3.4.73 \(\int x^4 (a+b x)^{3/2} (A+B x) \, dx\)

Optimal. Leaf size=151 \[ \frac {2 a^4 (a+b x)^{5/2} (A b-a B)}{5 b^6}-\frac {2 a^3 (a+b x)^{7/2} (4 A b-5 a B)}{7 b^6}+\frac {4 a^2 (a+b x)^{9/2} (3 A b-5 a B)}{9 b^6}+\frac {2 (a+b x)^{13/2} (A b-5 a B)}{13 b^6}-\frac {4 a (a+b x)^{11/2} (2 A b-5 a B)}{11 b^6}+\frac {2 B (a+b x)^{15/2}}{15 b^6} \]

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 151, 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} \begin {gather*} \frac {4 a^2 (a+b x)^{9/2} (3 A b-5 a B)}{9 b^6}-\frac {2 a^3 (a+b x)^{7/2} (4 A b-5 a B)}{7 b^6}+\frac {2 a^4 (a+b x)^{5/2} (A b-a B)}{5 b^6}+\frac {2 (a+b x)^{13/2} (A b-5 a B)}{13 b^6}-\frac {4 a (a+b x)^{11/2} (2 A b-5 a B)}{11 b^6}+\frac {2 B (a+b x)^{15/2}}{15 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*x)^(3/2)*(A + B*x),x]

[Out]

(2*a^4*(A*b - a*B)*(a + b*x)^(5/2))/(5*b^6) - (2*a^3*(4*A*b - 5*a*B)*(a + b*x)^(7/2))/(7*b^6) + (4*a^2*(3*A*b
- 5*a*B)*(a + b*x)^(9/2))/(9*b^6) - (4*a*(2*A*b - 5*a*B)*(a + b*x)^(11/2))/(11*b^6) + (2*(A*b - 5*a*B)*(a + b*
x)^(13/2))/(13*b^6) + (2*B*(a + b*x)^(15/2))/(15*b^6)

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^4 (a+b x)^{3/2} (A+B x) \, dx &=\int \left (-\frac {a^4 (-A b+a B) (a+b x)^{3/2}}{b^5}+\frac {a^3 (-4 A b+5 a B) (a+b x)^{5/2}}{b^5}-\frac {2 a^2 (-3 A b+5 a B) (a+b x)^{7/2}}{b^5}+\frac {2 a (-2 A b+5 a B) (a+b x)^{9/2}}{b^5}+\frac {(A b-5 a B) (a+b x)^{11/2}}{b^5}+\frac {B (a+b x)^{13/2}}{b^5}\right ) \, dx\\ &=\frac {2 a^4 (A b-a B) (a+b x)^{5/2}}{5 b^6}-\frac {2 a^3 (4 A b-5 a B) (a+b x)^{7/2}}{7 b^6}+\frac {4 a^2 (3 A b-5 a B) (a+b x)^{9/2}}{9 b^6}-\frac {4 a (2 A b-5 a B) (a+b x)^{11/2}}{11 b^6}+\frac {2 (A b-5 a B) (a+b x)^{13/2}}{13 b^6}+\frac {2 B (a+b x)^{15/2}}{15 b^6}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.11, size = 103, normalized size = 0.68 \begin {gather*} \frac {2 (a+b x)^{5/2} \left (-256 a^5 B+128 a^4 b (3 A+5 B x)-160 a^3 b^2 x (6 A+7 B x)+1680 a^2 b^3 x^2 (A+B x)-210 a b^4 x^3 (12 A+11 B x)+231 b^5 x^4 (15 A+13 B x)\right )}{45045 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*x)^(3/2)*(A + B*x),x]

[Out]

(2*(a + b*x)^(5/2)*(-256*a^5*B + 1680*a^2*b^3*x^2*(A + B*x) + 128*a^4*b*(3*A + 5*B*x) - 160*a^3*b^2*x*(6*A + 7
*B*x) - 210*a*b^4*x^3*(12*A + 11*B*x) + 231*b^5*x^4*(15*A + 13*B*x)))/(45045*b^6)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.06, size = 137, normalized size = 0.91 \begin {gather*} \frac {2 (a+b x)^{5/2} \left (-9009 a^5 B+9009 a^4 A b+32175 a^4 B (a+b x)-25740 a^3 A b (a+b x)-50050 a^3 B (a+b x)^2+30030 a^2 A b (a+b x)^2+40950 a^2 B (a+b x)^3-16380 a A b (a+b x)^3+3465 A b (a+b x)^4-17325 a B (a+b x)^4+3003 B (a+b x)^5\right )}{45045 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^4*(a + b*x)^(3/2)*(A + B*x),x]

[Out]

(2*(a + b*x)^(5/2)*(9009*a^4*A*b - 9009*a^5*B - 25740*a^3*A*b*(a + b*x) + 32175*a^4*B*(a + b*x) + 30030*a^2*A*
b*(a + b*x)^2 - 50050*a^3*B*(a + b*x)^2 - 16380*a*A*b*(a + b*x)^3 + 40950*a^2*B*(a + b*x)^3 + 3465*A*b*(a + b*
x)^4 - 17325*a*B*(a + b*x)^4 + 3003*B*(a + b*x)^5))/(45045*b^6)

________________________________________________________________________________________

fricas [A]  time = 1.22, size = 167, normalized size = 1.11 \begin {gather*} \frac {2 \, {\left (3003 \, B b^{7} x^{7} - 256 \, B a^{7} + 384 \, A a^{6} b + 231 \, {\left (16 \, B a b^{6} + 15 \, A b^{7}\right )} x^{6} + 63 \, {\left (B a^{2} b^{5} + 70 \, A a b^{6}\right )} x^{5} - 35 \, {\left (2 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5}\right )} x^{4} + 40 \, {\left (2 \, B a^{4} b^{3} - 3 \, A a^{3} b^{4}\right )} x^{3} - 48 \, {\left (2 \, B a^{5} b^{2} - 3 \, A a^{4} b^{3}\right )} x^{2} + 64 \, {\left (2 \, B a^{6} b - 3 \, A a^{5} b^{2}\right )} x\right )} \sqrt {b x + a}}{45045 \, b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x+a)^(3/2)*(B*x+A),x, algorithm="fricas")

[Out]

2/45045*(3003*B*b^7*x^7 - 256*B*a^7 + 384*A*a^6*b + 231*(16*B*a*b^6 + 15*A*b^7)*x^6 + 63*(B*a^2*b^5 + 70*A*a*b
^6)*x^5 - 35*(2*B*a^3*b^4 - 3*A*a^2*b^5)*x^4 + 40*(2*B*a^4*b^3 - 3*A*a^3*b^4)*x^3 - 48*(2*B*a^5*b^2 - 3*A*a^4*
b^3)*x^2 + 64*(2*B*a^6*b - 3*A*a^5*b^2)*x)*sqrt(b*x + a)/b^6

________________________________________________________________________________________

giac [B]  time = 1.30, size = 494, normalized size = 3.27 \begin {gather*} \frac {2 \, {\left (\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 )} A a^{2}}{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 )} B a^{2}}{b^{5}} + \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 )} A a}{b^{4}} + \frac {30 \, {\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 a}{b^{5}} + \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 )} A}{b^{4}} + \frac {7 \, {\left (429 \, {\left (b x + a\right )}^{\frac {15}{2}} - 3465 \, {\left (b x + a\right )}^{\frac {13}{2}} a + 12285 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{2} - 25025 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{3} + 32175 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{4} - 27027 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{5} + 15015 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{6} - 6435 \, \sqrt {b x + a} a^{7}\right )} B}{b^{5}}\right )}}{45045 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x+a)^(3/2)*(B*x+A),x, algorithm="giac")

[Out]

2/45045*(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 +
315*sqrt(b*x + a)*a^4)*A*a^2/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)*B*a^2/b^5 + 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)*A*a/b^4 + 30*(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*a/b^5 + 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)*A/b^4 + 7*(429*(b*x + a)^
(15/2) - 3465*(b*x + a)^(13/2)*a + 12285*(b*x + a)^(11/2)*a^2 - 25025*(b*x + a)^(9/2)*a^3 + 32175*(b*x + a)^(7
/2)*a^4 - 27027*(b*x + a)^(5/2)*a^5 + 15015*(b*x + a)^(3/2)*a^6 - 6435*sqrt(b*x + a)*a^7)*B/b^5)/b

________________________________________________________________________________________

maple [A]  time = 0.01, size = 119, normalized size = 0.79 \begin {gather*} \frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (3003 B \,b^{5} x^{5}+3465 A \,b^{5} x^{4}-2310 B a \,b^{4} x^{4}-2520 A a \,b^{4} x^{3}+1680 B \,a^{2} b^{3} x^{3}+1680 A \,a^{2} b^{3} x^{2}-1120 B \,a^{3} b^{2} x^{2}-960 A \,a^{3} b^{2} x +640 B \,a^{4} b x +384 A \,a^{4} b -256 B \,a^{5}\right )}{45045 b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(b*x+a)^(3/2)*(B*x+A),x)

[Out]

2/45045*(b*x+a)^(5/2)*(3003*B*b^5*x^5+3465*A*b^5*x^4-2310*B*a*b^4*x^4-2520*A*a*b^4*x^3+1680*B*a^2*b^3*x^3+1680
*A*a^2*b^3*x^2-1120*B*a^3*b^2*x^2-960*A*a^3*b^2*x+640*B*a^4*b*x+384*A*a^4*b-256*B*a^5)/b^6

________________________________________________________________________________________

maxima [A]  time = 0.84, size = 123, normalized size = 0.81 \begin {gather*} \frac {2 \, {\left (3003 \, {\left (b x + a\right )}^{\frac {15}{2}} B - 3465 \, {\left (5 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {13}{2}} + 8190 \, {\left (5 \, B a^{2} - 2 \, A a b\right )} {\left (b x + a\right )}^{\frac {11}{2}} - 10010 \, {\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {9}{2}} + 6435 \, {\left (5 \, B a^{4} - 4 \, A a^{3} b\right )} {\left (b x + a\right )}^{\frac {7}{2}} - 9009 \, {\left (B a^{5} - A a^{4} b\right )} {\left (b x + a\right )}^{\frac {5}{2}}\right )}}{45045 \, b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x+a)^(3/2)*(B*x+A),x, algorithm="maxima")

[Out]

2/45045*(3003*(b*x + a)^(15/2)*B - 3465*(5*B*a - A*b)*(b*x + a)^(13/2) + 8190*(5*B*a^2 - 2*A*a*b)*(b*x + a)^(1
1/2) - 10010*(5*B*a^3 - 3*A*a^2*b)*(b*x + a)^(9/2) + 6435*(5*B*a^4 - 4*A*a^3*b)*(b*x + a)^(7/2) - 9009*(B*a^5
- A*a^4*b)*(b*x + a)^(5/2))/b^6

________________________________________________________________________________________

mupad [B]  time = 0.06, size = 137, normalized size = 0.91 \begin {gather*} \frac {\left (20\,B\,a^2-8\,A\,a\,b\right )\,{\left (a+b\,x\right )}^{11/2}}{11\,b^6}+\frac {2\,B\,{\left (a+b\,x\right )}^{15/2}}{15\,b^6}+\frac {\left (2\,A\,b-10\,B\,a\right )\,{\left (a+b\,x\right )}^{13/2}}{13\,b^6}-\frac {\left (2\,B\,a^5-2\,A\,a^4\,b\right )\,{\left (a+b\,x\right )}^{5/2}}{5\,b^6}+\frac {\left (10\,B\,a^4-8\,A\,a^3\,b\right )\,{\left (a+b\,x\right )}^{7/2}}{7\,b^6}-\frac {\left (20\,B\,a^3-12\,A\,a^2\,b\right )\,{\left (a+b\,x\right )}^{9/2}}{9\,b^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(A + B*x)*(a + b*x)^(3/2),x)

[Out]

((20*B*a^2 - 8*A*a*b)*(a + b*x)^(11/2))/(11*b^6) + (2*B*(a + b*x)^(15/2))/(15*b^6) + ((2*A*b - 10*B*a)*(a + b*
x)^(13/2))/(13*b^6) - ((2*B*a^5 - 2*A*a^4*b)*(a + b*x)^(5/2))/(5*b^6) + ((10*B*a^4 - 8*A*a^3*b)*(a + b*x)^(7/2
))/(7*b^6) - ((20*B*a^3 - 12*A*a^2*b)*(a + b*x)^(9/2))/(9*b^6)

________________________________________________________________________________________

sympy [B]  time = 13.49, size = 355, normalized size = 2.35 \begin {gather*} \frac {2 A 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 A \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}} + \frac {2 B a \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^{6}} + \frac {2 B \left (\frac {a^{6} \left (a + b x\right )^{\frac {3}{2}}}{3} - \frac {6 a^{5} \left (a + b x\right )^{\frac {5}{2}}}{5} + \frac {15 a^{4} \left (a + b x\right )^{\frac {7}{2}}}{7} - \frac {20 a^{3} \left (a + b x\right )^{\frac {9}{2}}}{9} + \frac {15 a^{2} \left (a + b x\right )^{\frac {11}{2}}}{11} - \frac {6 a \left (a + b x\right )^{\frac {13}{2}}}{13} + \frac {\left (a + b x\right )^{\frac {15}{2}}}{15}\right )}{b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(b*x+a)**(3/2)*(B*x+A),x)

[Out]

2*A*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*A*(-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 + 2*B*a*(-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**6 + 2*B*(a**6*(a + b*x)**(3/2)/3 - 6*a**5*(a + b*x)**(5/2)/5 + 15*a**
4*(a + b*x)**(7/2)/7 - 20*a**3*(a + b*x)**(9/2)/9 + 15*a**2*(a + b*x)**(11/2)/11 - 6*a*(a + b*x)**(13/2)/13 +
(a + b*x)**(15/2)/15)/b**6

________________________________________________________________________________________