Optimal. Leaf size=163 \[ \frac {(b d-a e)^3 (B d-A e) (d+e x)^6}{6 e^5}-\frac {(b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^7}{7 e^5}+\frac {3 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^8}{8 e^5}-\frac {b^2 (4 b B d-A b e-3 a B e) (d+e x)^9}{9 e^5}+\frac {b^3 B (d+e x)^{10}}{10 e^5} \]
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Rubi [A]
time = 0.27, antiderivative size = 163, 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 {b^2 (d+e x)^9 (-3 a B e-A b e+4 b B d)}{9 e^5}+\frac {3 b (d+e x)^8 (b d-a e) (-a B e-A b e+2 b B d)}{8 e^5}-\frac {(d+e x)^7 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5}+\frac {(d+e x)^6 (b d-a e)^3 (B d-A e)}{6 e^5}+\frac {b^3 B (d+e x)^{10}}{10 e^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rubi steps
\begin {align*} \int (a+b x)^3 (A+B x) (d+e x)^5 \, dx &=\int \left (\frac {(-b d+a e)^3 (-B d+A e) (d+e x)^5}{e^4}+\frac {(-b d+a e)^2 (-4 b B d+3 A b e+a B e) (d+e x)^6}{e^4}-\frac {3 b (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^7}{e^4}+\frac {b^2 (-4 b B d+A b e+3 a B e) (d+e x)^8}{e^4}+\frac {b^3 B (d+e x)^9}{e^4}\right ) \, dx\\ &=\frac {(b d-a e)^3 (B d-A e) (d+e x)^6}{6 e^5}-\frac {(b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^7}{7 e^5}+\frac {3 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^8}{8 e^5}-\frac {b^2 (4 b B d-A b e-3 a B e) (d+e x)^9}{9 e^5}+\frac {b^3 B (d+e x)^{10}}{10 e^5}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(471\) vs. \(2(163)=326\).
time = 0.11, size = 471, normalized size = 2.89 \begin {gather*} a^3 A d^5 x+\frac {1}{2} a^2 d^4 (3 A b d+a B d+5 a A e) x^2+\frac {1}{3} a d^3 \left (a B d (3 b d+5 a e)+A \left (3 b^2 d^2+15 a b d e+10 a^2 e^2\right )\right ) x^3+\frac {1}{4} d^2 \left (a B d \left (3 b^2 d^2+15 a b d e+10 a^2 e^2\right )+A \left (b^3 d^3+15 a b^2 d^2 e+30 a^2 b d e^2+10 a^3 e^3\right )\right ) x^4+\frac {1}{5} d \left (30 a^2 b d e^2 (B d+A e)+5 a^3 e^3 (2 B d+A e)+15 a b^2 d^2 e (B d+2 A e)+b^3 d^3 (B d+5 A e)\right ) x^5+\frac {1}{6} e \left (30 a b^2 d^2 e (B d+A e)+15 a^2 b d e^2 (2 B d+A e)+a^3 e^3 (5 B d+A e)+5 b^3 d^3 (B d+2 A e)\right ) x^6+\frac {1}{7} e^2 \left (a^3 B e^3+10 b^3 d^2 (B d+A e)+15 a b^2 d e (2 B d+A e)+3 a^2 b e^2 (5 B d+A e)\right ) x^7+\frac {1}{8} b e^3 \left (3 a^2 B e^2+5 b^2 d (2 B d+A e)+3 a b e (5 B d+A e)\right ) x^8+\frac {1}{9} b^2 e^4 (5 b B d+A b e+3 a B e) x^9+\frac {1}{10} b^3 B e^5 x^{10} \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. \(528\) vs.
\(2(153)=306\).
time = 0.07, size = 529, normalized size = 3.25 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. 528 vs.
\(2 (163) = 326\).
time = 0.32, size = 528, normalized size = 3.24 \begin {gather*} \frac {1}{10} \, B b^{3} x^{10} e^{5} + A a^{3} d^{5} x + \frac {1}{9} \, {\left (5 \, B b^{3} d e^{4} + 3 \, B a b^{2} e^{5} + A b^{3} e^{5}\right )} x^{9} + \frac {1}{8} \, {\left (10 \, B b^{3} d^{2} e^{3} + 3 \, B a^{2} b e^{5} + 3 \, A a b^{2} e^{5} + 5 \, {\left (3 \, B a b^{2} e^{4} + A b^{3} e^{4}\right )} d\right )} x^{8} + \frac {1}{7} \, {\left (10 \, B b^{3} d^{3} e^{2} + B a^{3} e^{5} + 3 \, A a^{2} b e^{5} + 10 \, {\left (3 \, B a b^{2} e^{3} + A b^{3} e^{3}\right )} d^{2} + 15 \, {\left (B a^{2} b e^{4} + A a b^{2} e^{4}\right )} d\right )} x^{7} + \frac {1}{6} \, {\left (5 \, B b^{3} d^{4} e + A a^{3} e^{5} + 10 \, {\left (3 \, B a b^{2} e^{2} + A b^{3} e^{2}\right )} d^{3} + 30 \, {\left (B a^{2} b e^{3} + A a b^{2} e^{3}\right )} d^{2} + 5 \, {\left (B a^{3} e^{4} + 3 \, A a^{2} b e^{4}\right )} d\right )} x^{6} + \frac {1}{5} \, {\left (B b^{3} d^{5} + 5 \, A a^{3} d e^{4} + 5 \, {\left (3 \, B a b^{2} e + A b^{3} e\right )} d^{4} + 30 \, {\left (B a^{2} b e^{2} + A a b^{2} e^{2}\right )} d^{3} + 10 \, {\left (B a^{3} e^{3} + 3 \, A a^{2} b e^{3}\right )} d^{2}\right )} x^{5} + \frac {1}{4} \, {\left (10 \, A a^{3} d^{2} e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} + 15 \, {\left (B a^{2} b e + A a b^{2} e\right )} d^{4} + 10 \, {\left (B a^{3} e^{2} + 3 \, A a^{2} b e^{2}\right )} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (10 \, A a^{3} d^{3} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{5} + 5 \, {\left (B a^{3} e + 3 \, A a^{2} b e\right )} d^{4}\right )} x^{3} + \frac {1}{2} \, {\left (5 \, A a^{3} d^{4} e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{5}\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. 519 vs.
\(2 (163) = 326\).
time = 0.67, size = 519, normalized size = 3.18 \begin {gather*} \frac {1}{5} \, B b^{3} d^{5} x^{5} + A a^{3} d^{5} x + \frac {1}{4} \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} x^{4} + {\left (B a^{2} b + A a b^{2}\right )} d^{5} x^{3} + \frac {1}{2} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{5} x^{2} + \frac {1}{2520} \, {\left (252 \, B b^{3} x^{10} + 420 \, A a^{3} x^{6} + 280 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{9} + 945 \, {\left (B a^{2} b + A a b^{2}\right )} x^{8} + 360 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{7}\right )} e^{5} + \frac {1}{504} \, {\left (280 \, B b^{3} d x^{9} + 504 \, A a^{3} d x^{5} + 315 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d x^{8} + 1080 \, {\left (B a^{2} b + A a b^{2}\right )} d x^{7} + 420 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d x^{6}\right )} e^{4} + \frac {1}{28} \, {\left (35 \, B b^{3} d^{2} x^{8} + 70 \, A a^{3} d^{2} x^{4} + 40 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} x^{7} + 140 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} x^{6} + 56 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} x^{5}\right )} e^{3} + \frac {1}{42} \, {\left (60 \, B b^{3} d^{3} x^{7} + 140 \, A a^{3} d^{3} x^{3} + 70 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} x^{6} + 252 \, {\left (B a^{2} b + A a b^{2}\right )} d^{3} x^{5} + 105 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} x^{4}\right )} e^{2} + \frac {1}{12} \, {\left (10 \, B b^{3} d^{4} x^{6} + 30 \, A a^{3} d^{4} x^{2} + 12 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} x^{5} + 45 \, {\left (B a^{2} b + A a b^{2}\right )} d^{4} x^{4} + 20 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4} 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. 678 vs.
\(2 (155) = 310\).
time = 0.05, size = 678, normalized size = 4.16 \begin {gather*} A a^{3} d^{5} x + \frac {B b^{3} e^{5} x^{10}}{10} + x^{9} \left (\frac {A b^{3} e^{5}}{9} + \frac {B a b^{2} e^{5}}{3} + \frac {5 B b^{3} d e^{4}}{9}\right ) + x^{8} \cdot \left (\frac {3 A a b^{2} e^{5}}{8} + \frac {5 A b^{3} d e^{4}}{8} + \frac {3 B a^{2} b e^{5}}{8} + \frac {15 B a b^{2} d e^{4}}{8} + \frac {5 B b^{3} d^{2} e^{3}}{4}\right ) + x^{7} \cdot \left (\frac {3 A a^{2} b e^{5}}{7} + \frac {15 A a b^{2} d e^{4}}{7} + \frac {10 A b^{3} d^{2} e^{3}}{7} + \frac {B a^{3} e^{5}}{7} + \frac {15 B a^{2} b d e^{4}}{7} + \frac {30 B a b^{2} d^{2} e^{3}}{7} + \frac {10 B b^{3} d^{3} e^{2}}{7}\right ) + x^{6} \left (\frac {A a^{3} e^{5}}{6} + \frac {5 A a^{2} b d e^{4}}{2} + 5 A a b^{2} d^{2} e^{3} + \frac {5 A b^{3} d^{3} e^{2}}{3} + \frac {5 B a^{3} d e^{4}}{6} + 5 B a^{2} b d^{2} e^{3} + 5 B a b^{2} d^{3} e^{2} + \frac {5 B b^{3} d^{4} e}{6}\right ) + x^{5} \left (A a^{3} d e^{4} + 6 A a^{2} b d^{2} e^{3} + 6 A a b^{2} d^{3} e^{2} + A b^{3} d^{4} e + 2 B a^{3} d^{2} e^{3} + 6 B a^{2} b d^{3} e^{2} + 3 B a b^{2} d^{4} e + \frac {B b^{3} d^{5}}{5}\right ) + x^{4} \cdot \left (\frac {5 A a^{3} d^{2} e^{3}}{2} + \frac {15 A a^{2} b d^{3} e^{2}}{2} + \frac {15 A a b^{2} d^{4} e}{4} + \frac {A b^{3} d^{5}}{4} + \frac {5 B a^{3} d^{3} e^{2}}{2} + \frac {15 B a^{2} b d^{4} e}{4} + \frac {3 B a b^{2} d^{5}}{4}\right ) + x^{3} \cdot \left (\frac {10 A a^{3} d^{3} e^{2}}{3} + 5 A a^{2} b d^{4} e + A a b^{2} d^{5} + \frac {5 B a^{3} d^{4} e}{3} + B a^{2} b d^{5}\right ) + x^{2} \cdot \left (\frac {5 A a^{3} d^{4} e}{2} + \frac {3 A a^{2} b d^{5}}{2} + \frac {B a^{3} 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. 634 vs.
\(2 (163) = 326\).
time = 1.89, size = 634, normalized size = 3.89 \begin {gather*} \frac {1}{10} \, B b^{3} x^{10} e^{5} + \frac {5}{9} \, B b^{3} d x^{9} e^{4} + \frac {5}{4} \, B b^{3} d^{2} x^{8} e^{3} + \frac {10}{7} \, B b^{3} d^{3} x^{7} e^{2} + \frac {5}{6} \, B b^{3} d^{4} x^{6} e + \frac {1}{5} \, B b^{3} d^{5} x^{5} + \frac {1}{3} \, B a b^{2} x^{9} e^{5} + \frac {1}{9} \, A b^{3} x^{9} e^{5} + \frac {15}{8} \, B a b^{2} d x^{8} e^{4} + \frac {5}{8} \, A b^{3} d x^{8} e^{4} + \frac {30}{7} \, B a b^{2} d^{2} x^{7} e^{3} + \frac {10}{7} \, A b^{3} d^{2} x^{7} e^{3} + 5 \, B a b^{2} d^{3} x^{6} e^{2} + \frac {5}{3} \, A b^{3} d^{3} x^{6} e^{2} + 3 \, B a b^{2} d^{4} x^{5} e + A b^{3} d^{4} x^{5} e + \frac {3}{4} \, B a b^{2} d^{5} x^{4} + \frac {1}{4} \, A b^{3} d^{5} x^{4} + \frac {3}{8} \, B a^{2} b x^{8} e^{5} + \frac {3}{8} \, A a b^{2} x^{8} e^{5} + \frac {15}{7} \, B a^{2} b d x^{7} e^{4} + \frac {15}{7} \, A a b^{2} d x^{7} e^{4} + 5 \, B a^{2} b d^{2} x^{6} e^{3} + 5 \, A a b^{2} d^{2} x^{6} e^{3} + 6 \, B a^{2} b d^{3} x^{5} e^{2} + 6 \, A a b^{2} d^{3} x^{5} e^{2} + \frac {15}{4} \, B a^{2} b d^{4} x^{4} e + \frac {15}{4} \, A a b^{2} d^{4} x^{4} e + B a^{2} b d^{5} x^{3} + A a b^{2} d^{5} x^{3} + \frac {1}{7} \, B a^{3} x^{7} e^{5} + \frac {3}{7} \, A a^{2} b x^{7} e^{5} + \frac {5}{6} \, B a^{3} d x^{6} e^{4} + \frac {5}{2} \, A a^{2} b d x^{6} e^{4} + 2 \, B a^{3} d^{2} x^{5} e^{3} + 6 \, A a^{2} b d^{2} x^{5} e^{3} + \frac {5}{2} \, B a^{3} d^{3} x^{4} e^{2} + \frac {15}{2} \, A a^{2} b d^{3} x^{4} e^{2} + \frac {5}{3} \, B a^{3} d^{4} x^{3} e + 5 \, A a^{2} b d^{4} x^{3} e + \frac {1}{2} \, B a^{3} d^{5} x^{2} + \frac {3}{2} \, A a^{2} b d^{5} x^{2} + \frac {1}{6} \, A a^{3} x^{6} e^{5} + A a^{3} d x^{5} e^{4} + \frac {5}{2} \, A a^{3} d^{2} x^{4} e^{3} + \frac {10}{3} \, A a^{3} d^{3} x^{3} e^{2} + \frac {5}{2} \, A a^{3} d^{4} x^{2} e + A a^{3} d^{5} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.20, size = 544, normalized size = 3.34 \begin {gather*} x^3\,\left (\frac {5\,B\,a^3\,d^4\,e}{3}+\frac {10\,A\,a^3\,d^3\,e^2}{3}+B\,a^2\,b\,d^5+5\,A\,a^2\,b\,d^4\,e+A\,a\,b^2\,d^5\right )+x^8\,\left (\frac {3\,B\,a^2\,b\,e^5}{8}+\frac {15\,B\,a\,b^2\,d\,e^4}{8}+\frac {3\,A\,a\,b^2\,e^5}{8}+\frac {5\,B\,b^3\,d^2\,e^3}{4}+\frac {5\,A\,b^3\,d\,e^4}{8}\right )+x^4\,\left (\frac {5\,B\,a^3\,d^3\,e^2}{2}+\frac {5\,A\,a^3\,d^2\,e^3}{2}+\frac {15\,B\,a^2\,b\,d^4\,e}{4}+\frac {15\,A\,a^2\,b\,d^3\,e^2}{2}+\frac {3\,B\,a\,b^2\,d^5}{4}+\frac {15\,A\,a\,b^2\,d^4\,e}{4}+\frac {A\,b^3\,d^5}{4}\right )+x^7\,\left (\frac {B\,a^3\,e^5}{7}+\frac {15\,B\,a^2\,b\,d\,e^4}{7}+\frac {3\,A\,a^2\,b\,e^5}{7}+\frac {30\,B\,a\,b^2\,d^2\,e^3}{7}+\frac {15\,A\,a\,b^2\,d\,e^4}{7}+\frac {10\,B\,b^3\,d^3\,e^2}{7}+\frac {10\,A\,b^3\,d^2\,e^3}{7}\right )+x^5\,\left (2\,B\,a^3\,d^2\,e^3+A\,a^3\,d\,e^4+6\,B\,a^2\,b\,d^3\,e^2+6\,A\,a^2\,b\,d^2\,e^3+3\,B\,a\,b^2\,d^4\,e+6\,A\,a\,b^2\,d^3\,e^2+\frac {B\,b^3\,d^5}{5}+A\,b^3\,d^4\,e\right )+x^6\,\left (\frac {5\,B\,a^3\,d\,e^4}{6}+\frac {A\,a^3\,e^5}{6}+5\,B\,a^2\,b\,d^2\,e^3+\frac {5\,A\,a^2\,b\,d\,e^4}{2}+5\,B\,a\,b^2\,d^3\,e^2+5\,A\,a\,b^2\,d^2\,e^3+\frac {5\,B\,b^3\,d^4\,e}{6}+\frac {5\,A\,b^3\,d^3\,e^2}{3}\right )+\frac {a^2\,d^4\,x^2\,\left (5\,A\,a\,e+3\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^2\,e^4\,x^9\,\left (A\,b\,e+3\,B\,a\,e+5\,B\,b\,d\right )}{9}+A\,a^3\,d^5\,x+\frac {B\,b^3\,e^5\,x^{10}}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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