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

Optimal. Leaf size=95 \[ \frac {2 a^2 (a+b x)^{7/2} (A b-a B)}{7 b^4}+\frac {2 (a+b x)^{11/2} (A b-3 a B)}{11 b^4}-\frac {2 a (a+b x)^{9/2} (2 A b-3 a B)}{9 b^4}+\frac {2 B (a+b x)^{13/2}}{13 b^4} \]

[Out]

2/7*a^2*(A*b-B*a)*(b*x+a)^(7/2)/b^4-2/9*a*(2*A*b-3*B*a)*(b*x+a)^(9/2)/b^4+2/11*(A*b-3*B*a)*(b*x+a)^(11/2)/b^4+
2/13*B*(b*x+a)^(13/2)/b^4

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Rubi [A]  time = 0.03, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {77} \[ \frac {2 a^2 (a+b x)^{7/2} (A b-a B)}{7 b^4}+\frac {2 (a+b x)^{11/2} (A b-3 a B)}{11 b^4}-\frac {2 a (a+b x)^{9/2} (2 A b-3 a B)}{9 b^4}+\frac {2 B (a+b x)^{13/2}}{13 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)^(5/2)*(A + B*x),x]

[Out]

(2*a^2*(A*b - a*B)*(a + b*x)^(7/2))/(7*b^4) - (2*a*(2*A*b - 3*a*B)*(a + b*x)^(9/2))/(9*b^4) + (2*(A*b - 3*a*B)
*(a + b*x)^(11/2))/(11*b^4) + (2*B*(a + b*x)^(13/2))/(13*b^4)

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^2 (a+b x)^{5/2} (A+B x) \, dx &=\int \left (-\frac {a^2 (-A b+a B) (a+b x)^{5/2}}{b^3}+\frac {a (-2 A b+3 a B) (a+b x)^{7/2}}{b^3}+\frac {(A b-3 a B) (a+b x)^{9/2}}{b^3}+\frac {B (a+b x)^{11/2}}{b^3}\right ) \, dx\\ &=\frac {2 a^2 (A b-a B) (a+b x)^{7/2}}{7 b^4}-\frac {2 a (2 A b-3 a B) (a+b x)^{9/2}}{9 b^4}+\frac {2 (A b-3 a B) (a+b x)^{11/2}}{11 b^4}+\frac {2 B (a+b x)^{13/2}}{13 b^4}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 68, normalized size = 0.72 \[ \frac {2 (a+b x)^{7/2} \left (-48 a^3 B+8 a^2 b (13 A+21 B x)-14 a b^2 x (26 A+27 B x)+63 b^3 x^2 (13 A+11 B x)\right )}{9009 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a + b*x)^(7/2)*(-48*a^3*B + 63*b^3*x^2*(13*A + 11*B*x) + 8*a^2*b*(13*A + 21*B*x) - 14*a*b^2*x*(26*A + 27*B
*x)))/(9009*b^4)

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fricas [A]  time = 0.70, size = 143, normalized size = 1.51 \[ \frac {2 \, {\left (693 \, B b^{6} x^{6} - 48 \, B a^{6} + 104 \, A a^{5} b + 63 \, {\left (27 \, B a b^{5} + 13 \, A b^{6}\right )} x^{5} + 7 \, {\left (159 \, B a^{2} b^{4} + 299 \, A a b^{5}\right )} x^{4} + {\left (15 \, B a^{3} b^{3} + 1469 \, A a^{2} b^{4}\right )} x^{3} - 3 \, {\left (6 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{2} + 4 \, {\left (6 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x\right )} \sqrt {b x + a}}{9009 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9009*(693*B*b^6*x^6 - 48*B*a^6 + 104*A*a^5*b + 63*(27*B*a*b^5 + 13*A*b^6)*x^5 + 7*(159*B*a^2*b^4 + 299*A*a*b
^5)*x^4 + (15*B*a^3*b^3 + 1469*A*a^2*b^4)*x^3 - 3*(6*B*a^4*b^2 - 13*A*a^3*b^3)*x^2 + 4*(6*B*a^5*b - 13*A*a^4*b
^2)*x)*sqrt(b*x + a)/b^4

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giac [B]  time = 1.34, size = 516, normalized size = 5.43 \[ \frac {2 \, {\left (\frac {3003 \, {\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^{3}}{b^{2}} + \frac {1287 \, {\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^{3}}{b^{3}} + \frac {3861 \, {\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^{2}}{b^{2}} + \frac {429 \, {\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^{2}}{b^{3}} + \frac {429 \, {\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 a}{b^{2}} + \frac {195 \, {\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 a}{b^{3}} + \frac {65 \, {\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 )} A}{b^{2}} + \frac {15 \, {\left (231 \, {\left (b x + a\right )}^{\frac {13}{2}} - 1638 \, {\left (b x + a\right )}^{\frac {11}{2}} a + 5005 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{2} - 8580 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{3} + 9009 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{4} - 6006 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{5} + 3003 \, \sqrt {b x + a} a^{6}\right )} B}{b^{3}}\right )}}{45045 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3003*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*sqrt(b*x + a)*a^2)*A*a^3/b^2 + 1287*(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^3/b^3 + 3861*(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^2/b^2 + 429*(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^2/b
^3 + 429*(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*a/b^2 + 195*(63*(b*x + a)^(11/2) - 385*(b*x + a)^(9/2)*a + 990*(b*x + a)^(7/2)*a^2 - 138
6*(b*x + a)^(5/2)*a^3 + 1155*(b*x + a)^(3/2)*a^4 - 693*sqrt(b*x + a)*a^5)*B*a/b^3 + 65*(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*sq
rt(b*x + a)*a^5)*A/b^2 + 15*(231*(b*x + a)^(13/2) - 1638*(b*x + a)^(11/2)*a + 5005*(b*x + a)^(9/2)*a^2 - 8580*
(b*x + a)^(7/2)*a^3 + 9009*(b*x + a)^(5/2)*a^4 - 6006*(b*x + a)^(3/2)*a^5 + 3003*sqrt(b*x + a)*a^6)*B/b^3)/b

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maple [A]  time = 0.01, size = 71, normalized size = 0.75 \[ \frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (693 B \,b^{3} x^{3}+819 A \,b^{3} x^{2}-378 B a \,b^{2} x^{2}-364 A a \,b^{2} x +168 B \,a^{2} b x +104 A \,a^{2} b -48 B \,a^{3}\right )}{9009 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9009*(b*x+a)^(7/2)*(693*B*b^3*x^3+819*A*b^3*x^2-378*B*a*b^2*x^2-364*A*a*b^2*x+168*B*a^2*b*x+104*A*a^2*b-48*B
*a^3)/b^4

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maxima [A]  time = 0.87, size = 77, normalized size = 0.81 \[ \frac {2 \, {\left (693 \, {\left (b x + a\right )}^{\frac {13}{2}} B - 819 \, {\left (3 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {11}{2}} + 1001 \, {\left (3 \, B a^{2} - 2 \, A a b\right )} {\left (b x + a\right )}^{\frac {9}{2}} - 1287 \, {\left (B a^{3} - A a^{2} b\right )} {\left (b x + a\right )}^{\frac {7}{2}}\right )}}{9009 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9009*(693*(b*x + a)^(13/2)*B - 819*(3*B*a - A*b)*(b*x + a)^(11/2) + 1001*(3*B*a^2 - 2*A*a*b)*(b*x + a)^(9/2)
- 1287*(B*a^3 - A*a^2*b)*(b*x + a)^(7/2))/b^4

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mupad [B]  time = 0.07, size = 85, normalized size = 0.89 \[ \frac {\left (6\,B\,a^2-4\,A\,a\,b\right )\,{\left (a+b\,x\right )}^{9/2}}{9\,b^4}+\frac {2\,B\,{\left (a+b\,x\right )}^{13/2}}{13\,b^4}+\frac {\left (2\,A\,b-6\,B\,a\right )\,{\left (a+b\,x\right )}^{11/2}}{11\,b^4}-\frac {\left (2\,B\,a^3-2\,A\,a^2\,b\right )\,{\left (a+b\,x\right )}^{7/2}}{7\,b^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(A + B*x)*(a + b*x)^(5/2),x)

[Out]

((6*B*a^2 - 4*A*a*b)*(a + b*x)^(9/2))/(9*b^4) + (2*B*(a + b*x)^(13/2))/(13*b^4) + ((2*A*b - 6*B*a)*(a + b*x)^(
11/2))/(11*b^4) - ((2*B*a^3 - 2*A*a^2*b)*(a + b*x)^(7/2))/(7*b^4)

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sympy [A]  time = 4.02, size = 292, normalized size = 3.07 \[ \begin {cases} \frac {16 A a^{5} \sqrt {a + b x}}{693 b^{3}} - \frac {8 A a^{4} x \sqrt {a + b x}}{693 b^{2}} + \frac {2 A a^{3} x^{2} \sqrt {a + b x}}{231 b} + \frac {226 A a^{2} x^{3} \sqrt {a + b x}}{693} + \frac {46 A a b x^{4} \sqrt {a + b x}}{99} + \frac {2 A b^{2} x^{5} \sqrt {a + b x}}{11} - \frac {32 B a^{6} \sqrt {a + b x}}{3003 b^{4}} + \frac {16 B a^{5} x \sqrt {a + b x}}{3003 b^{3}} - \frac {4 B a^{4} x^{2} \sqrt {a + b x}}{1001 b^{2}} + \frac {10 B a^{3} x^{3} \sqrt {a + b x}}{3003 b} + \frac {106 B a^{2} x^{4} \sqrt {a + b x}}{429} + \frac {54 B a b x^{5} \sqrt {a + b x}}{143} + \frac {2 B b^{2} x^{6} \sqrt {a + b x}}{13} & \text {for}\: b \neq 0 \\a^{\frac {5}{2}} \left (\frac {A x^{3}}{3} + \frac {B x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((16*A*a**5*sqrt(a + b*x)/(693*b**3) - 8*A*a**4*x*sqrt(a + b*x)/(693*b**2) + 2*A*a**3*x**2*sqrt(a + b
*x)/(231*b) + 226*A*a**2*x**3*sqrt(a + b*x)/693 + 46*A*a*b*x**4*sqrt(a + b*x)/99 + 2*A*b**2*x**5*sqrt(a + b*x)
/11 - 32*B*a**6*sqrt(a + b*x)/(3003*b**4) + 16*B*a**5*x*sqrt(a + b*x)/(3003*b**3) - 4*B*a**4*x**2*sqrt(a + b*x
)/(1001*b**2) + 10*B*a**3*x**3*sqrt(a + b*x)/(3003*b) + 106*B*a**2*x**4*sqrt(a + b*x)/429 + 54*B*a*b*x**5*sqrt
(a + b*x)/143 + 2*B*b**2*x**6*sqrt(a + b*x)/13, Ne(b, 0)), (a**(5/2)*(A*x**3/3 + B*x**4/4), True))

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