Optimal. Leaf size=120 \[ -\frac {(b d-a e)^2 (B d-A e) (d+e x)^4}{4 e^4}+\frac {(b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^5}{5 e^4}-\frac {b (3 b B d-A b e-2 a B e) (d+e x)^6}{6 e^4}+\frac {b^2 B (d+e x)^7}{7 e^4} \]
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Rubi [A]
time = 0.11, antiderivative size = 120, 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 (d+e x)^6 (-2 a B e-A b e+3 b B d)}{6 e^4}+\frac {(d+e x)^5 (b d-a e) (-a B e-2 A b e+3 b B d)}{5 e^4}-\frac {(d+e x)^4 (b d-a e)^2 (B d-A e)}{4 e^4}+\frac {b^2 B (d+e x)^7}{7 e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rubi steps
\begin {align*} \int (a+b x)^2 (A+B x) (d+e x)^3 \, dx &=\int \left (\frac {(-b d+a e)^2 (-B d+A e) (d+e x)^3}{e^3}+\frac {(-b d+a e) (-3 b B d+2 A b e+a B e) (d+e x)^4}{e^3}+\frac {b (-3 b B d+A b e+2 a B e) (d+e x)^5}{e^3}+\frac {b^2 B (d+e x)^6}{e^3}\right ) \, dx\\ &=-\frac {(b d-a e)^2 (B d-A e) (d+e x)^4}{4 e^4}+\frac {(b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^5}{5 e^4}-\frac {b (3 b B d-A b e-2 a B e) (d+e x)^6}{6 e^4}+\frac {b^2 B (d+e x)^7}{7 e^4}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 216, normalized size = 1.80 \begin {gather*} a^2 A d^3 x+\frac {1}{2} a d^2 (2 A b d+a B d+3 a A e) x^2+\frac {1}{3} d \left (a B d (2 b d+3 a e)+A \left (b^2 d^2+6 a b d e+3 a^2 e^2\right )\right ) x^3+\frac {1}{4} \left (6 a b d e (B d+A e)+a^2 e^2 (3 B d+A e)+b^2 d^2 (B d+3 A e)\right ) x^4+\frac {1}{5} e \left (a^2 B e^2+3 b^2 d (B d+A e)+2 a b e (3 B d+A e)\right ) x^5+\frac {1}{6} b e^2 (3 b B d+A b e+2 a B e) x^6+\frac {1}{7} b^2 B e^3 x^7 \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. \(236\) vs.
\(2(112)=224\).
time = 0.07, size = 237, normalized size = 1.98
method | result | size |
default | \(\frac {b^{2} B \,e^{3} x^{7}}{7}+\frac {\left (\left (b^{2} A +2 a b B \right ) e^{3}+3 b^{2} B d \,e^{2}\right ) x^{6}}{6}+\frac {\left (\left (2 a b A +a^{2} B \right ) e^{3}+3 \left (b^{2} A +2 a b B \right ) d \,e^{2}+3 b^{2} B \,d^{2} e \right ) x^{5}}{5}+\frac {\left (a^{2} A \,e^{3}+3 \left (2 a b A +a^{2} B \right ) d \,e^{2}+3 \left (b^{2} A +2 a b B \right ) d^{2} e +b^{2} B \,d^{3}\right ) x^{4}}{4}+\frac {\left (3 a^{2} A d \,e^{2}+3 \left (2 a b A +a^{2} B \right ) d^{2} e +\left (b^{2} A +2 a b B \right ) d^{3}\right ) x^{3}}{3}+\frac {\left (3 a^{2} A \,d^{2} e +\left (2 a b A +a^{2} B \right ) d^{3}\right ) x^{2}}{2}+a^{2} A \,d^{3} x\) | \(237\) |
norman | \(\frac {b^{2} B \,e^{3} x^{7}}{7}+\left (\frac {1}{6} A \,b^{2} e^{3}+\frac {1}{3} B a b \,e^{3}+\frac {1}{2} b^{2} B d \,e^{2}\right ) x^{6}+\left (\frac {2}{5} A a b \,e^{3}+\frac {3}{5} A \,b^{2} d \,e^{2}+\frac {1}{5} B \,a^{2} e^{3}+\frac {6}{5} B a b d \,e^{2}+\frac {3}{5} b^{2} B \,d^{2} e \right ) x^{5}+\left (\frac {1}{4} a^{2} A \,e^{3}+\frac {3}{2} A a b d \,e^{2}+\frac {3}{4} A \,b^{2} d^{2} e +\frac {3}{4} B \,a^{2} d \,e^{2}+\frac {3}{2} B a b \,d^{2} e +\frac {1}{4} b^{2} B \,d^{3}\right ) x^{4}+\left (a^{2} A d \,e^{2}+2 A a b \,d^{2} e +\frac {1}{3} A \,b^{2} d^{3}+B \,a^{2} d^{2} e +\frac {2}{3} B a b \,d^{3}\right ) x^{3}+\left (\frac {3}{2} a^{2} A \,d^{2} e +A a b \,d^{3}+\frac {1}{2} B \,a^{2} d^{3}\right ) x^{2}+a^{2} A \,d^{3} x\) | \(247\) |
gosper | \(\frac {1}{7} b^{2} B \,e^{3} x^{7}+\frac {1}{6} x^{6} A \,b^{2} e^{3}+\frac {1}{3} x^{6} B a b \,e^{3}+\frac {1}{2} x^{6} b^{2} B d \,e^{2}+\frac {2}{5} x^{5} A a b \,e^{3}+\frac {3}{5} x^{5} A \,b^{2} d \,e^{2}+\frac {1}{5} x^{5} B \,a^{2} e^{3}+\frac {6}{5} x^{5} B a b d \,e^{2}+\frac {3}{5} x^{5} b^{2} B \,d^{2} e +\frac {1}{4} x^{4} a^{2} A \,e^{3}+\frac {3}{2} x^{4} A a b d \,e^{2}+\frac {3}{4} x^{4} A \,b^{2} d^{2} e +\frac {3}{4} x^{4} B \,a^{2} d \,e^{2}+\frac {3}{2} x^{4} B a b \,d^{2} e +\frac {1}{4} x^{4} b^{2} B \,d^{3}+x^{3} a^{2} A d \,e^{2}+2 x^{3} A a b \,d^{2} e +\frac {1}{3} x^{3} A \,b^{2} d^{3}+x^{3} B \,a^{2} d^{2} e +\frac {2}{3} x^{3} B a b \,d^{3}+\frac {3}{2} x^{2} a^{2} A \,d^{2} e +x^{2} A a b \,d^{3}+\frac {1}{2} x^{2} B \,a^{2} d^{3}+a^{2} A \,d^{3} x\) | \(288\) |
risch | \(\frac {1}{7} b^{2} B \,e^{3} x^{7}+\frac {1}{6} x^{6} A \,b^{2} e^{3}+\frac {1}{3} x^{6} B a b \,e^{3}+\frac {1}{2} x^{6} b^{2} B d \,e^{2}+\frac {2}{5} x^{5} A a b \,e^{3}+\frac {3}{5} x^{5} A \,b^{2} d \,e^{2}+\frac {1}{5} x^{5} B \,a^{2} e^{3}+\frac {6}{5} x^{5} B a b d \,e^{2}+\frac {3}{5} x^{5} b^{2} B \,d^{2} e +\frac {1}{4} x^{4} a^{2} A \,e^{3}+\frac {3}{2} x^{4} A a b d \,e^{2}+\frac {3}{4} x^{4} A \,b^{2} d^{2} e +\frac {3}{4} x^{4} B \,a^{2} d \,e^{2}+\frac {3}{2} x^{4} B a b \,d^{2} e +\frac {1}{4} x^{4} b^{2} B \,d^{3}+x^{3} a^{2} A d \,e^{2}+2 x^{3} A a b \,d^{2} e +\frac {1}{3} x^{3} A \,b^{2} d^{3}+x^{3} B \,a^{2} d^{2} e +\frac {2}{3} x^{3} B a b \,d^{3}+\frac {3}{2} x^{2} a^{2} A \,d^{2} e +x^{2} A a b \,d^{3}+\frac {1}{2} x^{2} B \,a^{2} d^{3}+a^{2} A \,d^{3} x\) | \(288\) |
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. 240 vs.
\(2 (119) = 238\).
time = 0.30, size = 240, normalized size = 2.00 \begin {gather*} \frac {1}{7} \, B b^{2} x^{7} e^{3} + A a^{2} d^{3} x + \frac {1}{6} \, {\left (3 \, B b^{2} d e^{2} + 2 \, B a b e^{3} + A b^{2} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, B b^{2} d^{2} e + B a^{2} e^{3} + 2 \, A a b e^{3} + 3 \, {\left (2 \, B a b e^{2} + A b^{2} e^{2}\right )} d\right )} x^{5} + \frac {1}{4} \, {\left (B b^{2} d^{3} + A a^{2} e^{3} + 3 \, {\left (2 \, B a b e + A b^{2} e\right )} d^{2} + 3 \, {\left (B a^{2} e^{2} + 2 \, A a b e^{2}\right )} d\right )} x^{4} + \frac {1}{3} \, {\left (3 \, A a^{2} d e^{2} + {\left (2 \, B a b + A b^{2}\right )} d^{3} + 3 \, {\left (B a^{2} e + 2 \, A a b e\right )} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, A a^{2} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d^{3}\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. 241 vs.
\(2 (119) = 238\).
time = 0.39, size = 241, normalized size = 2.01 \begin {gather*} \frac {1}{4} \, B b^{2} d^{3} x^{4} + A a^{2} d^{3} x + \frac {1}{3} \, {\left (2 \, B a b + A b^{2}\right )} d^{3} x^{3} + \frac {1}{2} \, {\left (B a^{2} + 2 \, A a b\right )} d^{3} x^{2} + \frac {1}{420} \, {\left (60 \, B b^{2} x^{7} + 105 \, A a^{2} x^{4} + 70 \, {\left (2 \, B a b + A b^{2}\right )} x^{6} + 84 \, {\left (B a^{2} + 2 \, A a b\right )} x^{5}\right )} e^{3} + \frac {1}{20} \, {\left (10 \, B b^{2} d x^{6} + 20 \, A a^{2} d x^{3} + 12 \, {\left (2 \, B a b + A b^{2}\right )} d x^{5} + 15 \, {\left (B a^{2} + 2 \, A a b\right )} d x^{4}\right )} e^{2} + \frac {1}{20} \, {\left (12 \, B b^{2} d^{2} x^{5} + 30 \, A a^{2} d^{2} x^{2} + 15 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} x^{4} + 20 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} 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. 296 vs.
\(2 (114) = 228\).
time = 0.04, size = 296, normalized size = 2.47 \begin {gather*} A a^{2} d^{3} x + \frac {B b^{2} e^{3} x^{7}}{7} + x^{6} \left (\frac {A b^{2} e^{3}}{6} + \frac {B a b e^{3}}{3} + \frac {B b^{2} d e^{2}}{2}\right ) + x^{5} \cdot \left (\frac {2 A a b e^{3}}{5} + \frac {3 A b^{2} d e^{2}}{5} + \frac {B a^{2} e^{3}}{5} + \frac {6 B a b d e^{2}}{5} + \frac {3 B b^{2} d^{2} e}{5}\right ) + x^{4} \left (\frac {A a^{2} e^{3}}{4} + \frac {3 A a b d e^{2}}{2} + \frac {3 A b^{2} d^{2} e}{4} + \frac {3 B a^{2} d e^{2}}{4} + \frac {3 B a b d^{2} e}{2} + \frac {B b^{2} d^{3}}{4}\right ) + x^{3} \left (A a^{2} d e^{2} + 2 A a b d^{2} e + \frac {A b^{2} d^{3}}{3} + B a^{2} d^{2} e + \frac {2 B a b d^{3}}{3}\right ) + x^{2} \cdot \left (\frac {3 A a^{2} d^{2} e}{2} + A a b d^{3} + \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 281 vs.
\(2 (119) = 238\).
time = 2.98, size = 281, normalized size = 2.34 \begin {gather*} \frac {1}{7} \, B b^{2} x^{7} e^{3} + \frac {1}{2} \, B b^{2} d x^{6} e^{2} + \frac {3}{5} \, B b^{2} d^{2} x^{5} e + \frac {1}{4} \, B b^{2} d^{3} x^{4} + \frac {1}{3} \, B a b x^{6} e^{3} + \frac {1}{6} \, A b^{2} x^{6} e^{3} + \frac {6}{5} \, B a b d x^{5} e^{2} + \frac {3}{5} \, A b^{2} d x^{5} e^{2} + \frac {3}{2} \, B a b d^{2} x^{4} e + \frac {3}{4} \, A b^{2} d^{2} x^{4} e + \frac {2}{3} \, B a b d^{3} x^{3} + \frac {1}{3} \, A b^{2} d^{3} x^{3} + \frac {1}{5} \, B a^{2} x^{5} e^{3} + \frac {2}{5} \, A a b x^{5} e^{3} + \frac {3}{4} \, B a^{2} d x^{4} e^{2} + \frac {3}{2} \, A a b d x^{4} e^{2} + B a^{2} d^{2} x^{3} e + 2 \, A a b d^{2} x^{3} e + \frac {1}{2} \, B a^{2} d^{3} x^{2} + A a b 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 = 0.09, size = 231, normalized size = 1.92 \begin {gather*} x^4\,\left (\frac {3\,B\,a^2\,d\,e^2}{4}+\frac {A\,a^2\,e^3}{4}+\frac {3\,B\,a\,b\,d^2\,e}{2}+\frac {3\,A\,a\,b\,d\,e^2}{2}+\frac {B\,b^2\,d^3}{4}+\frac {3\,A\,b^2\,d^2\,e}{4}\right )+x^3\,\left (B\,a^2\,d^2\,e+A\,a^2\,d\,e^2+\frac {2\,B\,a\,b\,d^3}{3}+2\,A\,a\,b\,d^2\,e+\frac {A\,b^2\,d^3}{3}\right )+x^5\,\left (\frac {B\,a^2\,e^3}{5}+\frac {6\,B\,a\,b\,d\,e^2}{5}+\frac {2\,A\,a\,b\,e^3}{5}+\frac {3\,B\,b^2\,d^2\,e}{5}+\frac {3\,A\,b^2\,d\,e^2}{5}\right )+A\,a^2\,d^3\,x+\frac {a\,d^2\,x^2\,\left (3\,A\,a\,e+2\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b\,e^2\,x^6\,\left (A\,b\,e+2\,B\,a\,e+3\,B\,b\,d\right )}{6}+\frac {B\,b^2\,e^3\,x^7}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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