Optimal. Leaf size=118 \[ \frac {(A b-a B) (b d-a e)^2 (a+b x)^7}{7 b^4}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^8}{8 b^4}+\frac {e (2 b B d+A b e-3 a B e) (a+b x)^9}{9 b^4}+\frac {B e^2 (a+b x)^{10}}{10 b^4} \]
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Rubi [A]
time = 0.22, antiderivative size = 118, 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 = {78}
\begin {gather*} \frac {e (a+b x)^9 (-3 a B e+A b e+2 b B d)}{9 b^4}+\frac {(a+b x)^8 (b d-a e) (-3 a B e+2 A b e+b B d)}{8 b^4}+\frac {(a+b x)^7 (A b-a B) (b d-a e)^2}{7 b^4}+\frac {B e^2 (a+b x)^{10}}{10 b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rubi steps
\begin {align*} \int (a+b x)^6 (A+B x) (d+e x)^2 \, dx &=\int \left (\frac {(A b-a B) (b d-a e)^2 (a+b x)^6}{b^3}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^7}{b^3}+\frac {e (2 b B d+A b e-3 a B e) (a+b x)^8}{b^3}+\frac {B e^2 (a+b x)^9}{b^3}\right ) \, dx\\ &=\frac {(A b-a B) (b d-a e)^2 (a+b x)^7}{7 b^4}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^8}{8 b^4}+\frac {e (2 b B d+A b e-3 a B e) (a+b x)^9}{9 b^4}+\frac {B e^2 (a+b x)^{10}}{10 b^4}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(386\) vs. \(2(118)=236\).
time = 0.15, size = 386, normalized size = 3.27 \begin {gather*} \frac {x \left (210 a^6 \left (4 A \left (3 d^2+3 d e x+e^2 x^2\right )+B x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right )+252 a^5 b x \left (5 A \left (6 d^2+8 d e x+3 e^2 x^2\right )+2 B x \left (10 d^2+15 d e x+6 e^2 x^2\right )\right )+630 a^4 b^2 x^2 \left (2 A \left (10 d^2+15 d e x+6 e^2 x^2\right )+B x \left (15 d^2+24 d e x+10 e^2 x^2\right )\right )+120 a^3 b^3 x^3 \left (7 A \left (15 d^2+24 d e x+10 e^2 x^2\right )+4 B x \left (21 d^2+35 d e x+15 e^2 x^2\right )\right )+45 a^2 b^4 x^4 \left (8 A \left (21 d^2+35 d e x+15 e^2 x^2\right )+5 B x \left (28 d^2+48 d e x+21 e^2 x^2\right )\right )+30 a b^5 x^5 \left (3 A \left (28 d^2+48 d e x+21 e^2 x^2\right )+2 B x \left (36 d^2+63 d e x+28 e^2 x^2\right )\right )+b^6 x^6 \left (10 A \left (36 d^2+63 d e x+28 e^2 x^2\right )+7 B x \left (45 d^2+80 d e x+36 e^2 x^2\right )\right )\right )}{2520} \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. \(468\) vs.
\(2(110)=220\).
time = 0.07, size = 469, normalized size = 3.97 Too large to display
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. 484 vs.
\(2 (116) = 232\).
time = 0.30, size = 484, normalized size = 4.10 \begin {gather*} \frac {1}{10} \, B b^{6} x^{10} e^{2} + A a^{6} d^{2} x + \frac {1}{9} \, {\left (2 \, B b^{6} d e + 6 \, B a b^{5} e^{2} + A b^{6} e^{2}\right )} x^{9} + \frac {1}{8} \, {\left (B b^{6} d^{2} + 15 \, B a^{2} b^{4} e^{2} + 6 \, A a b^{5} e^{2} + 2 \, {\left (6 \, B a b^{5} e + A b^{6} e\right )} d\right )} x^{8} + \frac {1}{7} \, {\left (20 \, B a^{3} b^{3} e^{2} + 15 \, A a^{2} b^{4} e^{2} + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} + 6 \, {\left (5 \, B a^{2} b^{4} e + 2 \, A a b^{5} e\right )} d\right )} x^{7} + \frac {1}{6} \, {\left (15 \, B a^{4} b^{2} e^{2} + 20 \, A a^{3} b^{3} e^{2} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} + 10 \, {\left (4 \, B a^{3} b^{3} e + 3 \, A a^{2} b^{4} e\right )} d\right )} x^{6} + \frac {1}{5} \, {\left (6 \, B a^{5} b e^{2} + 15 \, A a^{4} b^{2} e^{2} + 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} + 10 \, {\left (3 \, B a^{4} b^{2} e + 4 \, A a^{3} b^{3} e\right )} d\right )} x^{5} + \frac {1}{4} \, {\left (B a^{6} e^{2} + 6 \, A a^{5} b e^{2} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} + 6 \, {\left (2 \, B a^{5} b e + 5 \, A a^{4} b^{2} e\right )} d\right )} x^{4} + \frac {1}{3} \, {\left (A a^{6} e^{2} + 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} + 2 \, {\left (B a^{6} e + 6 \, A a^{5} b e\right )} d\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a^{6} d e + {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{2}\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. 476 vs.
\(2 (116) = 232\).
time = 1.01, size = 476, normalized size = 4.03 \begin {gather*} \frac {1}{8} \, B b^{6} d^{2} x^{8} + A a^{6} d^{2} x + \frac {1}{7} \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} x^{7} + \frac {1}{2} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} x^{6} + {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} x^{5} + \frac {5}{4} \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} x^{4} + {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} x^{3} + \frac {1}{2} \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{2} x^{2} + \frac {1}{2520} \, {\left (252 \, B b^{6} x^{10} + 840 \, A a^{6} x^{3} + 280 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{9} + 945 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{8} + 1800 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{7} + 2100 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{6} + 1512 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{5} + 630 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x^{4}\right )} e^{2} + \frac {1}{252} \, {\left (56 \, B b^{6} d x^{9} + 252 \, A a^{6} d x^{2} + 63 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d x^{8} + 216 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d x^{7} + 420 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d x^{6} + 504 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d x^{5} + 378 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d x^{4} + 168 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d x^{3}\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. 568 vs.
\(2 (116) = 232\).
time = 0.05, size = 568, normalized size = 4.81 \begin {gather*} A a^{6} d^{2} x + \frac {B b^{6} e^{2} x^{10}}{10} + x^{9} \left (\frac {A b^{6} e^{2}}{9} + \frac {2 B a b^{5} e^{2}}{3} + \frac {2 B b^{6} d e}{9}\right ) + x^{8} \cdot \left (\frac {3 A a b^{5} e^{2}}{4} + \frac {A b^{6} d e}{4} + \frac {15 B a^{2} b^{4} e^{2}}{8} + \frac {3 B a b^{5} d e}{2} + \frac {B b^{6} d^{2}}{8}\right ) + x^{7} \cdot \left (\frac {15 A a^{2} b^{4} e^{2}}{7} + \frac {12 A a b^{5} d e}{7} + \frac {A b^{6} d^{2}}{7} + \frac {20 B a^{3} b^{3} e^{2}}{7} + \frac {30 B a^{2} b^{4} d e}{7} + \frac {6 B a b^{5} d^{2}}{7}\right ) + x^{6} \cdot \left (\frac {10 A a^{3} b^{3} e^{2}}{3} + 5 A a^{2} b^{4} d e + A a b^{5} d^{2} + \frac {5 B a^{4} b^{2} e^{2}}{2} + \frac {20 B a^{3} b^{3} d e}{3} + \frac {5 B a^{2} b^{4} d^{2}}{2}\right ) + x^{5} \cdot \left (3 A a^{4} b^{2} e^{2} + 8 A a^{3} b^{3} d e + 3 A a^{2} b^{4} d^{2} + \frac {6 B a^{5} b e^{2}}{5} + 6 B a^{4} b^{2} d e + 4 B a^{3} b^{3} d^{2}\right ) + x^{4} \cdot \left (\frac {3 A a^{5} b e^{2}}{2} + \frac {15 A a^{4} b^{2} d e}{2} + 5 A a^{3} b^{3} d^{2} + \frac {B a^{6} e^{2}}{4} + 3 B a^{5} b d e + \frac {15 B a^{4} b^{2} d^{2}}{4}\right ) + x^{3} \left (\frac {A a^{6} e^{2}}{3} + 4 A a^{5} b d e + 5 A a^{4} b^{2} d^{2} + \frac {2 B a^{6} d e}{3} + 2 B a^{5} b d^{2}\right ) + x^{2} \left (A a^{6} d e + 3 A a^{5} b d^{2} + \frac {B a^{6} d^{2}}{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. 552 vs.
\(2 (116) = 232\).
time = 2.53, size = 552, normalized size = 4.68 \begin {gather*} \frac {1}{10} \, B b^{6} x^{10} e^{2} + \frac {2}{9} \, B b^{6} d x^{9} e + \frac {1}{8} \, B b^{6} d^{2} x^{8} + \frac {2}{3} \, B a b^{5} x^{9} e^{2} + \frac {1}{9} \, A b^{6} x^{9} e^{2} + \frac {3}{2} \, B a b^{5} d x^{8} e + \frac {1}{4} \, A b^{6} d x^{8} e + \frac {6}{7} \, B a b^{5} d^{2} x^{7} + \frac {1}{7} \, A b^{6} d^{2} x^{7} + \frac {15}{8} \, B a^{2} b^{4} x^{8} e^{2} + \frac {3}{4} \, A a b^{5} x^{8} e^{2} + \frac {30}{7} \, B a^{2} b^{4} d x^{7} e + \frac {12}{7} \, A a b^{5} d x^{7} e + \frac {5}{2} \, B a^{2} b^{4} d^{2} x^{6} + A a b^{5} d^{2} x^{6} + \frac {20}{7} \, B a^{3} b^{3} x^{7} e^{2} + \frac {15}{7} \, A a^{2} b^{4} x^{7} e^{2} + \frac {20}{3} \, B a^{3} b^{3} d x^{6} e + 5 \, A a^{2} b^{4} d x^{6} e + 4 \, B a^{3} b^{3} d^{2} x^{5} + 3 \, A a^{2} b^{4} d^{2} x^{5} + \frac {5}{2} \, B a^{4} b^{2} x^{6} e^{2} + \frac {10}{3} \, A a^{3} b^{3} x^{6} e^{2} + 6 \, B a^{4} b^{2} d x^{5} e + 8 \, A a^{3} b^{3} d x^{5} e + \frac {15}{4} \, B a^{4} b^{2} d^{2} x^{4} + 5 \, A a^{3} b^{3} d^{2} x^{4} + \frac {6}{5} \, B a^{5} b x^{5} e^{2} + 3 \, A a^{4} b^{2} x^{5} e^{2} + 3 \, B a^{5} b d x^{4} e + \frac {15}{2} \, A a^{4} b^{2} d x^{4} e + 2 \, B a^{5} b d^{2} x^{3} + 5 \, A a^{4} b^{2} d^{2} x^{3} + \frac {1}{4} \, B a^{6} x^{4} e^{2} + \frac {3}{2} \, A a^{5} b x^{4} e^{2} + \frac {2}{3} \, B a^{6} d x^{3} e + 4 \, A a^{5} b d x^{3} e + \frac {1}{2} \, B a^{6} d^{2} x^{2} + 3 \, A a^{5} b d^{2} x^{2} + \frac {1}{3} \, A a^{6} x^{3} e^{2} + A a^{6} d x^{2} e + A a^{6} d^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.20, size = 457, normalized size = 3.87 \begin {gather*} x^4\,\left (\frac {B\,a^6\,e^2}{4}+3\,B\,a^5\,b\,d\,e+\frac {3\,A\,a^5\,b\,e^2}{2}+\frac {15\,B\,a^4\,b^2\,d^2}{4}+\frac {15\,A\,a^4\,b^2\,d\,e}{2}+5\,A\,a^3\,b^3\,d^2\right )+x^7\,\left (\frac {20\,B\,a^3\,b^3\,e^2}{7}+\frac {30\,B\,a^2\,b^4\,d\,e}{7}+\frac {15\,A\,a^2\,b^4\,e^2}{7}+\frac {6\,B\,a\,b^5\,d^2}{7}+\frac {12\,A\,a\,b^5\,d\,e}{7}+\frac {A\,b^6\,d^2}{7}\right )+x^5\,\left (\frac {6\,B\,a^5\,b\,e^2}{5}+6\,B\,a^4\,b^2\,d\,e+3\,A\,a^4\,b^2\,e^2+4\,B\,a^3\,b^3\,d^2+8\,A\,a^3\,b^3\,d\,e+3\,A\,a^2\,b^4\,d^2\right )+x^6\,\left (\frac {5\,B\,a^4\,b^2\,e^2}{2}+\frac {20\,B\,a^3\,b^3\,d\,e}{3}+\frac {10\,A\,a^3\,b^3\,e^2}{3}+\frac {5\,B\,a^2\,b^4\,d^2}{2}+5\,A\,a^2\,b^4\,d\,e+A\,a\,b^5\,d^2\right )+x^3\,\left (\frac {2\,B\,a^6\,d\,e}{3}+\frac {A\,a^6\,e^2}{3}+2\,B\,a^5\,b\,d^2+4\,A\,a^5\,b\,d\,e+5\,A\,a^4\,b^2\,d^2\right )+x^8\,\left (\frac {15\,B\,a^2\,b^4\,e^2}{8}+\frac {3\,B\,a\,b^5\,d\,e}{2}+\frac {3\,A\,a\,b^5\,e^2}{4}+\frac {B\,b^6\,d^2}{8}+\frac {A\,b^6\,d\,e}{4}\right )+A\,a^6\,d^2\,x+\frac {a^5\,d\,x^2\,\left (2\,A\,a\,e+6\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^5\,e\,x^9\,\left (A\,b\,e+6\,B\,a\,e+2\,B\,b\,d\right )}{9}+\frac {B\,b^6\,e^2\,x^{10}}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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