3.399 \(\int x^3 (a+b x)^{3/2} (A+B x) \, dx\)

Optimal. Leaf size=122 \[ -\frac {2 a^3 (a+b x)^{5/2} (A b-a B)}{5 b^5}+\frac {2 a^2 (a+b x)^{7/2} (3 A b-4 a B)}{7 b^5}+\frac {2 (a+b x)^{11/2} (A b-4 a B)}{11 b^5}-\frac {2 a (a+b x)^{9/2} (A b-2 a B)}{3 b^5}+\frac {2 B (a+b x)^{13/2}}{13 b^5} \]

[Out]

-2/5*a^3*(A*b-B*a)*(b*x+a)^(5/2)/b^5+2/7*a^2*(3*A*b-4*B*a)*(b*x+a)^(7/2)/b^5-2/3*a*(A*b-2*B*a)*(b*x+a)^(9/2)/b
^5+2/11*(A*b-4*B*a)*(b*x+a)^(11/2)/b^5+2/13*B*(b*x+a)^(13/2)/b^5

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 122, 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} \[ \frac {2 a^2 (a+b x)^{7/2} (3 A b-4 a B)}{7 b^5}-\frac {2 a^3 (a+b x)^{5/2} (A b-a B)}{5 b^5}+\frac {2 (a+b x)^{11/2} (A b-4 a B)}{11 b^5}-\frac {2 a (a+b x)^{9/2} (A b-2 a B)}{3 b^5}+\frac {2 B (a+b x)^{13/2}}{13 b^5} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x)^(3/2)*(A + B*x),x]

[Out]

(-2*a^3*(A*b - a*B)*(a + b*x)^(5/2))/(5*b^5) + (2*a^2*(3*A*b - 4*a*B)*(a + b*x)^(7/2))/(7*b^5) - (2*a*(A*b - 2
*a*B)*(a + b*x)^(9/2))/(3*b^5) + (2*(A*b - 4*a*B)*(a + b*x)^(11/2))/(11*b^5) + (2*B*(a + b*x)^(13/2))/(13*b^5)

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

________________________________________________________________________________________

Mathematica [A]  time = 0.09, size = 87, normalized size = 0.71 \[ \frac {2 (a+b x)^{5/2} \left (128 a^4 B-16 a^3 b (13 A+20 B x)+40 a^2 b^2 x (13 A+14 B x)-70 a b^3 x^2 (13 A+12 B x)+105 b^4 x^3 (13 A+11 B x)\right )}{15015 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x)^(3/2)*(A + B*x),x]

[Out]

(2*(a + b*x)^(5/2)*(128*a^4*B + 105*b^4*x^3*(13*A + 11*B*x) - 70*a*b^3*x^2*(13*A + 12*B*x) + 40*a^2*b^2*x*(13*
A + 14*B*x) - 16*a^3*b*(13*A + 20*B*x)))/(15015*b^5)

________________________________________________________________________________________

fricas [A]  time = 0.69, size = 143, normalized size = 1.17 \[ \frac {2 \, {\left (1155 \, B b^{6} x^{6} + 128 \, B a^{6} - 208 \, A a^{5} b + 105 \, {\left (14 \, B a b^{5} + 13 \, A b^{6}\right )} x^{5} + 35 \, {\left (B a^{2} b^{4} + 52 \, A a b^{5}\right )} x^{4} - 5 \, {\left (8 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{3} + 6 \, {\left (8 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{2} - 8 \, {\left (8 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x\right )} \sqrt {b x + a}}{15015 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^(3/2)*(B*x+A),x, algorithm="fricas")

[Out]

2/15015*(1155*B*b^6*x^6 + 128*B*a^6 - 208*A*a^5*b + 105*(14*B*a*b^5 + 13*A*b^6)*x^5 + 35*(B*a^2*b^4 + 52*A*a*b
^5)*x^4 - 5*(8*B*a^3*b^3 - 13*A*a^2*b^4)*x^3 + 6*(8*B*a^4*b^2 - 13*A*a^3*b^3)*x^2 - 8*(8*B*a^5*b - 13*A*a^4*b^
2)*x)*sqrt(b*x + a)/b^5

________________________________________________________________________________________

giac [B]  time = 1.21, size = 422, normalized size = 3.46 \[ \frac {2 \, {\left (\frac {1287 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} A a^{2}}{b^{3}} + \frac {143 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} B a^{2}}{b^{4}} + \frac {286 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} A a}{b^{3}} + \frac {130 \, {\left (63 \, {\left (b x + a\right )}^{\frac {11}{2}} - 385 \, {\left (b x + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} - 693 \, \sqrt {b x + a} a^{5}\right )} B a}{b^{4}} + \frac {65 \, {\left (63 \, {\left (b x + a\right )}^{\frac {11}{2}} - 385 \, {\left (b x + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} - 693 \, \sqrt {b x + a} a^{5}\right )} A}{b^{3}} + \frac {15 \, {\left (231 \, {\left (b x + a\right )}^{\frac {13}{2}} - 1638 \, {\left (b x + a\right )}^{\frac {11}{2}} a + 5005 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{2} - 8580 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{3} + 9009 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{4} - 6006 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{5} + 3003 \, \sqrt {b x + a} a^{6}\right )} B}{b^{4}}\right )}}{45045 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^(3/2)*(B*x+A),x, algorithm="giac")

[Out]

2/45045*(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)*A*a^2
/b^3 + 143*(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 + 3
15*sqrt(b*x + a)*a^4)*B*a^2/b^4 + 286*(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^3 + 130*(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*a/b^4
+ 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*sqrt(b*x + a)*a^5)*A/b^3 + 15*(231*(b*x + a)^(13/2) - 1638*(b*x + a)^(11/2)*a + 500
5*(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^4)/b

________________________________________________________________________________________

maple [A]  time = 0.01, size = 95, normalized size = 0.78 \[ -\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (-1155 B \,x^{4} b^{4}-1365 A \,b^{4} x^{3}+840 B a \,b^{3} x^{3}+910 A a \,b^{3} x^{2}-560 B \,a^{2} b^{2} x^{2}-520 A \,a^{2} b^{2} x +320 B \,a^{3} b x +208 A \,a^{3} b -128 B \,a^{4}\right )}{15015 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x+a)^(3/2)*(B*x+A),x)

[Out]

-2/15015*(b*x+a)^(5/2)*(-1155*B*b^4*x^4-1365*A*b^4*x^3+840*B*a*b^3*x^3+910*A*a*b^3*x^2-560*B*a^2*b^2*x^2-520*A
*a^2*b^2*x+320*B*a^3*b*x+208*A*a^3*b-128*B*a^4)/b^5

________________________________________________________________________________________

maxima [A]  time = 0.88, size = 100, normalized size = 0.82 \[ \frac {2 \, {\left (1155 \, {\left (b x + a\right )}^{\frac {13}{2}} B - 1365 \, {\left (4 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {11}{2}} + 5005 \, {\left (2 \, B a^{2} - A a b\right )} {\left (b x + a\right )}^{\frac {9}{2}} - 2145 \, {\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {7}{2}} + 3003 \, {\left (B a^{4} - A a^{3} b\right )} {\left (b x + a\right )}^{\frac {5}{2}}\right )}}{15015 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^(3/2)*(B*x+A),x, algorithm="maxima")

[Out]

2/15015*(1155*(b*x + a)^(13/2)*B - 1365*(4*B*a - A*b)*(b*x + a)^(11/2) + 5005*(2*B*a^2 - A*a*b)*(b*x + a)^(9/2
) - 2145*(4*B*a^3 - 3*A*a^2*b)*(b*x + a)^(7/2) + 3003*(B*a^4 - A*a^3*b)*(b*x + a)^(5/2))/b^5

________________________________________________________________________________________

mupad [B]  time = 0.36, size = 111, normalized size = 0.91 \[ \frac {\left (12\,B\,a^2-6\,A\,a\,b\right )\,{\left (a+b\,x\right )}^{9/2}}{9\,b^5}+\frac {2\,B\,{\left (a+b\,x\right )}^{13/2}}{13\,b^5}+\frac {\left (2\,A\,b-8\,B\,a\right )\,{\left (a+b\,x\right )}^{11/2}}{11\,b^5}+\frac {\left (2\,B\,a^4-2\,A\,a^3\,b\right )\,{\left (a+b\,x\right )}^{5/2}}{5\,b^5}-\frac {\left (8\,B\,a^3-6\,A\,a^2\,b\right )\,{\left (a+b\,x\right )}^{7/2}}{7\,b^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(A + B*x)*(a + b*x)^(3/2),x)

[Out]

((12*B*a^2 - 6*A*a*b)*(a + b*x)^(9/2))/(9*b^5) + (2*B*(a + b*x)^(13/2))/(13*b^5) + ((2*A*b - 8*B*a)*(a + b*x)^
(11/2))/(11*b^5) + ((2*B*a^4 - 2*A*a^3*b)*(a + b*x)^(5/2))/(5*b^5) - ((8*B*a^3 - 6*A*a^2*b)*(a + b*x)^(7/2))/(
7*b^5)

________________________________________________________________________________________

sympy [B]  time = 11.85, size = 298, normalized size = 2.44 \[ \frac {2 A a \left (- \frac {a^{3} \left (a + b x\right )^{\frac {3}{2}}}{3} + \frac {3 a^{2} \left (a + b x\right )^{\frac {5}{2}}}{5} - \frac {3 a \left (a + b x\right )^{\frac {7}{2}}}{7} + \frac {\left (a + b x\right )^{\frac {9}{2}}}{9}\right )}{b^{4}} + \frac {2 A \left (\frac {a^{4} \left (a + b x\right )^{\frac {3}{2}}}{3} - \frac {4 a^{3} \left (a + b x\right )^{\frac {5}{2}}}{5} + \frac {6 a^{2} \left (a + b x\right )^{\frac {7}{2}}}{7} - \frac {4 a \left (a + b x\right )^{\frac {9}{2}}}{9} + \frac {\left (a + b x\right )^{\frac {11}{2}}}{11}\right )}{b^{4}} + \frac {2 B a \left (\frac {a^{4} \left (a + b x\right )^{\frac {3}{2}}}{3} - \frac {4 a^{3} \left (a + b x\right )^{\frac {5}{2}}}{5} + \frac {6 a^{2} \left (a + b x\right )^{\frac {7}{2}}}{7} - \frac {4 a \left (a + b x\right )^{\frac {9}{2}}}{9} + \frac {\left (a + b x\right )^{\frac {11}{2}}}{11}\right )}{b^{5}} + \frac {2 B \left (- \frac {a^{5} \left (a + b x\right )^{\frac {3}{2}}}{3} + a^{4} \left (a + b x\right )^{\frac {5}{2}} - \frac {10 a^{3} \left (a + b x\right )^{\frac {7}{2}}}{7} + \frac {10 a^{2} \left (a + b x\right )^{\frac {9}{2}}}{9} - \frac {5 a \left (a + b x\right )^{\frac {11}{2}}}{11} + \frac {\left (a + b x\right )^{\frac {13}{2}}}{13}\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x+a)**(3/2)*(B*x+A),x)

[Out]

2*A*a*(-a**3*(a + b*x)**(3/2)/3 + 3*a**2*(a + b*x)**(5/2)/5 - 3*a*(a + b*x)**(7/2)/7 + (a + b*x)**(9/2)/9)/b**
4 + 2*A*(a**4*(a + b*x)**(3/2)/3 - 4*a**3*(a + b*x)**(5/2)/5 + 6*a**2*(a + b*x)**(7/2)/7 - 4*a*(a + b*x)**(9/2
)/9 + (a + b*x)**(11/2)/11)/b**4 + 2*B*a*(a**4*(a + b*x)**(3/2)/3 - 4*a**3*(a + b*x)**(5/2)/5 + 6*a**2*(a + b*
x)**(7/2)/7 - 4*a*(a + b*x)**(9/2)/9 + (a + b*x)**(11/2)/11)/b**5 + 2*B*(-a**5*(a + b*x)**(3/2)/3 + a**4*(a +
b*x)**(5/2) - 10*a**3*(a + b*x)**(7/2)/7 + 10*a**2*(a + b*x)**(9/2)/9 - 5*a*(a + b*x)**(11/2)/11 + (a + b*x)**
(13/2)/13)/b**5

________________________________________________________________________________________