Optimal. Leaf size=206 \[ -\frac {(B d-A e) \left (c d^2+a e^2\right )^2 (d+e x)^4}{4 e^6}+\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right ) (d+e x)^5}{5 e^6}-\frac {c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right ) (d+e x)^6}{3 e^6}+\frac {2 c \left (5 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^7}{7 e^6}-\frac {c^2 (5 B d-A e) (d+e x)^8}{8 e^6}+\frac {B c^2 (d+e x)^9}{9 e^6} \]
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Rubi [A]
time = 0.15, antiderivative size = 206, 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 {2 c (d+e x)^7 \left (a B e^2-2 A c d e+5 B c d^2\right )}{7 e^6}+\frac {(d+e x)^5 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{5 e^6}-\frac {(d+e x)^4 \left (a e^2+c d^2\right )^2 (B d-A e)}{4 e^6}-\frac {c (d+e x)^6 \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{3 e^6}-\frac {c^2 (d+e x)^8 (5 B d-A e)}{8 e^6}+\frac {B c^2 (d+e x)^9}{9 e^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 786
Rubi steps
\begin {align*} \int (A+B x) (d+e x)^3 \left (a+c x^2\right )^2 \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2+a e^2\right )^2 (d+e x)^3}{e^5}+\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right ) (d+e x)^4}{e^5}+\frac {2 c \left (-5 B c d^3+3 A c d^2 e-3 a B d e^2+a A e^3\right ) (d+e x)^5}{e^5}-\frac {2 c \left (-5 B c d^2+2 A c d e-a B e^2\right ) (d+e x)^6}{e^5}+\frac {c^2 (-5 B d+A e) (d+e x)^7}{e^5}+\frac {B c^2 (d+e x)^8}{e^5}\right ) \, dx\\ &=-\frac {(B d-A e) \left (c d^2+a e^2\right )^2 (d+e x)^4}{4 e^6}+\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right ) (d+e x)^5}{5 e^6}-\frac {c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right ) (d+e x)^6}{3 e^6}+\frac {2 c \left (5 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^7}{7 e^6}-\frac {c^2 (5 B d-A e) (d+e x)^8}{8 e^6}+\frac {B c^2 (d+e x)^9}{9 e^6}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 244, normalized size = 1.18 \begin {gather*} a^2 A d^3 x+\frac {1}{2} a^2 d^2 (B d+3 A e) x^2+\frac {1}{3} a d \left (2 A c d^2+3 a B d e+3 a A e^2\right ) x^3+\frac {1}{4} a \left (2 B c d^3+6 A c d^2 e+3 a B d e^2+a A e^3\right ) x^4+\frac {1}{5} \left (A c^2 d^3+6 a B c d^2 e+6 a A c d e^2+a^2 B e^3\right ) x^5+\frac {1}{6} c \left (B c d^3+3 A c d^2 e+6 a B d e^2+2 a A e^3\right ) x^6+\frac {1}{7} c e \left (3 B c d^2+3 A c d e+2 a B e^2\right ) x^7+\frac {1}{8} c^2 e^2 (3 B d+A e) x^8+\frac {1}{9} B c^2 e^3 x^9 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.69, size = 252, normalized size = 1.22
method | result | size |
default | \(\frac {B \,c^{2} e^{3} x^{9}}{9}+\frac {\left (A \,e^{3}+3 B d \,e^{2}\right ) c^{2} x^{8}}{8}+\frac {\left (\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) c^{2}+2 B \,e^{3} a c \right ) x^{7}}{7}+\frac {\left (\left (3 A \,d^{2} e +B \,d^{3}\right ) c^{2}+2 \left (A \,e^{3}+3 B d \,e^{2}\right ) a c \right ) x^{6}}{6}+\frac {\left (A \,d^{3} c^{2}+2 \left (3 A d \,e^{2}+3 B \,d^{2} e \right ) a c +a^{2} B \,e^{3}\right ) x^{5}}{5}+\frac {\left (2 \left (3 A \,d^{2} e +B \,d^{3}\right ) a c +\left (A \,e^{3}+3 B d \,e^{2}\right ) a^{2}\right ) x^{4}}{4}+\frac {\left (2 A \,d^{3} a c +\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) a^{2}\right ) x^{3}}{3}+\frac {\left (3 A \,d^{2} e +B \,d^{3}\right ) a^{2} x^{2}}{2}+A \,d^{3} a^{2} x\) | \(252\) |
norman | \(\frac {B \,c^{2} e^{3} x^{9}}{9}+\left (\frac {1}{8} A \,c^{2} e^{3}+\frac {3}{8} B \,c^{2} d \,e^{2}\right ) x^{8}+\left (\frac {3}{7} A \,c^{2} d \,e^{2}+\frac {2}{7} B \,e^{3} a c +\frac {3}{7} B \,c^{2} d^{2} e \right ) x^{7}+\left (\frac {1}{3} A a c \,e^{3}+\frac {1}{2} A \,c^{2} d^{2} e +B a c d \,e^{2}+\frac {1}{6} B \,c^{2} d^{3}\right ) x^{6}+\left (\frac {6}{5} A a c d \,e^{2}+\frac {1}{5} A \,d^{3} c^{2}+\frac {1}{5} a^{2} B \,e^{3}+\frac {6}{5} B a c \,d^{2} e \right ) x^{5}+\left (\frac {1}{4} a^{2} A \,e^{3}+\frac {3}{2} A a c \,d^{2} e +\frac {3}{4} B \,a^{2} d \,e^{2}+\frac {1}{2} B a c \,d^{3}\right ) x^{4}+\left (A \,a^{2} d \,e^{2}+\frac {2}{3} A \,d^{3} a c +B \,a^{2} d^{2} e \right ) x^{3}+\left (\frac {3}{2} A \,a^{2} d^{2} e +\frac {1}{2} B \,a^{2} d^{3}\right ) x^{2}+A \,d^{3} a^{2} x\) | \(257\) |
gosper | \(\frac {1}{9} B \,c^{2} e^{3} x^{9}+\frac {1}{8} x^{8} A \,c^{2} e^{3}+\frac {3}{8} x^{8} B \,c^{2} d \,e^{2}+\frac {3}{7} x^{7} A \,c^{2} d \,e^{2}+\frac {2}{7} x^{7} B \,e^{3} a c +\frac {3}{7} x^{7} B \,c^{2} d^{2} e +\frac {1}{3} x^{6} A a c \,e^{3}+\frac {1}{2} x^{6} A \,c^{2} d^{2} e +x^{6} B a c d \,e^{2}+\frac {1}{6} x^{6} B \,c^{2} d^{3}+\frac {6}{5} x^{5} A a c d \,e^{2}+\frac {1}{5} x^{5} A \,d^{3} c^{2}+\frac {1}{5} x^{5} a^{2} B \,e^{3}+\frac {6}{5} x^{5} B a c \,d^{2} e +\frac {1}{4} x^{4} a^{2} A \,e^{3}+\frac {3}{2} x^{4} A a c \,d^{2} e +\frac {3}{4} x^{4} B \,a^{2} d \,e^{2}+\frac {1}{2} x^{4} B a c \,d^{3}+x^{3} A \,a^{2} d \,e^{2}+\frac {2}{3} x^{3} A \,d^{3} a c +x^{3} B \,a^{2} d^{2} e +\frac {3}{2} x^{2} A \,a^{2} d^{2} e +\frac {1}{2} x^{2} B \,a^{2} d^{3}+A \,d^{3} a^{2} x\) | \(288\) |
risch | \(\frac {1}{9} B \,c^{2} e^{3} x^{9}+\frac {1}{8} x^{8} A \,c^{2} e^{3}+\frac {3}{8} x^{8} B \,c^{2} d \,e^{2}+\frac {3}{7} x^{7} A \,c^{2} d \,e^{2}+\frac {2}{7} x^{7} B \,e^{3} a c +\frac {3}{7} x^{7} B \,c^{2} d^{2} e +\frac {1}{3} x^{6} A a c \,e^{3}+\frac {1}{2} x^{6} A \,c^{2} d^{2} e +x^{6} B a c d \,e^{2}+\frac {1}{6} x^{6} B \,c^{2} d^{3}+\frac {6}{5} x^{5} A a c d \,e^{2}+\frac {1}{5} x^{5} A \,d^{3} c^{2}+\frac {1}{5} x^{5} a^{2} B \,e^{3}+\frac {6}{5} x^{5} B a c \,d^{2} e +\frac {1}{4} x^{4} a^{2} A \,e^{3}+\frac {3}{2} x^{4} A a c \,d^{2} e +\frac {3}{4} x^{4} B \,a^{2} d \,e^{2}+\frac {1}{2} x^{4} B a c \,d^{3}+x^{3} A \,a^{2} d \,e^{2}+\frac {2}{3} x^{3} A \,d^{3} a c +x^{3} B \,a^{2} d^{2} e +\frac {3}{2} x^{2} A \,a^{2} d^{2} e +\frac {1}{2} x^{2} B \,a^{2} d^{3}+A \,d^{3} a^{2} x\) | \(288\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 254, normalized size = 1.23 \begin {gather*} \frac {1}{9} \, B c^{2} x^{9} e^{3} + \frac {1}{8} \, {\left (3 \, B c^{2} d e^{2} + A c^{2} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (3 \, B c^{2} d^{2} e + 3 \, A c^{2} d e^{2} + 2 \, B a c e^{3}\right )} x^{7} + A a^{2} d^{3} x + \frac {1}{6} \, {\left (B c^{2} d^{3} + 3 \, A c^{2} d^{2} e + 6 \, B a c d e^{2} + 2 \, A a c e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (A c^{2} d^{3} + 6 \, B a c d^{2} e + 6 \, A a c d e^{2} + B a^{2} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (2 \, B a c d^{3} + 6 \, A a c d^{2} e + 3 \, B a^{2} d e^{2} + A a^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (2 \, A a c d^{3} + 3 \, B a^{2} d^{2} e + 3 \, A a^{2} d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{2} d^{3} + 3 \, A a^{2} d^{2} e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.05, size = 263, normalized size = 1.28 \begin {gather*} \frac {1}{6} \, B c^{2} d^{3} x^{6} + \frac {1}{5} \, A c^{2} d^{3} x^{5} + \frac {1}{2} \, B a c d^{3} x^{4} + \frac {2}{3} \, A a c d^{3} x^{3} + \frac {1}{2} \, B a^{2} d^{3} x^{2} + A a^{2} d^{3} x + \frac {1}{2520} \, {\left (280 \, B c^{2} x^{9} + 315 \, A c^{2} x^{8} + 720 \, B a c x^{7} + 840 \, A a c x^{6} + 504 \, B a^{2} x^{5} + 630 \, A a^{2} x^{4}\right )} e^{3} + \frac {1}{280} \, {\left (105 \, B c^{2} d x^{8} + 120 \, A c^{2} d x^{7} + 280 \, B a c d x^{6} + 336 \, A a c d x^{5} + 210 \, B a^{2} d x^{4} + 280 \, A a^{2} d x^{3}\right )} e^{2} + \frac {1}{70} \, {\left (30 \, B c^{2} d^{2} x^{7} + 35 \, A c^{2} d^{2} x^{6} + 84 \, B a c d^{2} x^{5} + 105 \, A a c d^{2} x^{4} + 70 \, B a^{2} d^{2} x^{3} + 105 \, A a^{2} d^{2} x^{2}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 303, normalized size = 1.47 \begin {gather*} A a^{2} d^{3} x + \frac {B c^{2} e^{3} x^{9}}{9} + x^{8} \left (\frac {A c^{2} e^{3}}{8} + \frac {3 B c^{2} d e^{2}}{8}\right ) + x^{7} \cdot \left (\frac {3 A c^{2} d e^{2}}{7} + \frac {2 B a c e^{3}}{7} + \frac {3 B c^{2} d^{2} e}{7}\right ) + x^{6} \left (\frac {A a c e^{3}}{3} + \frac {A c^{2} d^{2} e}{2} + B a c d e^{2} + \frac {B c^{2} d^{3}}{6}\right ) + x^{5} \cdot \left (\frac {6 A a c d e^{2}}{5} + \frac {A c^{2} d^{3}}{5} + \frac {B a^{2} e^{3}}{5} + \frac {6 B a c d^{2} e}{5}\right ) + x^{4} \left (\frac {A a^{2} e^{3}}{4} + \frac {3 A a c d^{2} e}{2} + \frac {3 B a^{2} d e^{2}}{4} + \frac {B a c d^{3}}{2}\right ) + x^{3} \left (A a^{2} d e^{2} + \frac {2 A a c d^{3}}{3} + B a^{2} d^{2} e\right ) + x^{2} \cdot \left (\frac {3 A a^{2} d^{2} e}{2} + \frac {B a^{2} d^{3}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.03, size = 281, normalized size = 1.36 \begin {gather*} \frac {1}{9} \, B c^{2} x^{9} e^{3} + \frac {3}{8} \, B c^{2} d x^{8} e^{2} + \frac {3}{7} \, B c^{2} d^{2} x^{7} e + \frac {1}{6} \, B c^{2} d^{3} x^{6} + \frac {1}{8} \, A c^{2} x^{8} e^{3} + \frac {3}{7} \, A c^{2} d x^{7} e^{2} + \frac {1}{2} \, A c^{2} d^{2} x^{6} e + \frac {1}{5} \, A c^{2} d^{3} x^{5} + \frac {2}{7} \, B a c x^{7} e^{3} + B a c d x^{6} e^{2} + \frac {6}{5} \, B a c d^{2} x^{5} e + \frac {1}{2} \, B a c d^{3} x^{4} + \frac {1}{3} \, A a c x^{6} e^{3} + \frac {6}{5} \, A a c d x^{5} e^{2} + \frac {3}{2} \, A a c d^{2} x^{4} e + \frac {2}{3} \, A a c d^{3} x^{3} + \frac {1}{5} \, B a^{2} x^{5} e^{3} + \frac {3}{4} \, B a^{2} d x^{4} e^{2} + B a^{2} d^{2} x^{3} e + \frac {1}{2} \, B a^{2} d^{3} x^{2} + \frac {1}{4} \, A a^{2} x^{4} e^{3} + A a^{2} d x^{3} e^{2} + \frac {3}{2} \, A a^{2} d^{2} x^{2} e + A a^{2} d^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.75, size = 229, normalized size = 1.11 \begin {gather*} x^5\,\left (\frac {B\,a^2\,e^3}{5}+\frac {6\,B\,a\,c\,d^2\,e}{5}+\frac {6\,A\,a\,c\,d\,e^2}{5}+\frac {A\,c^2\,d^3}{5}\right )+\frac {a\,x^4\,\left (2\,B\,c\,d^3+6\,A\,c\,d^2\,e+3\,B\,a\,d\,e^2+A\,a\,e^3\right )}{4}+\frac {c\,x^6\,\left (B\,c\,d^3+3\,A\,c\,d^2\,e+6\,B\,a\,d\,e^2+2\,A\,a\,e^3\right )}{6}+\frac {a^2\,d^2\,x^2\,\left (3\,A\,e+B\,d\right )}{2}+\frac {c^2\,e^2\,x^8\,\left (A\,e+3\,B\,d\right )}{8}+A\,a^2\,d^3\,x+\frac {a\,d\,x^3\,\left (2\,A\,c\,d^2+3\,B\,a\,d\,e+3\,A\,a\,e^2\right )}{3}+\frac {c\,e\,x^7\,\left (3\,B\,c\,d^2+3\,A\,c\,d\,e+2\,B\,a\,e^2\right )}{7}+\frac {B\,c^2\,e^3\,x^9}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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