Optimal. Leaf size=334 \[ -\frac {(B d-A e) \left (c d^2+a e^2\right )^3 (d+e x)^5}{5 e^8}+\frac {\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right ) (d+e x)^6}{6 e^8}-\frac {3 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right ) (d+e x)^7}{7 e^8}-\frac {c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right ) (d+e x)^8}{8 e^8}-\frac {c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right ) (d+e x)^9}{9 e^8}+\frac {3 c^2 \left (7 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{10}}{10 e^8}-\frac {c^3 (7 B d-A e) (d+e x)^{11}}{11 e^8}+\frac {B c^3 (d+e x)^{12}}{12 e^8} \]
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Rubi [A]
time = 0.31, 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 = {786}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 786
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 = 0.09, size = 436, normalized size = 1.31 \begin {gather*} a^3 A d^4 x+\frac {1}{2} a^3 d^3 (B d+4 A e) x^2+\frac {1}{3} a^2 d^2 \left (3 A c d^2+4 a B d e+6 a A e^2\right ) x^3+\frac {1}{4} a^2 d \left (3 B c d^3+12 A c d^2 e+6 a B d e^2+4 a A e^3\right ) x^4+\frac {1}{5} a \left (4 a B d e \left (3 c d^2+a e^2\right )+A \left (3 c^2 d^4+18 a c d^2 e^2+a^2 e^4\right )\right ) x^5+\frac {1}{6} a \left (12 A c d e \left (c d^2+a e^2\right )+B \left (3 c^2 d^4+18 a c d^2 e^2+a^2 e^4\right )\right ) x^6+\frac {1}{7} c \left (12 a B d e \left (c d^2+a e^2\right )+A \left (c^2 d^4+18 a c d^2 e^2+3 a^2 e^4\right )\right ) x^7+\frac {1}{8} c \left (4 A c d e \left (c d^2+3 a e^2\right )+B \left (c^2 d^4+18 a c d^2 e^2+3 a^2 e^4\right )\right ) x^8+\frac {1}{9} c^2 e \left (4 B c d^3+6 A c d^2 e+12 a B d e^2+3 a A e^3\right ) x^9+\frac {1}{10} c^2 e^2 \left (6 B c d^2+4 A c d e+3 a B e^2\right ) x^{10}+\frac {1}{11} c^3 e^3 (4 B d+A e) x^{11}+\frac {1}{12} B c^3 e^4 x^{12} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.69, size = 455, normalized size = 1.36 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 462, normalized size = 1.38 \begin {gather*} \frac {1}{12} \, B c^{3} x^{12} e^{4} + \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.65, size = 477, normalized size = 1.43 \begin {gather*} \frac {1}{8} \, B c^{3} d^{4} x^{8} + \frac {1}{7} \, A c^{3} d^{4} x^{7} + \frac {1}{2} \, B a c^{2} d^{4} x^{6} + \frac {3}{5} \, A a c^{2} d^{4} x^{5} + \frac {3}{4} \, B a^{2} c d^{4} x^{4} + A a^{2} c d^{4} x^{3} + \frac {1}{2} \, B a^{3} d^{4} x^{2} + A a^{3} d^{4} x + \frac {1}{9240} \, {\left (770 \, B c^{3} x^{12} + 840 \, A c^{3} x^{11} + 2772 \, B a c^{2} x^{10} + 3080 \, A a c^{2} x^{9} + 3465 \, B a^{2} c x^{8} + 3960 \, A a^{2} c x^{7} + 1540 \, B a^{3} x^{6} + 1848 \, A a^{3} x^{5}\right )} e^{4} + \frac {1}{2310} \, {\left (840 \, B c^{3} d x^{11} + 924 \, A c^{3} d x^{10} + 3080 \, B a c^{2} d x^{9} + 3465 \, A a c^{2} d x^{8} + 3960 \, B a^{2} c d x^{7} + 4620 \, A a^{2} c d x^{6} + 1848 \, B a^{3} d x^{5} + 2310 \, A a^{3} d x^{4}\right )} e^{3} + \frac {1}{420} \, {\left (252 \, B c^{3} d^{2} x^{10} + 280 \, A c^{3} d^{2} x^{9} + 945 \, B a c^{2} d^{2} x^{8} + 1080 \, A a c^{2} d^{2} x^{7} + 1260 \, B a^{2} c d^{2} x^{6} + 1512 \, A a^{2} c d^{2} x^{5} + 630 \, B a^{3} d^{2} x^{4} + 840 \, A a^{3} d^{2} x^{3}\right )} e^{2} + \frac {1}{630} \, {\left (280 \, B c^{3} d^{3} x^{9} + 315 \, A c^{3} d^{3} x^{8} + 1080 \, B a c^{2} d^{3} x^{7} + 1260 \, A a c^{2} d^{3} x^{6} + 1512 \, B a^{2} c d^{3} x^{5} + 1890 \, A a^{2} c d^{3} x^{4} + 840 \, B a^{3} d^{3} x^{3} + 1260 \, A a^{3} d^{3} x^{2}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 564, normalized size = 1.69 \begin {gather*} 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} \cdot \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} \cdot \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} \cdot \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} \cdot \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} \cdot \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} \cdot \left (2 A a^{3} d^{3} e + \frac {B a^{3} d^{4}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.30, size = 520, normalized size = 1.56 \begin {gather*} \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 \end {gather*}
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 \begin {gather*} 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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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