, A a c d^{2} e^{3} + 12 \, B a^{2} x e^{5} + 24 \, A a b x e^{5} + 2 \, B a^{2} d e^{4} + 4 \, A a b d e^{4} + 10 \, A a^{2} e^{5}\right )} e^{\left (-6\right )}}{60 \, {\left (x e + d\right )}^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.16, size = 485, normalized size = 1.61 \begin {gather*} -\frac {\frac {2\,B\,a^2\,d\,e^4+10\,A\,a^2\,e^5+2\,B\,a\,b\,d^2\,e^3+4\,A\,a\,b\,d\,e^4+2\,B\,a\,c\,d^3\,e^2+2\,A\,a\,c\,d^2\,e^3+B\,b^2\,d^3\,e^2+A\,b^2\,d^2\,e^3+4\,B\,b\,c\,d^4\,e+2\,A\,b\,c\,d^3\,e^2+10\,B\,c^2\,d^5+2\,A\,c^2\,d^4\,e}{60\,e^6}+\frac {x^3\,\left (B\,b^2\,e^2+4\,B\,b\,c\,d\,e+2\,A\,b\,c\,e^2+10\,B\,c^2\,d^2+2\,A\,c^2\,d\,e+2\,B\,a\,c\,e^2\right )}{3\,e^3}+\frac {x^2\,\left (B\,b^2\,d\,e^2+A\,b^2\,e^3+4\,B\,b\,c\,d^2\,e+2\,A\,b\,c\,d\,e^2+2\,B\,a\,b\,e^3+10\,B\,c^2\,d^3+2\,A\,c^2\,d^2\,e+2\,B\,a\,c\,d\,e^2+2\,A\,a\,c\,e^3\right )}{4\,e^4}+\frac {x\,\left (2\,B\,a^2\,e^4+2\,B\,a\,b\,d\,e^3+4\,A\,a\,b\,e^4+2\,B\,a\,c\,d^2\,e^2+2\,A\,a\,c\,d\,e^3+B\,b^2\,d^2\,e^2+A\,b^2\,d\,e^3+4\,B\,b\,c\,d^3\,e+2\,A\,b\,c\,d^2\,e^2+10\,B\,c^2\,d^4+2\,A\,c^2\,d^3\,e\right )}{10\,e^5}+\frac {c\,x^4\,\left (A\,c\,e+2\,B\,b\,e+5\,B\,c\,d\right )}{2\,e^2}+\frac {B\,c^2\,x^5}{e}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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