Optimal. Leaf size=75 \[ \frac {(a+b x)^6 (-2 a B e+A b e+b B d)}{6 b^3}+\frac {(a+b x)^5 (A b-a B) (b d-a e)}{5 b^3}+\frac {B e (a+b x)^7}{7 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {27, 77} \[ \frac {(a+b x)^6 (-2 a B e+A b e+b B d)}{6 b^3}+\frac {(a+b x)^5 (A b-a B) (b d-a e)}{5 b^3}+\frac {B e (a+b x)^7}{7 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 77
Rubi steps
\begin {align*} \int (A+B x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^4 (A+B x) (d+e x) \, dx\\ &=\int \left (\frac {(A b-a B) (b d-a e) (a+b x)^4}{b^2}+\frac {(b B d+A b e-2 a B e) (a+b x)^5}{b^2}+\frac {B e (a+b x)^6}{b^2}\right ) \, dx\\ &=\frac {(A b-a B) (b d-a e) (a+b x)^5}{5 b^3}+\frac {(b B d+A b e-2 a B e) (a+b x)^6}{6 b^3}+\frac {B e (a+b x)^7}{7 b^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.05, size = 172, normalized size = 2.29 \[ a^4 A d x+\frac {1}{2} a^3 x^2 (a A e+a B d+4 A b d)+\frac {1}{3} a^2 x^3 (2 A b (2 a e+3 b d)+a B (a e+4 b d))+\frac {1}{6} b^3 x^6 (4 a B e+A b e+b B d)+\frac {1}{5} b^2 x^5 (A b (4 a e+b d)+2 a B (3 a e+2 b d))+\frac {1}{2} a b x^4 (A b (3 a e+2 b d)+a B (2 a e+3 b d))+\frac {1}{7} b^4 B e x^7 \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.49, size = 216, normalized size = 2.88 \[ \frac {1}{7} x^{7} e b^{4} B + \frac {1}{6} x^{6} d b^{4} B + \frac {2}{3} x^{6} e b^{3} a B + \frac {1}{6} x^{6} e b^{4} A + \frac {4}{5} x^{5} d b^{3} a B + \frac {6}{5} x^{5} e b^{2} a^{2} B + \frac {1}{5} x^{5} d b^{4} A + \frac {4}{5} x^{5} e b^{3} a A + \frac {3}{2} x^{4} d b^{2} a^{2} B + x^{4} e b a^{3} B + x^{4} d b^{3} a A + \frac {3}{2} x^{4} e b^{2} a^{2} A + \frac {4}{3} x^{3} d b a^{3} B + \frac {1}{3} x^{3} e a^{4} B + 2 x^{3} d b^{2} a^{2} A + \frac {4}{3} x^{3} e b a^{3} A + \frac {1}{2} x^{2} d a^{4} B + 2 x^{2} d b a^{3} A + \frac {1}{2} x^{2} e a^{4} A + x d a^{4} A \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.19, size = 226, normalized size = 3.01 \[ \frac {1}{7} \, B b^{4} x^{7} e + \frac {1}{6} \, B b^{4} d x^{6} + \frac {2}{3} \, B a b^{3} x^{6} e + \frac {1}{6} \, A b^{4} x^{6} e + \frac {4}{5} \, B a b^{3} d x^{5} + \frac {1}{5} \, A b^{4} d x^{5} + \frac {6}{5} \, B a^{2} b^{2} x^{5} e + \frac {4}{5} \, A a b^{3} x^{5} e + \frac {3}{2} \, B a^{2} b^{2} d x^{4} + A a b^{3} d x^{4} + B a^{3} b x^{4} e + \frac {3}{2} \, A a^{2} b^{2} x^{4} e + \frac {4}{3} \, B a^{3} b d x^{3} + 2 \, A a^{2} b^{2} d x^{3} + \frac {1}{3} \, B a^{4} x^{3} e + \frac {4}{3} \, A a^{3} b x^{3} e + \frac {1}{2} \, B a^{4} d x^{2} + 2 \, A a^{3} b d x^{2} + \frac {1}{2} \, A a^{4} x^{2} e + A a^{4} d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.04, size = 176, normalized size = 2.35 \[ \frac {B \,b^{4} e \,x^{7}}{7}+A \,a^{4} d x +\frac {\left (4 B a \,b^{3} e +\left (A e +B d \right ) b^{4}\right ) x^{6}}{6}+\frac {\left (A \,b^{4} d +6 B \,a^{2} b^{2} e +4 \left (A e +B d \right ) a \,b^{3}\right ) x^{5}}{5}+\frac {\left (4 A a \,b^{3} d +4 B \,a^{3} b e +6 \left (A e +B d \right ) a^{2} b^{2}\right ) x^{4}}{4}+\frac {\left (6 A \,a^{2} b^{2} d +B \,a^{4} e +4 \left (A e +B d \right ) a^{3} b \right ) x^{3}}{3}+\frac {\left (4 A \,a^{3} b d +\left (A e +B d \right ) a^{4}\right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.52, size = 198, normalized size = 2.64 \[ \frac {1}{7} \, B b^{4} e x^{7} + A a^{4} d x + \frac {1}{6} \, {\left (B b^{4} d + {\left (4 \, B a b^{3} + A b^{4}\right )} e\right )} x^{6} + \frac {1}{5} \, {\left ({\left (4 \, B a b^{3} + A b^{4}\right )} d + 2 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e\right )} x^{5} + \frac {1}{2} \, {\left ({\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d + {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e\right )} x^{4} + \frac {1}{3} \, {\left (2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d + {\left (B a^{4} + 4 \, A a^{3} b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a^{4} e + {\left (B a^{4} + 4 \, A a^{3} b\right )} d\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.09, size = 182, normalized size = 2.43 \[ x^3\,\left (\frac {B\,a^4\,e}{3}+\frac {4\,A\,a^3\,b\,e}{3}+\frac {4\,B\,a^3\,b\,d}{3}+2\,A\,a^2\,b^2\,d\right )+x^5\,\left (\frac {A\,b^4\,d}{5}+\frac {4\,A\,a\,b^3\,e}{5}+\frac {4\,B\,a\,b^3\,d}{5}+\frac {6\,B\,a^2\,b^2\,e}{5}\right )+x^2\,\left (\frac {A\,a^4\,e}{2}+\frac {B\,a^4\,d}{2}+2\,A\,a^3\,b\,d\right )+x^6\,\left (\frac {A\,b^4\,e}{6}+\frac {B\,b^4\,d}{6}+\frac {2\,B\,a\,b^3\,e}{3}\right )+A\,a^4\,d\,x+\frac {a\,b\,x^4\,\left (2\,A\,b^2\,d+2\,B\,a^2\,e+3\,A\,a\,b\,e+3\,B\,a\,b\,d\right )}{2}+\frac {B\,b^4\,e\,x^7}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.10, size = 226, normalized size = 3.01 \[ A a^{4} d x + \frac {B b^{4} e x^{7}}{7} + x^{6} \left (\frac {A b^{4} e}{6} + \frac {2 B a b^{3} e}{3} + \frac {B b^{4} d}{6}\right ) + x^{5} \left (\frac {4 A a b^{3} e}{5} + \frac {A b^{4} d}{5} + \frac {6 B a^{2} b^{2} e}{5} + \frac {4 B a b^{3} d}{5}\right ) + x^{4} \left (\frac {3 A a^{2} b^{2} e}{2} + A a b^{3} d + B a^{3} b e + \frac {3 B a^{2} b^{2} d}{2}\right ) + x^{3} \left (\frac {4 A a^{3} b e}{3} + 2 A a^{2} b^{2} d + \frac {B a^{4} e}{3} + \frac {4 B a^{3} b d}{3}\right ) + x^{2} \left (\frac {A a^{4} e}{2} + 2 A a^{3} b d + \frac {B a^{4} d}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________