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]

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

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Rubi [A]  time = 0.02, antiderivative size = 67, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.05, 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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fricas [B]  time = 0.62, size = 118, normalized size = 1.76 \[ \frac {2 \, {\left (63 \, B b^{5} x^{5} + 8 \, B a^{5} - 22 \, A a^{4} b + 7 \, {\left (23 \, B a b^{4} + 11 \, A b^{5}\right )} x^{4} + {\left (113 \, B a^{2} b^{3} + 209 \, A a b^{4}\right )} x^{3} + 3 \, {\left (B a^{3} b^{2} + 55 \, A a^{2} b^{3}\right )} x^{2} - {\left (4 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} x\right )} \sqrt {b x + a}}{693 \, b^{3}} \]

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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giac [B]  time = 1.27, size = 418, normalized size = 6.24 \[ \frac {2 \, {\left (\frac {1155 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} A a^{3}}{b} + \frac {231 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} B a^{3}}{b^{2}} + \frac {693 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} A a^{2}}{b} + \frac {297 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} B a^{2}}{b^{2}} + \frac {297 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} A a}{b} + \frac {33 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} B a}{b^{2}} + \frac {11 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} A}{b} + \frac {5 \, {\left (63 \, {\left (b x + a\right )}^{\frac {11}{2}} - 385 \, {\left (b x + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} - 693 \, \sqrt {b x + a} a^{5}\right )} B}{b^{2}}\right )}}{3465 \, b} \]

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*(1155*((b*x + a)^(3/2) - 3*sqrt(b*x + a)*a)*A*a^3/b + 231*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 1
5*sqrt(b*x + a)*a^2)*B*a^3/b^2 + 693*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*sqrt(b*x + a)*a^2)*A*a^2/b
+ 297*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2 - 35*sqrt(b*x + a)*a^3)*B*a^2/b^2 +
297*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2 - 35*sqrt(b*x + a)*a^3)*A*a/b + 33*(35*
(b*x + a)^(9/2) - 180*(b*x + a)^(7/2)*a + 378*(b*x + a)^(5/2)*a^2 - 420*(b*x + a)^(3/2)*a^3 + 315*sqrt(b*x + a
)*a^4)*B*a/b^2 + 11*(35*(b*x + a)^(9/2) - 180*(b*x + a)^(7/2)*a + 378*(b*x + a)^(5/2)*a^2 - 420*(b*x + a)^(3/2
)*a^3 + 315*sqrt(b*x + a)*a^4)*A/b + 5*(63*(b*x + a)^(11/2) - 385*(b*x + a)^(9/2)*a + 990*(b*x + a)^(7/2)*a^2
- 1386*(b*x + a)^(5/2)*a^3 + 1155*(b*x + a)^(3/2)*a^4 - 693*sqrt(b*x + a)*a^5)*B/b^2)/b

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maple [A]  time = 0.00, size = 47, normalized size = 0.70 \[ -\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (-63 B \,b^{2} x^{2}-77 A \,b^{2} x +28 B a b x +22 A a b -8 B \,a^{2}\right )}{693 b^{3}} \]

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 = 0.89, size = 54, normalized size = 0.81 \[ \frac {2 \, {\left (63 \, {\left (b x + a\right )}^{\frac {11}{2}} B - 77 \, {\left (2 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {9}{2}} + 99 \, {\left (B a^{2} - A a b\right )} {\left (b x + a\right )}^{\frac {7}{2}}\right )}}{693 \, b^{3}} \]

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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mupad [B]  time = 0.38, size = 52, normalized size = 0.78 \[ \frac {2\,{\left (a+b\,x\right )}^{7/2}\,\left (99\,B\,a^2+63\,B\,{\left (a+b\,x\right )}^2-99\,A\,a\,b+77\,A\,b\,\left (a+b\,x\right )-154\,B\,a\,\left (a+b\,x\right )\right )}{693\,b^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(A + B*x)*(a + b*x)^(5/2),x)

[Out]

(2*(a + b*x)^(7/2)*(99*B*a^2 + 63*B*(a + b*x)^2 - 99*A*a*b + 77*A*b*(a + b*x) - 154*B*a*(a + b*x)))/(693*b^3)

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sympy [A]  time = 3.24, size = 245, normalized size = 3.66 \[ \begin {cases} - \frac {4 A a^{4} \sqrt {a + b x}}{63 b^{2}} + \frac {2 A a^{3} x \sqrt {a + b x}}{63 b} + \frac {10 A a^{2} x^{2} \sqrt {a + b x}}{21} + \frac {38 A a b x^{3} \sqrt {a + b x}}{63} + \frac {2 A b^{2} x^{4} \sqrt {a + b x}}{9} + \frac {16 B a^{5} \sqrt {a + b x}}{693 b^{3}} - \frac {8 B a^{4} x \sqrt {a + b x}}{693 b^{2}} + \frac {2 B a^{3} x^{2} \sqrt {a + b x}}{231 b} + \frac {226 B a^{2} x^{3} \sqrt {a + b x}}{693} + \frac {46 B a b x^{4} \sqrt {a + b x}}{99} + \frac {2 B b^{2} x^{5} \sqrt {a + b x}}{11} & \text {for}\: b \neq 0 \\a^{\frac {5}{2}} \left (\frac {A x^{2}}{2} + \frac {B x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]

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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