3.1721 \(\int (a+b x) (A+B x) \sqrt{d+e x} \, dx\)

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]

(2*(b*d - a*e)*(B*d - A*e)*(d + e*x)^(3/2))/(3*e^3) - (2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(5/2))/(5*e^3) +
(2*b*B*(d + e*x)^(7/2))/(7*e^3)

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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]

Int[(a + b*x)*(A + B*x)*Sqrt[d + e*x],x]

[Out]

(2*(b*d - a*e)*(B*d - A*e)*(d + e*x)^(3/2))/(3*e^3) - (2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(5/2))/(5*e^3) +
(2*b*B*(d + e*x)^(7/2))/(7*e^3)

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

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]

Integrate[(a + b*x)*(A + B*x)*Sqrt[d + e*x],x]

[Out]

(2*(d + e*x)^(3/2)*(7*A*b*e*(-2*d + 3*e*x) + 7*a*e*(-2*B*d + 5*A*e + 3*B*e*x) + b*B*(8*d^2 - 12*d*e*x + 15*e^2
*x^2)))/(105*e^3)

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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]

int((b*x+a)*(B*x+A)*(e*x+d)^(1/2),x)

[Out]

2/105*(e*x+d)^(3/2)*(15*B*b*e^2*x^2+21*A*b*e^2*x+21*B*a*e^2*x-12*B*b*d*e*x+35*A*a*e^2-14*A*b*d*e-14*B*a*d*e+8*
B*b*d^2)/e^3

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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]

integrate((b*x+a)*(B*x+A)*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/105*(15*(e*x + d)^(7/2)*B*b - 21*(2*B*b*d - (B*a + A*b)*e)*(e*x + d)^(5/2) + 35*(B*b*d^2 + A*a*e^2 - (B*a +
A*b)*d*e)*(e*x + d)^(3/2))/e^3

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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]

integrate((b*x+a)*(B*x+A)*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/105*(15*B*b*e^3*x^3 + 8*B*b*d^3 + 35*A*a*d*e^2 - 14*(B*a + A*b)*d^2*e + 3*(B*b*d*e^2 + 7*(B*a + A*b)*e^3)*x^
2 - (4*B*b*d^2*e - 35*A*a*e^3 - 7*(B*a + A*b)*d*e^2)*x)*sqrt(e*x + d)/e^3

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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]

integrate((b*x+a)*(B*x+A)*(e*x+d)**(1/2),x)

[Out]

2*(B*b*(d + e*x)**(7/2)/(7*e**2) + (d + e*x)**(5/2)*(A*b*e + B*a*e - 2*B*b*d)/(5*e**2) + (d + e*x)**(3/2)*(A*a
*e**2 - A*b*d*e - B*a*d*e + B*b*d**2)/(3*e**2))/e

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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]

integrate((b*x+a)*(B*x+A)*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/105*(7*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a*e^(-1) + 7*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*A*
b*e^(-1) + (15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*B*b*e^(-2) + 35*(x*e + d)^(3/2
)*A*a)*e^(-1)