3.411 \(\int x^3 (a+b x)^{5/2} (A+B x) \, dx\)

Optimal. Leaf size=122 \[ -\frac {2 a^3 (a+b x)^{7/2} (A b-a B)}{7 b^5}+\frac {2 a^2 (a+b x)^{9/2} (3 A b-4 a B)}{9 b^5}+\frac {2 (a+b x)^{13/2} (A b-4 a B)}{13 b^5}-\frac {6 a (a+b x)^{11/2} (A b-2 a B)}{11 b^5}+\frac {2 B (a+b x)^{15/2}}{15 b^5} \]

[Out]

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

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

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x)^(5/2)*(A + B*x),x]

[Out]

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

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Mathematica [A]  time = 0.09, size = 87, normalized size = 0.71 \[ \frac {2 (a+b x)^{7/2} \left (128 a^4 B-16 a^3 b (15 A+28 B x)+168 a^2 b^2 x (5 A+6 B x)-42 a b^3 x^2 (45 A+44 B x)+231 b^4 x^3 (15 A+13 B x)\right )}{45045 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x)^(5/2)*(A + B*x),x]

[Out]

(2*(a + b*x)^(7/2)*(128*a^4*B + 168*a^2*b^2*x*(5*A + 6*B*x) + 231*b^4*x^3*(15*A + 13*B*x) - 16*a^3*b*(15*A + 2
8*B*x) - 42*a*b^3*x^2*(45*A + 44*B*x)))/(45045*b^5)

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fricas [A]  time = 0.68, size = 167, normalized size = 1.37 \[ \frac {2 \, {\left (3003 \, B b^{7} x^{7} + 128 \, B a^{7} - 240 \, A a^{6} b + 231 \, {\left (31 \, B a b^{6} + 15 \, A b^{7}\right )} x^{6} + 63 \, {\left (71 \, B a^{2} b^{5} + 135 \, A a b^{6}\right )} x^{5} + 35 \, {\left (B a^{3} b^{4} + 159 \, A a^{2} b^{5}\right )} x^{4} - 5 \, {\left (8 \, B a^{4} b^{3} - 15 \, A a^{3} b^{4}\right )} x^{3} + 6 \, {\left (8 \, B a^{5} b^{2} - 15 \, A a^{4} b^{3}\right )} x^{2} - 8 \, {\left (8 \, B a^{6} b - 15 \, A a^{5} b^{2}\right )} x\right )} \sqrt {b x + a}}{45045 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^(5/2)*(B*x+A),x, algorithm="fricas")

[Out]

2/45045*(3003*B*b^7*x^7 + 128*B*a^7 - 240*A*a^6*b + 231*(31*B*a*b^6 + 15*A*b^7)*x^6 + 63*(71*B*a^2*b^5 + 135*A
*a*b^6)*x^5 + 35*(B*a^3*b^4 + 159*A*a^2*b^5)*x^4 - 5*(8*B*a^4*b^3 - 15*A*a^3*b^4)*x^3 + 6*(8*B*a^5*b^2 - 15*A*
a^4*b^3)*x^2 - 8*(8*B*a^6*b - 15*A*a^5*b^2)*x)*sqrt(b*x + a)/b^5

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giac [B]  time = 1.30, size = 612, normalized size = 5.02 \[ \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^{3}}{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^{3}}{b^{4}} + \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^{2}}{b^{3}} + \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^{2}}{b^{4}} + \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 )} A a}{b^{3}} + \frac {45 \, {\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^{4}} + \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^{3}} + \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^{4}}\right )}}{45045 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^(5/2)*(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^3
/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^3/b^4 + 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^2/b^3 + 195*(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^4 + 195*(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^3 + 45*(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^4 + 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^3 + 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^4)/b

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maple [A]  time = 0.01, size = 95, normalized size = 0.78 \[ -\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (-3003 B \,x^{4} b^{4}-3465 A \,b^{4} x^{3}+1848 B a \,b^{3} x^{3}+1890 A a \,b^{3} x^{2}-1008 B \,a^{2} b^{2} x^{2}-840 A \,a^{2} b^{2} x +448 B \,a^{3} b x +240 A \,a^{3} b -128 B \,a^{4}\right )}{45045 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x+a)^(5/2)*(B*x+A),x)

[Out]

-2/45045*(b*x+a)^(7/2)*(-3003*B*b^4*x^4-3465*A*b^4*x^3+1848*B*a*b^3*x^3+1890*A*a*b^3*x^2-1008*B*a^2*b^2*x^2-84
0*A*a^2*b^2*x+448*B*a^3*b*x+240*A*a^3*b-128*B*a^4)/b^5

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maxima [A]  time = 0.90, size = 100, normalized size = 0.82 \[ \frac {2 \, {\left (3003 \, {\left (b x + a\right )}^{\frac {15}{2}} B - 3465 \, {\left (4 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {13}{2}} + 12285 \, {\left (2 \, B a^{2} - A a b\right )} {\left (b x + a\right )}^{\frac {11}{2}} - 5005 \, {\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {9}{2}} + 6435 \, {\left (B a^{4} - A a^{3} b\right )} {\left (b x + a\right )}^{\frac {7}{2}}\right )}}{45045 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^(5/2)*(B*x+A),x, algorithm="maxima")

[Out]

2/45045*(3003*(b*x + a)^(15/2)*B - 3465*(4*B*a - A*b)*(b*x + a)^(13/2) + 12285*(2*B*a^2 - A*a*b)*(b*x + a)^(11
/2) - 5005*(4*B*a^3 - 3*A*a^2*b)*(b*x + a)^(9/2) + 6435*(B*a^4 - A*a^3*b)*(b*x + a)^(7/2))/b^5

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mupad [B]  time = 0.05, size = 111, normalized size = 0.91 \[ \frac {\left (12\,B\,a^2-6\,A\,a\,b\right )\,{\left (a+b\,x\right )}^{11/2}}{11\,b^5}+\frac {2\,B\,{\left (a+b\,x\right )}^{15/2}}{15\,b^5}+\frac {\left (2\,A\,b-8\,B\,a\right )\,{\left (a+b\,x\right )}^{13/2}}{13\,b^5}+\frac {\left (2\,B\,a^4-2\,A\,a^3\,b\right )\,{\left (a+b\,x\right )}^{7/2}}{7\,b^5}-\frac {\left (8\,B\,a^3-6\,A\,a^2\,b\right )\,{\left (a+b\,x\right )}^{9/2}}{9\,b^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(A + B*x)*(a + b*x)^(5/2),x)

[Out]

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

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sympy [B]  time = 18.33, size = 496, normalized size = 4.07 \[ \frac {2 A a^{2} \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 {4 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^{4}} + \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^{4}} + \frac {2 B a^{2} \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 {4 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^{5}} + \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^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x+a)**(5/2)*(B*x+A),x)

[Out]

2*A*a**2*(-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 + 4*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**4 + 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**4 + 2*B*a**2*(
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 + 4*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**5 + 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**5

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