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

Optimal. Leaf size=67 \[ \frac{2 (a+b x)^{9/2} (A b-2 a B)}{9 b^3}-\frac{2 a (a+b x)^{7/2} (A b-a B)}{7 b^3}+\frac{2 B (a+b x)^{11/2}}{11 b^3} \]

[Out]

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

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Rubi [A]  time = 0.0238322, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {77} \[ \frac{2 (a+b x)^{9/2} (A b-2 a B)}{9 b^3}-\frac{2 a (a+b x)^{7/2} (A b-a B)}{7 b^3}+\frac{2 B (a+b x)^{11/2}}{11 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)^(5/2)*(A + B*x),x]

[Out]

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

Mathematica [A]  time = 0.0363971, size = 49, normalized size = 0.73 \[ \frac{2 (a+b x)^{7/2} \left (8 a^2 B-2 a b (11 A+14 B x)+7 b^2 x (11 A+9 B x)\right )}{693 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a + b*x)^(7/2)*(8*a^2*B + 7*b^2*x*(11*A + 9*B*x) - 2*a*b*(11*A + 14*B*x)))/(693*b^3)

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Maple [A]  time = 0.004, size = 47, normalized size = 0.7 \begin{align*} -{\frac{-126\,{b}^{2}B{x}^{2}-154\,{b}^{2}Ax+56\,abBx+44\,Aba-16\,B{a}^{2}}{693\,{b}^{3}} \left ( bx+a \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.34349, size = 73, normalized size = 1.09 \begin{align*} \frac{2 \,{\left (63 \,{\left (b x + a\right )}^{\frac{11}{2}} B - 77 \,{\left (2 \, B a - A b\right )}{\left (b x + a\right )}^{\frac{9}{2}} + 99 \,{\left (B a^{2} - A a b\right )}{\left (b x + a\right )}^{\frac{7}{2}}\right )}}{693 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^(5/2)*(B*x+A),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.50012, size = 266, normalized size = 3.97 \begin{align*} \frac{2 \,{\left (63 \, B b^{5} x^{5} + 8 \, B a^{5} - 22 \, A a^{4} b + 7 \,{\left (23 \, B a b^{4} + 11 \, A b^{5}\right )} x^{4} +{\left (113 \, B a^{2} b^{3} + 209 \, A a b^{4}\right )} x^{3} + 3 \,{\left (B a^{3} b^{2} + 55 \, A a^{2} b^{3}\right )} x^{2} -{\left (4 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} x\right )} \sqrt{b x + a}}{693 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^(5/2)*(B*x+A),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.08946, size = 245, normalized size = 3.66 \begin{align*} \begin{cases} - \frac{4 A a^{4} \sqrt{a + b x}}{63 b^{2}} + \frac{2 A a^{3} x \sqrt{a + b x}}{63 b} + \frac{10 A a^{2} x^{2} \sqrt{a + b x}}{21} + \frac{38 A a b x^{3} \sqrt{a + b x}}{63} + \frac{2 A b^{2} x^{4} \sqrt{a + b x}}{9} + \frac{16 B a^{5} \sqrt{a + b x}}{693 b^{3}} - \frac{8 B a^{4} x \sqrt{a + b x}}{693 b^{2}} + \frac{2 B a^{3} x^{2} \sqrt{a + b x}}{231 b} + \frac{226 B a^{2} x^{3} \sqrt{a + b x}}{693} + \frac{46 B a b x^{4} \sqrt{a + b x}}{99} + \frac{2 B b^{2} x^{5} \sqrt{a + b x}}{11} & \text{for}\: b \neq 0 \\a^{\frac{5}{2}} \left (\frac{A x^{2}}{2} + \frac{B x^{3}}{3}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)**(5/2)*(B*x+A),x)

[Out]

Piecewise((-4*A*a**4*sqrt(a + b*x)/(63*b**2) + 2*A*a**3*x*sqrt(a + b*x)/(63*b) + 10*A*a**2*x**2*sqrt(a + b*x)/
21 + 38*A*a*b*x**3*sqrt(a + b*x)/63 + 2*A*b**2*x**4*sqrt(a + b*x)/9 + 16*B*a**5*sqrt(a + b*x)/(693*b**3) - 8*B
*a**4*x*sqrt(a + b*x)/(693*b**2) + 2*B*a**3*x**2*sqrt(a + b*x)/(231*b) + 226*B*a**2*x**3*sqrt(a + b*x)/693 + 4
6*B*a*b*x**4*sqrt(a + b*x)/99 + 2*B*b**2*x**5*sqrt(a + b*x)/11, Ne(b, 0)), (a**(5/2)*(A*x**2/2 + B*x**3/3), Tr
ue))

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Giac [B]  time = 1.13915, size = 374, normalized size = 5.58 \begin{align*} \frac{2 \,{\left (\frac{231 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a\right )} A a^{2}}{b} + \frac{33 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2}\right )} B a^{2}}{b^{2}} + \frac{66 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2}\right )} A a}{b} + \frac{22 \,{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3}\right )} B a}{b^{2}} + \frac{11 \,{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3}\right )} A}{b} + \frac{{\left (315 \,{\left (b x + a\right )}^{\frac{11}{2}} - 1540 \,{\left (b x + a\right )}^{\frac{9}{2}} a + 2970 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{2} - 2772 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{3} + 1155 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{4}\right )} B}{b^{2}}\right )}}{3465 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)^(5/2)*(B*x+A),x, algorithm="giac")

[Out]

2/3465*(231*(3*(b*x + a)^(5/2) - 5*(b*x + a)^(3/2)*a)*A*a^2/b + 33*(15*(b*x + a)^(7/2) - 42*(b*x + a)^(5/2)*a
+ 35*(b*x + a)^(3/2)*a^2)*B*a^2/b^2 + 66*(15*(b*x + a)^(7/2) - 42*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2)*
A*a/b + 22*(35*(b*x + a)^(9/2) - 135*(b*x + a)^(7/2)*a + 189*(b*x + a)^(5/2)*a^2 - 105*(b*x + a)^(3/2)*a^3)*B*
a/b^2 + 11*(35*(b*x + a)^(9/2) - 135*(b*x + a)^(7/2)*a + 189*(b*x + a)^(5/2)*a^2 - 105*(b*x + a)^(3/2)*a^3)*A/
b + (315*(b*x + a)^(11/2) - 1540*(b*x + a)^(9/2)*a + 2970*(b*x + a)^(7/2)*a^2 - 2772*(b*x + a)^(5/2)*a^3 + 115
5*(b*x + a)^(3/2)*a^4)*B/b^2)/b