3.1718 \(\int (a+b x) (A+B x) (d+e x)^{7/2} \, dx\)

Optimal. Leaf size=83 \[ -\frac{2 (d+e x)^{11/2} (-a B e-A b e+2 b B d)}{11 e^3}+\frac{2 (d+e x)^{9/2} (b d-a e) (B d-A e)}{9 e^3}+\frac{2 b B (d+e x)^{13/2}}{13 e^3} \]

[Out]

(2*(b*d - a*e)*(B*d - A*e)*(d + e*x)^(9/2))/(9*e^3) - (2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(11/2))/(11*e^3)
+ (2*b*B*(d + e*x)^(13/2))/(13*e^3)

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Rubi [A]  time = 0.0439221, 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} \[ -\frac{2 (d+e x)^{11/2} (-a B e-A b e+2 b B d)}{11 e^3}+\frac{2 (d+e x)^{9/2} (b d-a e) (B d-A e)}{9 e^3}+\frac{2 b B (d+e x)^{13/2}}{13 e^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(A + B*x)*(d + e*x)^(7/2),x]

[Out]

(2*(b*d - a*e)*(B*d - A*e)*(d + e*x)^(9/2))/(9*e^3) - (2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(11/2))/(11*e^3)
+ (2*b*B*(d + e*x)^(13/2))/(13*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)^{7/2} \, dx &=\int \left (\frac{(-b d+a e) (-B d+A e) (d+e x)^{7/2}}{e^2}+\frac{(-2 b B d+A b e+a B e) (d+e x)^{9/2}}{e^2}+\frac{b B (d+e x)^{11/2}}{e^2}\right ) \, dx\\ &=\frac{2 (b d-a e) (B d-A e) (d+e x)^{9/2}}{9 e^3}-\frac{2 (2 b B d-A b e-a B e) (d+e x)^{11/2}}{11 e^3}+\frac{2 b B (d+e x)^{13/2}}{13 e^3}\\ \end{align*}

Mathematica [A]  time = 0.0743422, size = 70, normalized size = 0.84 \[ \frac{2 (d+e x)^{9/2} \left (13 a e (11 A e-2 B d+9 B e x)+13 A b e (9 e x-2 d)+b B \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )}{1287 e^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(A + B*x)*(d + e*x)^(7/2),x]

[Out]

(2*(d + e*x)^(9/2)*(13*A*b*e*(-2*d + 9*e*x) + 13*a*e*(-2*B*d + 11*A*e + 9*B*e*x) + b*B*(8*d^2 - 36*d*e*x + 99*
e^2*x^2)))/(1287*e^3)

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Maple [A]  time = 0.003, size = 73, normalized size = 0.9 \begin{align*}{\frac{198\,bB{x}^{2}{e}^{2}+234\,Ab{e}^{2}x+234\,Ba{e}^{2}x-72\,Bbdex+286\,aA{e}^{2}-52\,Abde-52\,Bade+16\,bB{d}^{2}}{1287\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(B*x+A)*(e*x+d)^(7/2),x)

[Out]

2/1287*(e*x+d)^(9/2)*(99*B*b*e^2*x^2+117*A*b*e^2*x+117*B*a*e^2*x-36*B*b*d*e*x+143*A*a*e^2-26*A*b*d*e-26*B*a*d*
e+8*B*b*d^2)/e^3

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Maxima [A]  time = 1.10207, size = 101, normalized size = 1.22 \begin{align*} \frac{2 \,{\left (99 \,{\left (e x + d\right )}^{\frac{13}{2}} B b - 117 \,{\left (2 \, B b d -{\left (B a + A b\right )} e\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 143 \,{\left (B b d^{2} + A a e^{2} -{\left (B a + A b\right )} d e\right )}{\left (e x + d\right )}^{\frac{9}{2}}\right )}}{1287 \, 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)^(7/2),x, algorithm="maxima")

[Out]

2/1287*(99*(e*x + d)^(13/2)*B*b - 117*(2*B*b*d - (B*a + A*b)*e)*(e*x + d)^(11/2) + 143*(B*b*d^2 + A*a*e^2 - (B
*a + A*b)*d*e)*(e*x + d)^(9/2))/e^3

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Fricas [B]  time = 1.90534, size = 536, normalized size = 6.46 \begin{align*} \frac{2 \,{\left (99 \, B b e^{6} x^{6} + 8 \, B b d^{6} + 143 \, A a d^{4} e^{2} - 26 \,{\left (B a + A b\right )} d^{5} e + 9 \,{\left (40 \, B b d e^{5} + 13 \,{\left (B a + A b\right )} e^{6}\right )} x^{5} +{\left (458 \, B b d^{2} e^{4} + 143 \, A a e^{6} + 442 \,{\left (B a + A b\right )} d e^{5}\right )} x^{4} + 2 \,{\left (106 \, B b d^{3} e^{3} + 286 \, A a d e^{5} + 299 \,{\left (B a + A b\right )} d^{2} e^{4}\right )} x^{3} + 3 \,{\left (B b d^{4} e^{2} + 286 \, A a d^{2} e^{4} + 104 \,{\left (B a + A b\right )} d^{3} e^{3}\right )} x^{2} -{\left (4 \, B b d^{5} e - 572 \, A a d^{3} e^{3} - 13 \,{\left (B a + A b\right )} d^{4} e^{2}\right )} x\right )} \sqrt{e x + d}}{1287 \, 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)^(7/2),x, algorithm="fricas")

[Out]

2/1287*(99*B*b*e^6*x^6 + 8*B*b*d^6 + 143*A*a*d^4*e^2 - 26*(B*a + A*b)*d^5*e + 9*(40*B*b*d*e^5 + 13*(B*a + A*b)
*e^6)*x^5 + (458*B*b*d^2*e^4 + 143*A*a*e^6 + 442*(B*a + A*b)*d*e^5)*x^4 + 2*(106*B*b*d^3*e^3 + 286*A*a*d*e^5 +
299*(B*a + A*b)*d^2*e^4)*x^3 + 3*(B*b*d^4*e^2 + 286*A*a*d^2*e^4 + 104*(B*a + A*b)*d^3*e^3)*x^2 - (4*B*b*d^5*e
- 572*A*a*d^3*e^3 - 13*(B*a + A*b)*d^4*e^2)*x)*sqrt(e*x + d)/e^3

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Sympy [A]  time = 8.29217, size = 578, normalized size = 6.96 \begin{align*} \begin{cases} \frac{2 A a d^{4} \sqrt{d + e x}}{9 e} + \frac{8 A a d^{3} x \sqrt{d + e x}}{9} + \frac{4 A a d^{2} e x^{2} \sqrt{d + e x}}{3} + \frac{8 A a d e^{2} x^{3} \sqrt{d + e x}}{9} + \frac{2 A a e^{3} x^{4} \sqrt{d + e x}}{9} - \frac{4 A b d^{5} \sqrt{d + e x}}{99 e^{2}} + \frac{2 A b d^{4} x \sqrt{d + e x}}{99 e} + \frac{16 A b d^{3} x^{2} \sqrt{d + e x}}{33} + \frac{92 A b d^{2} e x^{3} \sqrt{d + e x}}{99} + \frac{68 A b d e^{2} x^{4} \sqrt{d + e x}}{99} + \frac{2 A b e^{3} x^{5} \sqrt{d + e x}}{11} - \frac{4 B a d^{5} \sqrt{d + e x}}{99 e^{2}} + \frac{2 B a d^{4} x \sqrt{d + e x}}{99 e} + \frac{16 B a d^{3} x^{2} \sqrt{d + e x}}{33} + \frac{92 B a d^{2} e x^{3} \sqrt{d + e x}}{99} + \frac{68 B a d e^{2} x^{4} \sqrt{d + e x}}{99} + \frac{2 B a e^{3} x^{5} \sqrt{d + e x}}{11} + \frac{16 B b d^{6} \sqrt{d + e x}}{1287 e^{3}} - \frac{8 B b d^{5} x \sqrt{d + e x}}{1287 e^{2}} + \frac{2 B b d^{4} x^{2} \sqrt{d + e x}}{429 e} + \frac{424 B b d^{3} x^{3} \sqrt{d + e x}}{1287} + \frac{916 B b d^{2} e x^{4} \sqrt{d + e x}}{1287} + \frac{80 B b d e^{2} x^{5} \sqrt{d + e x}}{143} + \frac{2 B b e^{3} x^{6} \sqrt{d + e x}}{13} & \text{for}\: e \neq 0 \\d^{\frac{7}{2}} \left (A a x + \frac{A b x^{2}}{2} + \frac{B a x^{2}}{2} + \frac{B b x^{3}}{3}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)*(e*x+d)**(7/2),x)

[Out]

Piecewise((2*A*a*d**4*sqrt(d + e*x)/(9*e) + 8*A*a*d**3*x*sqrt(d + e*x)/9 + 4*A*a*d**2*e*x**2*sqrt(d + e*x)/3 +
8*A*a*d*e**2*x**3*sqrt(d + e*x)/9 + 2*A*a*e**3*x**4*sqrt(d + e*x)/9 - 4*A*b*d**5*sqrt(d + e*x)/(99*e**2) + 2*
A*b*d**4*x*sqrt(d + e*x)/(99*e) + 16*A*b*d**3*x**2*sqrt(d + e*x)/33 + 92*A*b*d**2*e*x**3*sqrt(d + e*x)/99 + 68
*A*b*d*e**2*x**4*sqrt(d + e*x)/99 + 2*A*b*e**3*x**5*sqrt(d + e*x)/11 - 4*B*a*d**5*sqrt(d + e*x)/(99*e**2) + 2*
B*a*d**4*x*sqrt(d + e*x)/(99*e) + 16*B*a*d**3*x**2*sqrt(d + e*x)/33 + 92*B*a*d**2*e*x**3*sqrt(d + e*x)/99 + 68
*B*a*d*e**2*x**4*sqrt(d + e*x)/99 + 2*B*a*e**3*x**5*sqrt(d + e*x)/11 + 16*B*b*d**6*sqrt(d + e*x)/(1287*e**3) -
8*B*b*d**5*x*sqrt(d + e*x)/(1287*e**2) + 2*B*b*d**4*x**2*sqrt(d + e*x)/(429*e) + 424*B*b*d**3*x**3*sqrt(d + e
*x)/1287 + 916*B*b*d**2*e*x**4*sqrt(d + e*x)/1287 + 80*B*b*d*e**2*x**5*sqrt(d + e*x)/143 + 2*B*b*e**3*x**6*sqr
t(d + e*x)/13, Ne(e, 0)), (d**(7/2)*(A*a*x + A*b*x**2/2 + B*a*x**2/2 + B*b*x**3/3), True))

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Giac [B]  time = 2.839, size = 1058, normalized size = 12.75 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)*(e*x+d)^(7/2),x, algorithm="giac")

[Out]

2/45045*(3003*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a*d^3*e^(-1) + 3003*(3*(x*e + d)^(5/2) - 5*(x*e + d)
^(3/2)*d)*A*b*d^3*e^(-1) + 429*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*B*b*d^3*e^
(-2) + 15015*(x*e + d)^(3/2)*A*a*d^3 + 1287*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^
2)*B*a*d^2*e^(-1) + 1287*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*b*d^2*e^(-1) +
429*(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*d^2*
e^(-2) + 9009*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*A*a*d^2 + 429*(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*a*d*e^(-1) + 429*(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)*A*b*d*e^(-1) + 39*(315*(x*e + d)^(11/2) - 1540*
(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*B*b*d*e^(-
2) + 1287*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*a*d + 13*(315*(x*e + d)^(11/2
) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*B
*a*e^(-1) + 13*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2
)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*A*b*e^(-1) + 5*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e
+ d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*B*b*e^(-2) +
143*(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)*A*a)*e^(
-1)