Optimal. Leaf size=83 \[ -\frac{2 (d+e x)^{5/2} (-a B e-A b e+2 b B d)}{5 e^3}+\frac{2 (d+e x)^{3/2} (b d-a e) (B d-A e)}{3 e^3}+\frac{2 b B (d+e x)^{7/2}}{7 e^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0359425, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ -\frac{2 (d+e x)^{5/2} (-a B e-A b e+2 b B d)}{5 e^3}+\frac{2 (d+e x)^{3/2} (b d-a e) (B d-A e)}{3 e^3}+\frac{2 b B (d+e x)^{7/2}}{7 e^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 77
Rubi steps
\begin{align*} \int (a+b x) (A+B x) \sqrt{d+e x} \, dx &=\int \left (\frac{(-b d+a e) (-B d+A e) \sqrt{d+e x}}{e^2}+\frac{(-2 b B d+A b e+a B e) (d+e x)^{3/2}}{e^2}+\frac{b B (d+e x)^{5/2}}{e^2}\right ) \, dx\\ &=\frac{2 (b d-a e) (B d-A e) (d+e x)^{3/2}}{3 e^3}-\frac{2 (2 b B d-A b e-a B e) (d+e x)^{5/2}}{5 e^3}+\frac{2 b B (d+e x)^{7/2}}{7 e^3}\\ \end{align*}
Mathematica [A] time = 0.0517321, size = 70, normalized size = 0.84 \[ \frac{2 (d+e x)^{3/2} \left (7 a e (5 A e-2 B d+3 B e x)+7 A b e (3 e x-2 d)+b B \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )}{105 e^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 73, normalized size = 0.9 \begin{align*}{\frac{30\,bB{x}^{2}{e}^{2}+42\,Ab{e}^{2}x+42\,Ba{e}^{2}x-24\,Bbdex+70\,aA{e}^{2}-28\,Abde-28\,Bade+16\,bB{d}^{2}}{105\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.05887, size = 101, normalized size = 1.22 \begin{align*} \frac{2 \,{\left (15 \,{\left (e x + d\right )}^{\frac{7}{2}} B b - 21 \,{\left (2 \, B b d -{\left (B a + A b\right )} e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\left (B b d^{2} + A a e^{2} -{\left (B a + A b\right )} d e\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{105 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.85765, size = 250, normalized size = 3.01 \begin{align*} \frac{2 \,{\left (15 \, B b e^{3} x^{3} + 8 \, B b d^{3} + 35 \, A a d e^{2} - 14 \,{\left (B a + A b\right )} d^{2} e + 3 \,{\left (B b d e^{2} + 7 \,{\left (B a + A b\right )} e^{3}\right )} x^{2} -{\left (4 \, B b d^{2} e - 35 \, A a e^{3} - 7 \,{\left (B a + A b\right )} d e^{2}\right )} x\right )} \sqrt{e x + d}}{105 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.52274, size = 94, normalized size = 1.13 \begin{align*} \frac{2 \left (\frac{B b \left (d + e x\right )^{\frac{7}{2}}}{7 e^{2}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (A b e + B a e - 2 B b d\right )}{5 e^{2}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (A a e^{2} - A b d e - B a d e + B b d^{2}\right )}{3 e^{2}}\right )}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 3.10179, size = 153, normalized size = 1.84 \begin{align*} \frac{2}{105} \,{\left (7 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} B a e^{\left (-1\right )} + 7 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} A b e^{\left (-1\right )} +{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} B b e^{\left (-2\right )} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} A a\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]