Optimal. Leaf size=128 \[ -\frac {2 b (d+e x)^{7/2} (-2 a B e-A b e+3 b B d)}{7 e^4}+\frac {2 (d+e x)^{5/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{5 e^4}-\frac {2 (d+e x)^{3/2} (b d-a e)^2 (B d-A e)}{3 e^4}+\frac {2 b^2 B (d+e x)^{9/2}}{9 e^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {77} \[ -\frac {2 b (d+e x)^{7/2} (-2 a B e-A b e+3 b B d)}{7 e^4}+\frac {2 (d+e x)^{5/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{5 e^4}-\frac {2 (d+e x)^{3/2} (b d-a e)^2 (B d-A e)}{3 e^4}+\frac {2 b^2 B (d+e x)^{9/2}}{9 e^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 77
Rubi steps
\begin {align*} \int (a+b x)^2 (A+B x) \sqrt {d+e x} \, dx &=\int \left (\frac {(-b d+a e)^2 (-B d+A e) \sqrt {d+e x}}{e^3}+\frac {(-b d+a e) (-3 b B d+2 A b e+a B e) (d+e x)^{3/2}}{e^3}+\frac {b (-3 b B d+A b e+2 a B e) (d+e x)^{5/2}}{e^3}+\frac {b^2 B (d+e x)^{7/2}}{e^3}\right ) \, dx\\ &=-\frac {2 (b d-a e)^2 (B d-A e) (d+e x)^{3/2}}{3 e^4}+\frac {2 (b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^{5/2}}{5 e^4}-\frac {2 b (3 b B d-A b e-2 a B e) (d+e x)^{7/2}}{7 e^4}+\frac {2 b^2 B (d+e x)^{9/2}}{9 e^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.13, size = 107, normalized size = 0.84 \[ \frac {2 (d+e x)^{3/2} \left (-45 b (d+e x)^2 (-2 a B e-A b e+3 b B d)+63 (d+e x) (b d-a e) (-a B e-2 A b e+3 b B d)-105 (b d-a e)^2 (B d-A e)+35 b^2 B (d+e x)^3\right )}{315 e^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.57, size = 220, normalized size = 1.72 \[ \frac {2 \, {\left (35 \, B b^{2} e^{4} x^{4} - 16 \, B b^{2} d^{4} + 105 \, A a^{2} d e^{3} + 24 \, {\left (2 \, B a b + A b^{2}\right )} d^{3} e - 42 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{2} + 5 \, {\left (B b^{2} d e^{3} + 9 \, {\left (2 \, B a b + A b^{2}\right )} e^{4}\right )} x^{3} - 3 \, {\left (2 \, B b^{2} d^{2} e^{2} - 3 \, {\left (2 \, B a b + A b^{2}\right )} d e^{3} - 21 \, {\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} x^{2} + {\left (8 \, B b^{2} d^{3} e + 105 \, A a^{2} e^{4} - 12 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e^{2} + 21 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{3}\right )} x\right )} \sqrt {e x + d}}{315 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.68, size = 511, normalized size = 3.99 \[ \frac {2}{315} \, {\left (105 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} B a^{2} d e^{\left (-1\right )} + 210 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A a b d e^{\left (-1\right )} + 42 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} B a b d e^{\left (-2\right )} + 21 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} A b^{2} d e^{\left (-2\right )} + 9 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} B b^{2} d e^{\left (-3\right )} + 21 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} B a^{2} e^{\left (-1\right )} + 42 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} A a b e^{\left (-1\right )} + 18 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} B a b e^{\left (-2\right )} + 9 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} A b^{2} e^{\left (-2\right )} + {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} B b^{2} e^{\left (-3\right )} + 315 \, \sqrt {x e + d} A a^{2} d + 105 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A a^{2}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 169, normalized size = 1.32 \[ \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (35 b^{2} B \,x^{3} e^{3}+45 A \,b^{2} e^{3} x^{2}+90 B a b \,e^{3} x^{2}-30 B \,b^{2} d \,e^{2} x^{2}+126 A a b \,e^{3} x -36 A \,b^{2} d \,e^{2} x +63 B \,a^{2} e^{3} x -72 B a b d \,e^{2} x +24 B \,b^{2} d^{2} e x +105 a^{2} A \,e^{3}-84 A a b d \,e^{2}+24 A \,b^{2} d^{2} e -42 B \,a^{2} d \,e^{2}+48 B a b \,d^{2} e -16 B \,b^{2} d^{3}\right )}{315 e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.54, size = 159, normalized size = 1.24 \[ \frac {2 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} B b^{2} - 45 \, {\left (3 \, B b^{2} d - {\left (2 \, B a b + A b^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 63 \, {\left (3 \, B b^{2} d^{2} - 2 \, {\left (2 \, B a b + A b^{2}\right )} d e + {\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 105 \, {\left (B b^{2} d^{3} - A a^{2} e^{3} - {\left (2 \, B a b + A b^{2}\right )} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{315 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.07, size = 115, normalized size = 0.90 \[ \frac {{\left (d+e\,x\right )}^{7/2}\,\left (2\,A\,b^2\,e-6\,B\,b^2\,d+4\,B\,a\,b\,e\right )}{7\,e^4}+\frac {2\,B\,b^2\,{\left (d+e\,x\right )}^{9/2}}{9\,e^4}+\frac {2\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (2\,A\,b\,e+B\,a\,e-3\,B\,b\,d\right )}{5\,e^4}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{3/2}}{3\,e^4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 4.92, size = 201, normalized size = 1.57 \[ \frac {2 \left (\frac {B b^{2} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{3}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (A b^{2} e + 2 B a b e - 3 B b^{2} d\right )}{7 e^{3}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (2 A a b e^{2} - 2 A b^{2} d e + B a^{2} e^{2} - 4 B a b d e + 3 B b^{2} d^{2}\right )}{5 e^{3}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (A a^{2} e^{3} - 2 A a b d e^{2} + A b^{2} d^{2} e - B a^{2} d e^{2} + 2 B a b d^{2} e - B b^{2} d^{3}\right )}{3 e^{3}}\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________