3.1719 \(\int (a+b x) (A+B x) (d+e x)^{5/2} \, dx\)

Optimal. Leaf size=83 \[ -\frac{2 (d+e x)^{9/2} (-a B e-A b e+2 b B d)}{9 e^3}+\frac{2 (d+e x)^{7/2} (b d-a e) (B d-A e)}{7 e^3}+\frac{2 b B (d+e x)^{11/2}}{11 e^3} \]

[Out]

(2*(b*d - a*e)*(B*d - A*e)*(d + e*x)^(7/2))/(7*e^3) - (2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(9/2))/(9*e^3) +
(2*b*B*(d + e*x)^(11/2))/(11*e^3)

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Rubi [A]  time = 0.0350631, 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} \[ -\frac{2 (d+e x)^{9/2} (-a B e-A b e+2 b B d)}{9 e^3}+\frac{2 (d+e x)^{7/2} (b d-a e) (B d-A e)}{7 e^3}+\frac{2 b B (d+e x)^{11/2}}{11 e^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(A + B*x)*(d + e*x)^(5/2),x]

[Out]

(2*(b*d - a*e)*(B*d - A*e)*(d + e*x)^(7/2))/(7*e^3) - (2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(9/2))/(9*e^3) +
(2*b*B*(d + e*x)^(11/2))/(11*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)^{5/2} \, dx &=\int \left (\frac{(-b d+a e) (-B d+A e) (d+e x)^{5/2}}{e^2}+\frac{(-2 b B d+A b e+a B e) (d+e x)^{7/2}}{e^2}+\frac{b B (d+e x)^{9/2}}{e^2}\right ) \, dx\\ &=\frac{2 (b d-a e) (B d-A e) (d+e x)^{7/2}}{7 e^3}-\frac{2 (2 b B d-A b e-a B e) (d+e x)^{9/2}}{9 e^3}+\frac{2 b B (d+e x)^{11/2}}{11 e^3}\\ \end{align*}

Mathematica [A]  time = 0.0585851, size = 70, normalized size = 0.84 \[ \frac{2 (d+e x)^{7/2} \left (11 a e (9 A e-2 B d+7 B e x)+11 A b e (7 e x-2 d)+b B \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )}{693 e^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(A + B*x)*(d + e*x)^(5/2),x]

[Out]

(2*(d + e*x)^(7/2)*(11*A*b*e*(-2*d + 7*e*x) + 11*a*e*(-2*B*d + 9*A*e + 7*B*e*x) + b*B*(8*d^2 - 28*d*e*x + 63*e
^2*x^2)))/(693*e^3)

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Maple [A]  time = 0.003, size = 73, normalized size = 0.9 \begin{align*}{\frac{126\,bB{x}^{2}{e}^{2}+154\,Ab{e}^{2}x+154\,Ba{e}^{2}x-56\,Bbdex+198\,aA{e}^{2}-44\,Abde-44\,Bade+16\,bB{d}^{2}}{693\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(B*x+A)*(e*x+d)^(5/2),x)

[Out]

2/693*(e*x+d)^(7/2)*(63*B*b*e^2*x^2+77*A*b*e^2*x+77*B*a*e^2*x-28*B*b*d*e*x+99*A*a*e^2-22*A*b*d*e-22*B*a*d*e+8*
B*b*d^2)/e^3

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Maxima [A]  time = 1.04985, size = 101, normalized size = 1.22 \begin{align*} \frac{2 \,{\left (63 \,{\left (e x + d\right )}^{\frac{11}{2}} B b - 77 \,{\left (2 \, B b d -{\left (B a + A b\right )} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 99 \,{\left (B b d^{2} + A a e^{2} -{\left (B a + A b\right )} d e\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{693 \, 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)^(5/2),x, algorithm="maxima")

[Out]

2/693*(63*(e*x + d)^(11/2)*B*b - 77*(2*B*b*d - (B*a + A*b)*e)*(e*x + d)^(9/2) + 99*(B*b*d^2 + A*a*e^2 - (B*a +
A*b)*d*e)*(e*x + d)^(7/2))/e^3

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Fricas [B]  time = 1.82336, size = 435, normalized size = 5.24 \begin{align*} \frac{2 \,{\left (63 \, B b e^{5} x^{5} + 8 \, B b d^{5} + 99 \, A a d^{3} e^{2} - 22 \,{\left (B a + A b\right )} d^{4} e + 7 \,{\left (23 \, B b d e^{4} + 11 \,{\left (B a + A b\right )} e^{5}\right )} x^{4} +{\left (113 \, B b d^{2} e^{3} + 99 \, A a e^{5} + 209 \,{\left (B a + A b\right )} d e^{4}\right )} x^{3} + 3 \,{\left (B b d^{3} e^{2} + 99 \, A a d e^{4} + 55 \,{\left (B a + A b\right )} d^{2} e^{3}\right )} x^{2} -{\left (4 \, B b d^{4} e - 297 \, A a d^{2} e^{3} - 11 \,{\left (B a + A b\right )} d^{3} e^{2}\right )} x\right )} \sqrt{e x + d}}{693 \, 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)^(5/2),x, algorithm="fricas")

[Out]

2/693*(63*B*b*e^5*x^5 + 8*B*b*d^5 + 99*A*a*d^3*e^2 - 22*(B*a + A*b)*d^4*e + 7*(23*B*b*d*e^4 + 11*(B*a + A*b)*e
^5)*x^4 + (113*B*b*d^2*e^3 + 99*A*a*e^5 + 209*(B*a + A*b)*d*e^4)*x^3 + 3*(B*b*d^3*e^2 + 99*A*a*d*e^4 + 55*(B*a
+ A*b)*d^2*e^3)*x^2 - (4*B*b*d^4*e - 297*A*a*d^2*e^3 - 11*(B*a + A*b)*d^3*e^2)*x)*sqrt(e*x + d)/e^3

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Sympy [A]  time = 3.56146, size = 476, normalized size = 5.73 \begin{align*} \begin{cases} \frac{2 A a d^{3} \sqrt{d + e x}}{7 e} + \frac{6 A a d^{2} x \sqrt{d + e x}}{7} + \frac{6 A a d e x^{2} \sqrt{d + e x}}{7} + \frac{2 A a e^{2} x^{3} \sqrt{d + e x}}{7} - \frac{4 A b d^{4} \sqrt{d + e x}}{63 e^{2}} + \frac{2 A b d^{3} x \sqrt{d + e x}}{63 e} + \frac{10 A b d^{2} x^{2} \sqrt{d + e x}}{21} + \frac{38 A b d e x^{3} \sqrt{d + e x}}{63} + \frac{2 A b e^{2} x^{4} \sqrt{d + e x}}{9} - \frac{4 B a d^{4} \sqrt{d + e x}}{63 e^{2}} + \frac{2 B a d^{3} x \sqrt{d + e x}}{63 e} + \frac{10 B a d^{2} x^{2} \sqrt{d + e x}}{21} + \frac{38 B a d e x^{3} \sqrt{d + e x}}{63} + \frac{2 B a e^{2} x^{4} \sqrt{d + e x}}{9} + \frac{16 B b d^{5} \sqrt{d + e x}}{693 e^{3}} - \frac{8 B b d^{4} x \sqrt{d + e x}}{693 e^{2}} + \frac{2 B b d^{3} x^{2} \sqrt{d + e x}}{231 e} + \frac{226 B b d^{2} x^{3} \sqrt{d + e x}}{693} + \frac{46 B b d e x^{4} \sqrt{d + e x}}{99} + \frac{2 B b e^{2} x^{5} \sqrt{d + e x}}{11} & \text{for}\: e \neq 0 \\d^{\frac{5}{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)**(5/2),x)

[Out]

Piecewise((2*A*a*d**3*sqrt(d + e*x)/(7*e) + 6*A*a*d**2*x*sqrt(d + e*x)/7 + 6*A*a*d*e*x**2*sqrt(d + e*x)/7 + 2*
A*a*e**2*x**3*sqrt(d + e*x)/7 - 4*A*b*d**4*sqrt(d + e*x)/(63*e**2) + 2*A*b*d**3*x*sqrt(d + e*x)/(63*e) + 10*A*
b*d**2*x**2*sqrt(d + e*x)/21 + 38*A*b*d*e*x**3*sqrt(d + e*x)/63 + 2*A*b*e**2*x**4*sqrt(d + e*x)/9 - 4*B*a*d**4
*sqrt(d + e*x)/(63*e**2) + 2*B*a*d**3*x*sqrt(d + e*x)/(63*e) + 10*B*a*d**2*x**2*sqrt(d + e*x)/21 + 38*B*a*d*e*
x**3*sqrt(d + e*x)/63 + 2*B*a*e**2*x**4*sqrt(d + e*x)/9 + 16*B*b*d**5*sqrt(d + e*x)/(693*e**3) - 8*B*b*d**4*x*
sqrt(d + e*x)/(693*e**2) + 2*B*b*d**3*x**2*sqrt(d + e*x)/(231*e) + 226*B*b*d**2*x**3*sqrt(d + e*x)/693 + 46*B*
b*d*e*x**4*sqrt(d + e*x)/99 + 2*B*b*e**2*x**5*sqrt(d + e*x)/11, Ne(e, 0)), (d**(5/2)*(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 = 2.18995, size = 682, normalized size = 8.22 \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)^(5/2),x, algorithm="giac")

[Out]

2/3465*(231*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a*d^2*e^(-1) + 231*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3
/2)*d)*A*b*d^2*e^(-1) + 33*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*B*b*d^2*e^(-2)
+ 1155*(x*e + d)^(3/2)*A*a*d^2 + 66*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*B*a*
d*e^(-1) + 66*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*b*d*e^(-1) + 22*(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*e^(-2) + 462*(3*
(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*A*a*d + 11*(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*e^(-1) + 11*(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*e^(-1) + (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*e^(-2) + 33*(15*(x*e + d)^(7/2) -
42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*a)*e^(-1)