Optimal. Leaf size=334 \[ -\frac {c (d+e x)^8 \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{8 e^8}+\frac {3 c^2 (d+e x)^{10} \left (a B e^2-2 A c d e+7 B c d^2\right )}{10 e^8}-\frac {c^2 (d+e x)^9 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{9 e^8}+\frac {(d+e x)^6 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{6 e^8}-\frac {(d+e x)^5 \left (a e^2+c d^2\right )^3 (B d-A e)}{5 e^8}-\frac {3 c (d+e x)^7 \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{7 e^8}-\frac {c^3 (d+e x)^{11} (7 B d-A e)}{11 e^8}+\frac {B c^3 (d+e x)^{12}}{12 e^8} \]
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Rubi [A] time = 0.45, antiderivative size = 334, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {772} \[ -\frac {c (d+e x)^8 \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{8 e^8}+\frac {3 c^2 (d+e x)^{10} \left (a B e^2-2 A c d e+7 B c d^2\right )}{10 e^8}-\frac {c^2 (d+e x)^9 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{9 e^8}-\frac {3 c (d+e x)^7 \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{7 e^8}+\frac {(d+e x)^6 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{6 e^8}-\frac {(d+e x)^5 \left (a e^2+c d^2\right )^3 (B d-A e)}{5 e^8}-\frac {c^3 (d+e x)^{11} (7 B d-A e)}{11 e^8}+\frac {B c^3 (d+e x)^{12}}{12 e^8} \]
Antiderivative was successfully verified.
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Rule 772
Rubi steps
\begin {align*} \int (A+B x) (d+e x)^4 \left (a+c x^2\right )^3 \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2+a e^2\right )^3 (d+e x)^4}{e^7}+\frac {\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right ) (d+e x)^5}{e^7}+\frac {3 c \left (c d^2+a e^2\right ) \left (-7 B c d^3+5 A c d^2 e-3 a B d e^2+a A e^3\right ) (d+e x)^6}{e^7}-\frac {c \left (-35 B c^2 d^4+20 A c^2 d^3 e-30 a B c d^2 e^2+12 a A c d e^3-3 a^2 B e^4\right ) (d+e x)^7}{e^7}+\frac {c^2 \left (-35 B c d^3+15 A c d^2 e-15 a B d e^2+3 a A e^3\right ) (d+e x)^8}{e^7}-\frac {3 c^2 \left (-7 B c d^2+2 A c d e-a B e^2\right ) (d+e x)^9}{e^7}+\frac {c^3 (-7 B d+A e) (d+e x)^{10}}{e^7}+\frac {B c^3 (d+e x)^{11}}{e^7}\right ) \, dx\\ &=-\frac {(B d-A e) \left (c d^2+a e^2\right )^3 (d+e x)^5}{5 e^8}+\frac {\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right ) (d+e x)^6}{6 e^8}-\frac {3 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right ) (d+e x)^7}{7 e^8}-\frac {c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right ) (d+e x)^8}{8 e^8}-\frac {c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right ) (d+e x)^9}{9 e^8}+\frac {3 c^2 \left (7 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{10}}{10 e^8}-\frac {c^3 (7 B d-A e) (d+e x)^{11}}{11 e^8}+\frac {B c^3 (d+e x)^{12}}{12 e^8}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 436, normalized size = 1.31 \[ \frac {1}{2} a^3 d^3 x^2 (4 A e+B d)+a^3 A d^4 x+\frac {1}{8} c x^8 \left (B \left (3 a^2 e^4+18 a c d^2 e^2+c^2 d^4\right )+4 A c d e \left (3 a e^2+c d^2\right )\right )+\frac {1}{7} c x^7 \left (A \left (3 a^2 e^4+18 a c d^2 e^2+c^2 d^4\right )+12 a B d e \left (a e^2+c d^2\right )\right )+\frac {1}{6} a x^6 \left (B \left (a^2 e^4+18 a c d^2 e^2+3 c^2 d^4\right )+12 A c d e \left (a e^2+c d^2\right )\right )+\frac {1}{5} a x^5 \left (A \left (a^2 e^4+18 a c d^2 e^2+3 c^2 d^4\right )+4 a B d e \left (a e^2+3 c d^2\right )\right )+\frac {1}{3} a^2 d^2 x^3 \left (6 a A e^2+4 a B d e+3 A c d^2\right )+\frac {1}{4} a^2 d x^4 \left (4 a A e^3+6 a B d e^2+12 A c d^2 e+3 B c d^3\right )+\frac {1}{10} c^2 e^2 x^{10} \left (3 a B e^2+4 A c d e+6 B c d^2\right )+\frac {1}{9} c^2 e x^9 \left (3 a A e^3+12 a B d e^2+6 A c d^2 e+4 B c d^3\right )+\frac {1}{11} c^3 e^3 x^{11} (A e+4 B d)+\frac {1}{12} B c^3 e^4 x^{12} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 536, normalized size = 1.60 \[ \frac {1}{12} x^{12} e^{4} c^{3} B + \frac {4}{11} x^{11} e^{3} d c^{3} B + \frac {1}{11} x^{11} e^{4} c^{3} A + \frac {3}{5} x^{10} e^{2} d^{2} c^{3} B + \frac {3}{10} x^{10} e^{4} c^{2} a B + \frac {2}{5} x^{10} e^{3} d c^{3} A + \frac {4}{9} x^{9} e d^{3} c^{3} B + \frac {4}{3} x^{9} e^{3} d c^{2} a B + \frac {2}{3} x^{9} e^{2} d^{2} c^{3} A + \frac {1}{3} x^{9} e^{4} c^{2} a A + \frac {1}{8} x^{8} d^{4} c^{3} B + \frac {9}{4} x^{8} e^{2} d^{2} c^{2} a B + \frac {3}{8} x^{8} e^{4} c a^{2} B + \frac {1}{2} x^{8} e d^{3} c^{3} A + \frac {3}{2} x^{8} e^{3} d c^{2} a A + \frac {12}{7} x^{7} e d^{3} c^{2} a B + \frac {12}{7} x^{7} e^{3} d c a^{2} B + \frac {1}{7} x^{7} d^{4} c^{3} A + \frac {18}{7} x^{7} e^{2} d^{2} c^{2} a A + \frac {3}{7} x^{7} e^{4} c a^{2} A + \frac {1}{2} x^{6} d^{4} c^{2} a B + 3 x^{6} e^{2} d^{2} c a^{2} B + \frac {1}{6} x^{6} e^{4} a^{3} B + 2 x^{6} e d^{3} c^{2} a A + 2 x^{6} e^{3} d c a^{2} A + \frac {12}{5} x^{5} e d^{3} c a^{2} B + \frac {4}{5} x^{5} e^{3} d a^{3} B + \frac {3}{5} x^{5} d^{4} c^{2} a A + \frac {18}{5} x^{5} e^{2} d^{2} c a^{2} A + \frac {1}{5} x^{5} e^{4} a^{3} A + \frac {3}{4} x^{4} d^{4} c a^{2} B + \frac {3}{2} x^{4} e^{2} d^{2} a^{3} B + 3 x^{4} e d^{3} c a^{2} A + x^{4} e^{3} d a^{3} A + \frac {4}{3} x^{3} e d^{3} a^{3} B + x^{3} d^{4} c a^{2} A + 2 x^{3} e^{2} d^{2} a^{3} A + \frac {1}{2} x^{2} d^{4} a^{3} B + 2 x^{2} e d^{3} a^{3} A + x d^{4} a^{3} A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 520, normalized size = 1.56 \[ \frac {1}{12} \, B c^{3} x^{12} e^{4} + \frac {4}{11} \, B c^{3} d x^{11} e^{3} + \frac {3}{5} \, B c^{3} d^{2} x^{10} e^{2} + \frac {4}{9} \, B c^{3} d^{3} x^{9} e + \frac {1}{8} \, B c^{3} d^{4} x^{8} + \frac {1}{11} \, A c^{3} x^{11} e^{4} + \frac {2}{5} \, A c^{3} d x^{10} e^{3} + \frac {2}{3} \, A c^{3} d^{2} x^{9} e^{2} + \frac {1}{2} \, A c^{3} d^{3} x^{8} e + \frac {1}{7} \, A c^{3} d^{4} x^{7} + \frac {3}{10} \, B a c^{2} x^{10} e^{4} + \frac {4}{3} \, B a c^{2} d x^{9} e^{3} + \frac {9}{4} \, B a c^{2} d^{2} x^{8} e^{2} + \frac {12}{7} \, B a c^{2} d^{3} x^{7} e + \frac {1}{2} \, B a c^{2} d^{4} x^{6} + \frac {1}{3} \, A a c^{2} x^{9} e^{4} + \frac {3}{2} \, A a c^{2} d x^{8} e^{3} + \frac {18}{7} \, A a c^{2} d^{2} x^{7} e^{2} + 2 \, A a c^{2} d^{3} x^{6} e + \frac {3}{5} \, A a c^{2} d^{4} x^{5} + \frac {3}{8} \, B a^{2} c x^{8} e^{4} + \frac {12}{7} \, B a^{2} c d x^{7} e^{3} + 3 \, B a^{2} c d^{2} x^{6} e^{2} + \frac {12}{5} \, B a^{2} c d^{3} x^{5} e + \frac {3}{4} \, B a^{2} c d^{4} x^{4} + \frac {3}{7} \, A a^{2} c x^{7} e^{4} + 2 \, A a^{2} c d x^{6} e^{3} + \frac {18}{5} \, A a^{2} c d^{2} x^{5} e^{2} + 3 \, A a^{2} c d^{3} x^{4} e + A a^{2} c d^{4} x^{3} + \frac {1}{6} \, B a^{3} x^{6} e^{4} + \frac {4}{5} \, B a^{3} d x^{5} e^{3} + \frac {3}{2} \, B a^{3} d^{2} x^{4} e^{2} + \frac {4}{3} \, B a^{3} d^{3} x^{3} e + \frac {1}{2} \, B a^{3} d^{4} x^{2} + \frac {1}{5} \, A a^{3} x^{5} e^{4} + A a^{3} d x^{4} e^{3} + 2 \, A a^{3} d^{2} x^{3} e^{2} + 2 \, A a^{3} d^{3} x^{2} e + A a^{3} d^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 455, normalized size = 1.36 \[ \frac {B \,c^{3} e^{4} x^{12}}{12}+\frac {\left (A \,e^{4}+4 B d \,e^{3}\right ) c^{3} x^{11}}{11}+\frac {\left (3 B a \,c^{2} e^{4}+\left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) c^{3}\right ) x^{10}}{10}+A \,a^{3} d^{4} x +\frac {\left (3 \left (A \,e^{4}+4 B d \,e^{3}\right ) a \,c^{2}+\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) c^{3}\right ) x^{9}}{9}+\frac {\left (3 B \,a^{2} c \,e^{4}+3 \left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) a \,c^{2}+\left (4 A \,d^{3} e +B \,d^{4}\right ) c^{3}\right ) x^{8}}{8}+\frac {\left (A \,c^{3} d^{4}+3 \left (A \,e^{4}+4 B d \,e^{3}\right ) a^{2} c +3 \left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) a \,c^{2}\right ) x^{7}}{7}+\frac {\left (B \,a^{3} e^{4}+3 \left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) a^{2} c +3 \left (4 A \,d^{3} e +B \,d^{4}\right ) a \,c^{2}\right ) x^{6}}{6}+\frac {\left (4 A \,d^{3} e +B \,d^{4}\right ) a^{3} x^{2}}{2}+\frac {\left (3 A a \,c^{2} d^{4}+\left (A \,e^{4}+4 B d \,e^{3}\right ) a^{3}+3 \left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) a^{2} c \right ) x^{5}}{5}+\frac {\left (\left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) a^{3}+3 \left (4 A \,d^{3} e +B \,d^{4}\right ) a^{2} c \right ) x^{4}}{4}+\frac {\left (3 A \,a^{2} c \,d^{4}+\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) a^{3}\right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 478, normalized size = 1.43 \[ \frac {1}{12} \, B c^{3} e^{4} x^{12} + \frac {1}{11} \, {\left (4 \, B c^{3} d e^{3} + A c^{3} e^{4}\right )} x^{11} + \frac {1}{10} \, {\left (6 \, B c^{3} d^{2} e^{2} + 4 \, A c^{3} d e^{3} + 3 \, B a c^{2} e^{4}\right )} x^{10} + \frac {1}{9} \, {\left (4 \, B c^{3} d^{3} e + 6 \, A c^{3} d^{2} e^{2} + 12 \, B a c^{2} d e^{3} + 3 \, A a c^{2} e^{4}\right )} x^{9} + A a^{3} d^{4} x + \frac {1}{8} \, {\left (B c^{3} d^{4} + 4 \, A c^{3} d^{3} e + 18 \, B a c^{2} d^{2} e^{2} + 12 \, A a c^{2} d e^{3} + 3 \, B a^{2} c e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (A c^{3} d^{4} + 12 \, B a c^{2} d^{3} e + 18 \, A a c^{2} d^{2} e^{2} + 12 \, B a^{2} c d e^{3} + 3 \, A a^{2} c e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, B a c^{2} d^{4} + 12 \, A a c^{2} d^{3} e + 18 \, B a^{2} c d^{2} e^{2} + 12 \, A a^{2} c d e^{3} + B a^{3} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, A a c^{2} d^{4} + 12 \, B a^{2} c d^{3} e + 18 \, A a^{2} c d^{2} e^{2} + 4 \, B a^{3} d e^{3} + A a^{3} e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, B a^{2} c d^{4} + 12 \, A a^{2} c d^{3} e + 6 \, B a^{3} d^{2} e^{2} + 4 \, A a^{3} d e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, A a^{2} c d^{4} + 4 \, B a^{3} d^{3} e + 6 \, A a^{3} d^{2} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{3} d^{4} + 4 \, A a^{3} d^{3} e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 438, normalized size = 1.31 \[ x^5\,\left (\frac {4\,B\,a^3\,d\,e^3}{5}+\frac {A\,a^3\,e^4}{5}+\frac {12\,B\,a^2\,c\,d^3\,e}{5}+\frac {18\,A\,a^2\,c\,d^2\,e^2}{5}+\frac {3\,A\,a\,c^2\,d^4}{5}\right )+x^8\,\left (\frac {3\,B\,a^2\,c\,e^4}{8}+\frac {9\,B\,a\,c^2\,d^2\,e^2}{4}+\frac {3\,A\,a\,c^2\,d\,e^3}{2}+\frac {B\,c^3\,d^4}{8}+\frac {A\,c^3\,d^3\,e}{2}\right )+x^6\,\left (\frac {B\,a^3\,e^4}{6}+3\,B\,a^2\,c\,d^2\,e^2+2\,A\,a^2\,c\,d\,e^3+\frac {B\,a\,c^2\,d^4}{2}+2\,A\,a\,c^2\,d^3\,e\right )+x^7\,\left (\frac {12\,B\,a^2\,c\,d\,e^3}{7}+\frac {3\,A\,a^2\,c\,e^4}{7}+\frac {12\,B\,a\,c^2\,d^3\,e}{7}+\frac {18\,A\,a\,c^2\,d^2\,e^2}{7}+\frac {A\,c^3\,d^4}{7}\right )+\frac {a^3\,d^3\,x^2\,\left (4\,A\,e+B\,d\right )}{2}+\frac {c^3\,e^3\,x^{11}\,\left (A\,e+4\,B\,d\right )}{11}+\frac {a^2\,d^2\,x^3\,\left (3\,A\,c\,d^2+4\,B\,a\,d\,e+6\,A\,a\,e^2\right )}{3}+\frac {c^2\,e^2\,x^{10}\,\left (6\,B\,c\,d^2+4\,A\,c\,d\,e+3\,B\,a\,e^2\right )}{10}+A\,a^3\,d^4\,x+\frac {a^2\,d\,x^4\,\left (3\,B\,c\,d^3+12\,A\,c\,d^2\,e+6\,B\,a\,d\,e^2+4\,A\,a\,e^3\right )}{4}+\frac {c^2\,e\,x^9\,\left (4\,B\,c\,d^3+6\,A\,c\,d^2\,e+12\,B\,a\,d\,e^2+3\,A\,a\,e^3\right )}{9}+\frac {B\,c^3\,e^4\,x^{12}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 564, normalized size = 1.69 \[ A a^{3} d^{4} x + \frac {B c^{3} e^{4} x^{12}}{12} + x^{11} \left (\frac {A c^{3} e^{4}}{11} + \frac {4 B c^{3} d e^{3}}{11}\right ) + x^{10} \left (\frac {2 A c^{3} d e^{3}}{5} + \frac {3 B a c^{2} e^{4}}{10} + \frac {3 B c^{3} d^{2} e^{2}}{5}\right ) + x^{9} \left (\frac {A a c^{2} e^{4}}{3} + \frac {2 A c^{3} d^{2} e^{2}}{3} + \frac {4 B a c^{2} d e^{3}}{3} + \frac {4 B c^{3} d^{3} e}{9}\right ) + x^{8} \left (\frac {3 A a c^{2} d e^{3}}{2} + \frac {A c^{3} d^{3} e}{2} + \frac {3 B a^{2} c e^{4}}{8} + \frac {9 B a c^{2} d^{2} e^{2}}{4} + \frac {B c^{3} d^{4}}{8}\right ) + x^{7} \left (\frac {3 A a^{2} c e^{4}}{7} + \frac {18 A a c^{2} d^{2} e^{2}}{7} + \frac {A c^{3} d^{4}}{7} + \frac {12 B a^{2} c d e^{3}}{7} + \frac {12 B a c^{2} d^{3} e}{7}\right ) + x^{6} \left (2 A a^{2} c d e^{3} + 2 A a c^{2} d^{3} e + \frac {B a^{3} e^{4}}{6} + 3 B a^{2} c d^{2} e^{2} + \frac {B a c^{2} d^{4}}{2}\right ) + x^{5} \left (\frac {A a^{3} e^{4}}{5} + \frac {18 A a^{2} c d^{2} e^{2}}{5} + \frac {3 A a c^{2} d^{4}}{5} + \frac {4 B a^{3} d e^{3}}{5} + \frac {12 B a^{2} c d^{3} e}{5}\right ) + x^{4} \left (A a^{3} d e^{3} + 3 A a^{2} c d^{3} e + \frac {3 B a^{3} d^{2} e^{2}}{2} + \frac {3 B a^{2} c d^{4}}{4}\right ) + x^{3} \left (2 A a^{3} d^{2} e^{2} + A a^{2} c d^{4} + \frac {4 B a^{3} d^{3} e}{3}\right ) + x^{2} \left (2 A a^{3} d^{3} e + \frac {B a^{3} d^{4}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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