3.1720 \(\int (a+b x) (A+B x) (d+e x)^{3/2} \, dx\)

Optimal. Leaf size=83 \[ -\frac{2 (d+e x)^{7/2} (-a B e-A b e+2 b B d)}{7 e^3}+\frac{2 (d+e x)^{5/2} (b d-a e) (B d-A e)}{5 e^3}+\frac{2 b B (d+e x)^{9/2}}{9 e^3} \]

[Out]

(2*(b*d - a*e)*(B*d - A*e)*(d + e*x)^(5/2))/(5*e^3) - (2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(7/2))/(7*e^3) +
(2*b*B*(d + e*x)^(9/2))/(9*e^3)

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Rubi [A]  time = 0.0348659, 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} \[ -\frac{2 (d+e x)^{7/2} (-a B e-A b e+2 b B d)}{7 e^3}+\frac{2 (d+e x)^{5/2} (b d-a e) (B d-A e)}{5 e^3}+\frac{2 b B (d+e x)^{9/2}}{9 e^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(A + B*x)*(d + e*x)^(3/2),x]

[Out]

(2*(b*d - a*e)*(B*d - A*e)*(d + e*x)^(5/2))/(5*e^3) - (2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(7/2))/(7*e^3) +
(2*b*B*(d + e*x)^(9/2))/(9*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) (d+e x)^{3/2} \, dx &=\int \left (\frac{(-b d+a e) (-B d+A e) (d+e x)^{3/2}}{e^2}+\frac{(-2 b B d+A b e+a B e) (d+e x)^{5/2}}{e^2}+\frac{b B (d+e x)^{7/2}}{e^2}\right ) \, dx\\ &=\frac{2 (b d-a e) (B d-A e) (d+e x)^{5/2}}{5 e^3}-\frac{2 (2 b B d-A b e-a B e) (d+e x)^{7/2}}{7 e^3}+\frac{2 b B (d+e x)^{9/2}}{9 e^3}\\ \end{align*}

Mathematica [A]  time = 0.0563339, size = 70, normalized size = 0.84 \[ \frac{2 (d+e x)^{5/2} \left (9 a e (7 A e-2 B d+5 B e x)+9 A b e (5 e x-2 d)+b B \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )}{315 e^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(A + B*x)*(d + e*x)^(3/2),x]

[Out]

(2*(d + e*x)^(5/2)*(9*A*b*e*(-2*d + 5*e*x) + 9*a*e*(-2*B*d + 7*A*e + 5*B*e*x) + b*B*(8*d^2 - 20*d*e*x + 35*e^2
*x^2)))/(315*e^3)

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Maple [A]  time = 0.003, size = 73, normalized size = 0.9 \begin{align*}{\frac{70\,bB{x}^{2}{e}^{2}+90\,Ab{e}^{2}x+90\,Ba{e}^{2}x-40\,Bbdex+126\,aA{e}^{2}-36\,Abde-36\,Bade+16\,bB{d}^{2}}{315\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(B*x+A)*(e*x+d)^(3/2),x)

[Out]

2/315*(e*x+d)^(5/2)*(35*B*b*e^2*x^2+45*A*b*e^2*x+45*B*a*e^2*x-20*B*b*d*e*x+63*A*a*e^2-18*A*b*d*e-18*B*a*d*e+8*
B*b*d^2)/e^3

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Maxima [A]  time = 1.32124, size = 101, normalized size = 1.22 \begin{align*} \frac{2 \,{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} B b - 45 \,{\left (2 \, B b d -{\left (B a + A b\right )} e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 63 \,{\left (B b d^{2} + A a e^{2} -{\left (B a + A b\right )} d e\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{315 \, 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)^(3/2),x, algorithm="maxima")

[Out]

2/315*(35*(e*x + d)^(9/2)*B*b - 45*(2*B*b*d - (B*a + A*b)*e)*(e*x + d)^(7/2) + 63*(B*b*d^2 + A*a*e^2 - (B*a +
A*b)*d*e)*(e*x + d)^(5/2))/e^3

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Fricas [B]  time = 1.88055, size = 342, normalized size = 4.12 \begin{align*} \frac{2 \,{\left (35 \, B b e^{4} x^{4} + 8 \, B b d^{4} + 63 \, A a d^{2} e^{2} - 18 \,{\left (B a + A b\right )} d^{3} e + 5 \,{\left (10 \, B b d e^{3} + 9 \,{\left (B a + A b\right )} e^{4}\right )} x^{3} + 3 \,{\left (B b d^{2} e^{2} + 21 \, A a e^{4} + 24 \,{\left (B a + A b\right )} d e^{3}\right )} x^{2} -{\left (4 \, B b d^{3} e - 126 \, A a d e^{3} - 9 \,{\left (B a + A b\right )} d^{2} e^{2}\right )} x\right )} \sqrt{e x + d}}{315 \, 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)^(3/2),x, algorithm="fricas")

[Out]

2/315*(35*B*b*e^4*x^4 + 8*B*b*d^4 + 63*A*a*d^2*e^2 - 18*(B*a + A*b)*d^3*e + 5*(10*B*b*d*e^3 + 9*(B*a + A*b)*e^
4)*x^3 + 3*(B*b*d^2*e^2 + 21*A*a*e^4 + 24*(B*a + A*b)*d*e^3)*x^2 - (4*B*b*d^3*e - 126*A*a*d*e^3 - 9*(B*a + A*b
)*d^2*e^2)*x)*sqrt(e*x + d)/e^3

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Sympy [A]  time = 9.97825, size = 318, normalized size = 3.83 \begin{align*} A a d \left (\begin{cases} \sqrt{d} x & \text{for}\: e = 0 \\\frac{2 \left (d + e x\right )^{\frac{3}{2}}}{3 e} & \text{otherwise} \end{cases}\right ) + \frac{2 A a \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e} + \frac{2 A b d \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{2 A b \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{2}} + \frac{2 B a d \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{2 B a \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{2}} + \frac{2 B b d \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}} + \frac{2 B b \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{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)**(3/2),x)

[Out]

A*a*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*A*a*(-d*(d + e*x)**(3/2)/3 + (d +
e*x)**(5/2)/5)/e + 2*A*b*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 2*A*b*(d**2*(d + e*x)**(3/2)/3
- 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 2*B*a*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e*
*2 + 2*B*a*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 2*B*b*d*(d**2*(d + e
*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 2*B*b*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d
+ e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3

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Giac [B]  time = 1.62527, size = 377, normalized size = 4.54 \begin{align*} \frac{2}{315} \,{\left (21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} B a d e^{\left (-1\right )} + 21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} A b d e^{\left (-1\right )} + 3 \,{\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 d e^{\left (-2\right )} + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} A a d + 3 \,{\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 a e^{\left (-1\right )} + 3 \,{\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 )} A b e^{\left (-1\right )} +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} B b e^{\left (-2\right )} + 21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} 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)^(3/2),x, algorithm="giac")

[Out]

2/315*(21*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a*d*e^(-1) + 21*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d
)*A*b*d*e^(-1) + 3*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*B*b*d*e^(-2) + 105*(x*
e + d)^(3/2)*A*a*d + 3*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*B*a*e^(-1) + 3*(15
*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*b*e^(-1) + (35*(x*e + d)^(9/2) - 135*(x*e
+ d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*B*b*e^(-2) + 21*(3*(x*e + d)^(5/2) - 5*(x*e
+ d)^(3/2)*d)*A*a)*e^(-1)