Optimal. Leaf size=108 \[ -\frac {(B d-A e) \left (c d^2+a e^2\right ) (d+e x)^6}{6 e^4}+\frac {\left (3 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^7}{7 e^4}-\frac {c (3 B d-A e) (d+e x)^8}{8 e^4}+\frac {B c (d+e x)^9}{9 e^4} \]
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Rubi [A]
time = 0.13, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {786}
\begin {gather*} \frac {(d+e x)^7 \left (a B e^2-2 A c d e+3 B c d^2\right )}{7 e^4}-\frac {(d+e x)^6 \left (a e^2+c d^2\right ) (B d-A e)}{6 e^4}-\frac {c (d+e x)^8 (3 B d-A e)}{8 e^4}+\frac {B c (d+e x)^9}{9 e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 786
Rubi steps
\begin {align*} \int (A+B x) (d+e x)^5 \left (a+c x^2\right ) \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2+a e^2\right ) (d+e x)^5}{e^3}+\frac {\left (3 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^6}{e^3}+\frac {c (-3 B d+A e) (d+e x)^7}{e^3}+\frac {B c (d+e x)^8}{e^3}\right ) \, dx\\ &=-\frac {(B d-A e) \left (c d^2+a e^2\right ) (d+e x)^6}{6 e^4}+\frac {\left (3 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^7}{7 e^4}-\frac {c (3 B d-A e) (d+e x)^8}{8 e^4}+\frac {B c (d+e x)^9}{9 e^4}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(233\) vs. \(2(108)=216\).
time = 0.04, size = 233, normalized size = 2.16 \begin {gather*} a A d^5 x+\frac {1}{2} a d^4 (B d+5 A e) x^2+\frac {1}{3} d^3 \left (A c d^2+5 a B d e+10 a A e^2\right ) x^3+\frac {1}{4} d^2 \left (B c d^3+5 A c d^2 e+10 a B d e^2+10 a A e^3\right ) x^4+d e \left (B c d^3+2 A c d^2 e+2 a B d e^2+a A e^3\right ) x^5+\frac {1}{6} e^2 \left (10 B c d^3+10 A c d^2 e+5 a B d e^2+a A e^3\right ) x^6+\frac {1}{7} e^3 \left (10 B c d^2+5 A c d e+a B e^2\right ) x^7+\frac {1}{8} c e^4 (5 B d+A e) x^8+\frac {1}{9} B c e^5 x^9 \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(246\) vs.
\(2(100)=200\).
time = 0.63, size = 247, normalized size = 2.29
method | result | size |
norman | \(\frac {B \,e^{5} c \,x^{9}}{9}+\left (\frac {1}{8} A c \,e^{5}+\frac {5}{8} B c d \,e^{4}\right ) x^{8}+\left (\frac {5}{7} A c d \,e^{4}+\frac {1}{7} B \,e^{5} a +\frac {10}{7} B c \,d^{2} e^{3}\right ) x^{7}+\left (\frac {1}{6} A a \,e^{5}+\frac {5}{3} A c \,d^{2} e^{3}+\frac {5}{6} B a d \,e^{4}+\frac {5}{3} B c \,d^{3} e^{2}\right ) x^{6}+\left (A a d \,e^{4}+2 A c \,d^{3} e^{2}+2 B a \,d^{2} e^{3}+B c \,d^{4} e \right ) x^{5}+\left (\frac {5}{2} A a \,d^{2} e^{3}+\frac {5}{4} A c \,d^{4} e +\frac {5}{2} B a \,d^{3} e^{2}+\frac {1}{4} B c \,d^{5}\right ) x^{4}+\left (\frac {10}{3} A a \,d^{3} e^{2}+\frac {1}{3} A \,d^{5} c +\frac {5}{3} B a \,d^{4} e \right ) x^{3}+\left (\frac {5}{2} A a \,d^{4} e +\frac {1}{2} B a \,d^{5}\right ) x^{2}+A \,d^{5} a x\) | \(238\) |
default | \(\frac {B \,e^{5} c \,x^{9}}{9}+\frac {\left (A \,e^{5}+5 B d \,e^{4}\right ) c \,x^{8}}{8}+\frac {\left (\left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) c +B \,e^{5} a \right ) x^{7}}{7}+\frac {\left (\left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) c +\left (A \,e^{5}+5 B d \,e^{4}\right ) a \right ) x^{6}}{6}+\frac {\left (\left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) c +\left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) a \right ) x^{5}}{5}+\frac {\left (\left (5 A \,d^{4} e +B \,d^{5}\right ) c +\left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) a \right ) x^{4}}{4}+\frac {\left (A \,d^{5} c +\left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) a \right ) x^{3}}{3}+\frac {\left (5 A \,d^{4} e +B \,d^{5}\right ) a \,x^{2}}{2}+A \,d^{5} a x\) | \(247\) |
gosper | \(\frac {1}{9} B \,e^{5} c \,x^{9}+\frac {1}{8} x^{8} A c \,e^{5}+\frac {5}{8} x^{8} B c d \,e^{4}+\frac {5}{7} x^{7} A c d \,e^{4}+\frac {1}{7} x^{7} B \,e^{5} a +\frac {10}{7} x^{7} B c \,d^{2} e^{3}+\frac {1}{6} x^{6} A a \,e^{5}+\frac {5}{3} x^{6} A c \,d^{2} e^{3}+\frac {5}{6} x^{6} B a d \,e^{4}+\frac {5}{3} x^{6} B c \,d^{3} e^{2}+A a d \,e^{4} x^{5}+2 A c \,d^{3} e^{2} x^{5}+2 B a \,d^{2} e^{3} x^{5}+B c \,d^{4} e \,x^{5}+\frac {5}{2} x^{4} A a \,d^{2} e^{3}+\frac {5}{4} x^{4} A c \,d^{4} e +\frac {5}{2} x^{4} B a \,d^{3} e^{2}+\frac {1}{4} x^{4} B c \,d^{5}+\frac {10}{3} x^{3} A a \,d^{3} e^{2}+\frac {1}{3} x^{3} A \,d^{5} c +\frac {5}{3} x^{3} B a \,d^{4} e +\frac {5}{2} x^{2} A a \,d^{4} e +\frac {1}{2} x^{2} B a \,d^{5}+A \,d^{5} a x\) | \(269\) |
risch | \(\frac {1}{9} B \,e^{5} c \,x^{9}+\frac {1}{8} x^{8} A c \,e^{5}+\frac {5}{8} x^{8} B c d \,e^{4}+\frac {5}{7} x^{7} A c d \,e^{4}+\frac {1}{7} x^{7} B \,e^{5} a +\frac {10}{7} x^{7} B c \,d^{2} e^{3}+\frac {1}{6} x^{6} A a \,e^{5}+\frac {5}{3} x^{6} A c \,d^{2} e^{3}+\frac {5}{6} x^{6} B a d \,e^{4}+\frac {5}{3} x^{6} B c \,d^{3} e^{2}+A a d \,e^{4} x^{5}+2 A c \,d^{3} e^{2} x^{5}+2 B a \,d^{2} e^{3} x^{5}+B c \,d^{4} e \,x^{5}+\frac {5}{2} x^{4} A a \,d^{2} e^{3}+\frac {5}{4} x^{4} A c \,d^{4} e +\frac {5}{2} x^{4} B a \,d^{3} e^{2}+\frac {1}{4} x^{4} B c \,d^{5}+\frac {10}{3} x^{3} A a \,d^{3} e^{2}+\frac {1}{3} x^{3} A \,d^{5} c +\frac {5}{3} x^{3} B a \,d^{4} e +\frac {5}{2} x^{2} A a \,d^{4} e +\frac {1}{2} x^{2} B a \,d^{5}+A \,d^{5} a x\) | \(269\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 225 vs.
\(2 (101) = 202\).
time = 0.27, size = 225, normalized size = 2.08 \begin {gather*} \frac {1}{9} \, B c x^{9} e^{5} + \frac {1}{8} \, {\left (5 \, B c d e^{4} + A c e^{5}\right )} x^{8} + A a d^{5} x + \frac {1}{7} \, {\left (10 \, B c d^{2} e^{3} + 5 \, A c d e^{4} + B a e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (10 \, B c d^{3} e^{2} + 10 \, A c d^{2} e^{3} + 5 \, B a d e^{4} + A a e^{5}\right )} x^{6} + {\left (B c d^{4} e + 2 \, A c d^{3} e^{2} + 2 \, B a d^{2} e^{3} + A a d e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (B c d^{5} + 5 \, A c d^{4} e + 10 \, B a d^{3} e^{2} + 10 \, A a d^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (A c d^{5} + 5 \, B a d^{4} e + 10 \, A a d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a d^{5} + 5 \, A a d^{4} e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 243 vs.
\(2 (101) = 202\).
time = 3.58, size = 243, normalized size = 2.25 \begin {gather*} \frac {1}{4} \, B c d^{5} x^{4} + \frac {1}{3} \, A c d^{5} x^{3} + \frac {1}{2} \, B a d^{5} x^{2} + A a d^{5} x + \frac {1}{504} \, {\left (56 \, B c x^{9} + 63 \, A c x^{8} + 72 \, B a x^{7} + 84 \, A a x^{6}\right )} e^{5} + \frac {1}{168} \, {\left (105 \, B c d x^{8} + 120 \, A c d x^{7} + 140 \, B a d x^{6} + 168 \, A a d x^{5}\right )} e^{4} + \frac {1}{42} \, {\left (60 \, B c d^{2} x^{7} + 70 \, A c d^{2} x^{6} + 84 \, B a d^{2} x^{5} + 105 \, A a d^{2} x^{4}\right )} e^{3} + \frac {1}{6} \, {\left (10 \, B c d^{3} x^{6} + 12 \, A c d^{3} x^{5} + 15 \, B a d^{3} x^{4} + 20 \, A a d^{3} x^{3}\right )} e^{2} + \frac {1}{12} \, {\left (12 \, B c d^{4} x^{5} + 15 \, A c d^{4} x^{4} + 20 \, B a d^{4} x^{3} + 30 \, A a d^{4} x^{2}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 287 vs.
\(2 (105) = 210\).
time = 0.03, size = 287, normalized size = 2.66 \begin {gather*} A a d^{5} x + \frac {B c e^{5} x^{9}}{9} + x^{8} \left (\frac {A c e^{5}}{8} + \frac {5 B c d e^{4}}{8}\right ) + x^{7} \cdot \left (\frac {5 A c d e^{4}}{7} + \frac {B a e^{5}}{7} + \frac {10 B c d^{2} e^{3}}{7}\right ) + x^{6} \left (\frac {A a e^{5}}{6} + \frac {5 A c d^{2} e^{3}}{3} + \frac {5 B a d e^{4}}{6} + \frac {5 B c d^{3} e^{2}}{3}\right ) + x^{5} \left (A a d e^{4} + 2 A c d^{3} e^{2} + 2 B a d^{2} e^{3} + B c d^{4} e\right ) + x^{4} \cdot \left (\frac {5 A a d^{2} e^{3}}{2} + \frac {5 A c d^{4} e}{4} + \frac {5 B a d^{3} e^{2}}{2} + \frac {B c d^{5}}{4}\right ) + x^{3} \cdot \left (\frac {10 A a d^{3} e^{2}}{3} + \frac {A c d^{5}}{3} + \frac {5 B a d^{4} e}{3}\right ) + x^{2} \cdot \left (\frac {5 A a d^{4} e}{2} + \frac {B a d^{5}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 256 vs.
\(2 (101) = 202\).
time = 4.49, size = 256, normalized size = 2.37 \begin {gather*} \frac {1}{9} \, B c x^{9} e^{5} + \frac {5}{8} \, B c d x^{8} e^{4} + \frac {10}{7} \, B c d^{2} x^{7} e^{3} + \frac {5}{3} \, B c d^{3} x^{6} e^{2} + B c d^{4} x^{5} e + \frac {1}{4} \, B c d^{5} x^{4} + \frac {1}{8} \, A c x^{8} e^{5} + \frac {5}{7} \, A c d x^{7} e^{4} + \frac {5}{3} \, A c d^{2} x^{6} e^{3} + 2 \, A c d^{3} x^{5} e^{2} + \frac {5}{4} \, A c d^{4} x^{4} e + \frac {1}{3} \, A c d^{5} x^{3} + \frac {1}{7} \, B a x^{7} e^{5} + \frac {5}{6} \, B a d x^{6} e^{4} + 2 \, B a d^{2} x^{5} e^{3} + \frac {5}{2} \, B a d^{3} x^{4} e^{2} + \frac {5}{3} \, B a d^{4} x^{3} e + \frac {1}{2} \, B a d^{5} x^{2} + \frac {1}{6} \, A a x^{6} e^{5} + A a d x^{5} e^{4} + \frac {5}{2} \, A a d^{2} x^{4} e^{3} + \frac {10}{3} \, A a d^{3} x^{3} e^{2} + \frac {5}{2} \, A a d^{4} x^{2} e + A a d^{5} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 231, normalized size = 2.14 \begin {gather*} x^5\,\left (B\,c\,d^4\,e+2\,A\,c\,d^3\,e^2+2\,B\,a\,d^2\,e^3+A\,a\,d\,e^4\right )+x^3\,\left (\frac {A\,c\,d^5}{3}+\frac {5\,B\,a\,d^4\,e}{3}+\frac {10\,A\,a\,d^3\,e^2}{3}\right )+x^7\,\left (\frac {10\,B\,c\,d^2\,e^3}{7}+\frac {5\,A\,c\,d\,e^4}{7}+\frac {B\,a\,e^5}{7}\right )+x^4\,\left (\frac {B\,c\,d^5}{4}+\frac {5\,A\,c\,d^4\,e}{4}+\frac {5\,B\,a\,d^3\,e^2}{2}+\frac {5\,A\,a\,d^2\,e^3}{2}\right )+x^6\,\left (\frac {5\,B\,c\,d^3\,e^2}{3}+\frac {5\,A\,c\,d^2\,e^3}{3}+\frac {5\,B\,a\,d\,e^4}{6}+\frac {A\,a\,e^5}{6}\right )+A\,a\,d^5\,x+\frac {B\,c\,e^5\,x^9}{9}+\frac {a\,d^4\,x^2\,\left (5\,A\,e+B\,d\right )}{2}+\frac {c\,e^4\,x^8\,\left (A\,e+5\,B\,d\right )}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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