\(\int x^4 (a+b x)^{5/2} (A+B x) \, dx\) [410]

Optimal result

Integrand size = 18, antiderivative size = 151 \[ \int x^4 (a+b x)^{5/2} (A+B x) \, dx=\frac {2 a^4 (A b-a B) (a+b x)^{7/2}}{7 b^6}-\frac {2 a^3 (4 A b-5 a B) (a+b x)^{9/2}}{9 b^6}+\frac {4 a^2 (3 A b-5 a B) (a+b x)^{11/2}}{11 b^6}-\frac {4 a (2 A b-5 a B) (a+b x)^{13/2}}{13 b^6}+\frac {2 (A b-5 a B) (a+b x)^{15/2}}{15 b^6}+\frac {2 B (a+b x)^{17/2}}{17 b^6} \]

[Out]

2/7*a^4*(A*b-B*a)*(b*x+a)^(7/2)/b^6-2/9*a^3*(4*A*b-5*B*a)*(b*x+a)^(9/2)/b^6+4/11*a^2*(3*A*b-5*B*a)*(b*x+a)^(11
/2)/b^6-4/13*a*(2*A*b-5*B*a)*(b*x+a)^(13/2)/b^6+2/15*(A*b-5*B*a)*(b*x+a)^(15/2)/b^6+2/17*B*(b*x+a)^(17/2)/b^6

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {78} \[ \int x^4 (a+b x)^{5/2} (A+B x) \, dx=\frac {2 a^4 (a+b x)^{7/2} (A b-a B)}{7 b^6}-\frac {2 a^3 (a+b x)^{9/2} (4 A b-5 a B)}{9 b^6}+\frac {4 a^2 (a+b x)^{11/2} (3 A b-5 a B)}{11 b^6}+\frac {2 (a+b x)^{15/2} (A b-5 a B)}{15 b^6}-\frac {4 a (a+b x)^{13/2} (2 A b-5 a B)}{13 b^6}+\frac {2 B (a+b x)^{17/2}}{17 b^6} \]

[In]

Int[x^4*(a + b*x)^(5/2)*(A + B*x),x]

[Out]

(2*a^4*(A*b - a*B)*(a + b*x)^(7/2))/(7*b^6) - (2*a^3*(4*A*b - 5*a*B)*(a + b*x)^(9/2))/(9*b^6) + (4*a^2*(3*A*b
- 5*a*B)*(a + b*x)^(11/2))/(11*b^6) - (4*a*(2*A*b - 5*a*B)*(a + b*x)^(13/2))/(13*b^6) + (2*(A*b - 5*a*B)*(a +
b*x)^(15/2))/(15*b^6) + (2*B*(a + b*x)^(17/2))/(17*b^6)

Rule 78

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*} \text {integral}& = \int \left (-\frac {a^4 (-A b+a B) (a+b x)^{5/2}}{b^5}+\frac {a^3 (-4 A b+5 a B) (a+b x)^{7/2}}{b^5}-\frac {2 a^2 (-3 A b+5 a B) (a+b x)^{9/2}}{b^5}+\frac {2 a (-2 A b+5 a B) (a+b x)^{11/2}}{b^5}+\frac {(A b-5 a B) (a+b x)^{13/2}}{b^5}+\frac {B (a+b x)^{15/2}}{b^5}\right ) \, dx \\ & = \frac {2 a^4 (A b-a B) (a+b x)^{7/2}}{7 b^6}-\frac {2 a^3 (4 A b-5 a B) (a+b x)^{9/2}}{9 b^6}+\frac {4 a^2 (3 A b-5 a B) (a+b x)^{11/2}}{11 b^6}-\frac {4 a (2 A b-5 a B) (a+b x)^{13/2}}{13 b^6}+\frac {2 (A b-5 a B) (a+b x)^{15/2}}{15 b^6}+\frac {2 B (a+b x)^{17/2}}{17 b^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.70 \[ \int x^4 (a+b x)^{5/2} (A+B x) \, dx=\frac {2 (a+b x)^{7/2} \left (-1280 a^5 B+3003 b^5 x^4 (17 A+15 B x)+128 a^4 b (17 A+35 B x)-224 a^3 b^2 x (34 A+45 B x)+336 a^2 b^3 x^2 (51 A+55 B x)-462 a b^4 x^3 (68 A+65 B x)\right )}{765765 b^6} \]

[In]

Integrate[x^4*(a + b*x)^(5/2)*(A + B*x),x]

[Out]

(2*(a + b*x)^(7/2)*(-1280*a^5*B + 3003*b^5*x^4*(17*A + 15*B*x) + 128*a^4*b*(17*A + 35*B*x) - 224*a^3*b^2*x*(34
*A + 45*B*x) + 336*a^2*b^3*x^2*(51*A + 55*B*x) - 462*a*b^4*x^3*(68*A + 65*B*x)))/(765765*b^6)

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.61

method	result	size
pseudoelliptic	\(\frac {256 \left (b x +a \right )^{\frac {7}{2}} \left (\frac {3003 x^{4} \left (\frac {15 B x}{17}+A \right ) b^{5}}{128}-\frac {231 x^{3} \left (\frac {65 B x}{68}+A \right ) a \,b^{4}}{16}+\frac {63 x^{2} \left (\frac {55 B x}{51}+A \right ) a^{2} b^{3}}{8}-\frac {7 x \left (\frac {45 B x}{34}+A \right ) a^{3} b^{2}}{2}+a^{4} \left (\frac {35 B x}{17}+A \right ) b -\frac {10 a^{5} B}{17}\right )}{45045 b^{6}}\)	\(92\)
gosper	\(\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (45045 b^{5} B \,x^{5}+51051 A \,b^{5} x^{4}-30030 B a \,b^{4} x^{4}-31416 A a \,b^{4} x^{3}+18480 B \,a^{2} b^{3} x^{3}+17136 A \,a^{2} b^{3} x^{2}-10080 B \,a^{3} b^{2} x^{2}-7616 a^{3} b^{2} A x +4480 a^{4} b B x +2176 a^{4} b A -1280 a^{5} B \right )}{765765 b^{6}}\)	\(119\)
derivativedivides	\(\frac {\frac {2 B \left (b x +a \right )^{\frac {17}{2}}}{17}+\frac {2 \left (A b -5 B a \right ) \left (b x +a \right )^{\frac {15}{2}}}{15}+\frac {2 \left (6 a^{2} B -4 a \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {13}{2}}}{13}+\frac {2 \left (-4 a^{3} B +6 a^{2} \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {11}{2}}}{11}+\frac {2 \left (B \,a^{4}-4 a^{3} \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {2 a^{4} \left (A b -B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{7}}{b^{6}}\)	\(138\)
default	\(\frac {\frac {2 B \left (b x +a \right )^{\frac {17}{2}}}{17}+\frac {2 \left (A b -5 B a \right ) \left (b x +a \right )^{\frac {15}{2}}}{15}+\frac {2 \left (6 a^{2} B -4 a \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {13}{2}}}{13}+\frac {2 \left (-4 a^{3} B +6 a^{2} \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {11}{2}}}{11}+\frac {2 \left (B \,a^{4}-4 a^{3} \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {2 a^{4} \left (A b -B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{7}}{b^{6}}\)	\(138\)
trager	\(\frac {2 \left (45045 b^{8} B \,x^{8}+51051 A \,b^{8} x^{7}+105105 B a \,b^{7} x^{7}+121737 A a \,b^{7} x^{6}+63525 B \,a^{2} b^{6} x^{6}+76041 A \,a^{2} b^{6} x^{5}+315 B \,a^{3} b^{5} x^{5}+595 A \,a^{3} b^{5} x^{4}-350 B \,a^{4} b^{4} x^{4}-680 A \,a^{4} b^{4} x^{3}+400 B \,a^{5} b^{3} x^{3}+816 A \,a^{5} b^{3} x^{2}-480 B \,a^{6} b^{2} x^{2}-1088 A \,a^{6} b^{2} x +640 B \,a^{7} b x +2176 A \,a^{7} b -1280 B \,a^{8}\right ) \sqrt {b x +a}}{765765 b^{6}}\)	\(191\)
risch	\(\frac {2 \left (45045 b^{8} B \,x^{8}+51051 A \,b^{8} x^{7}+105105 B a \,b^{7} x^{7}+121737 A a \,b^{7} x^{6}+63525 B \,a^{2} b^{6} x^{6}+76041 A \,a^{2} b^{6} x^{5}+315 B \,a^{3} b^{5} x^{5}+595 A \,a^{3} b^{5} x^{4}-350 B \,a^{4} b^{4} x^{4}-680 A \,a^{4} b^{4} x^{3}+400 B \,a^{5} b^{3} x^{3}+816 A \,a^{5} b^{3} x^{2}-480 B \,a^{6} b^{2} x^{2}-1088 A \,a^{6} b^{2} x +640 B \,a^{7} b x +2176 A \,a^{7} b -1280 B \,a^{8}\right ) \sqrt {b x +a}}{765765 b^{6}}\)	\(191\)

[In]

int(x^4*(b*x+a)^(5/2)*(B*x+A),x,method=_RETURNVERBOSE)

[Out]

256/45045*(b*x+a)^(7/2)*(3003/128*x^4*(15/17*B*x+A)*b^5-231/16*x^3*(65/68*B*x+A)*a*b^4+63/8*x^2*(55/51*B*x+A)*
a^2*b^3-7/2*x*(45/34*B*x+A)*a^3*b^2+a^4*(35/17*B*x+A)*b-10/17*a^5*B)/b^6

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.27 \[ \int x^4 (a+b x)^{5/2} (A+B x) \, dx=\frac {2 \, {\left (45045 \, B b^{8} x^{8} - 1280 \, B a^{8} + 2176 \, A a^{7} b + 3003 \, {\left (35 \, B a b^{7} + 17 \, A b^{8}\right )} x^{7} + 231 \, {\left (275 \, B a^{2} b^{6} + 527 \, A a b^{7}\right )} x^{6} + 63 \, {\left (5 \, B a^{3} b^{5} + 1207 \, A a^{2} b^{6}\right )} x^{5} - 35 \, {\left (10 \, B a^{4} b^{4} - 17 \, A a^{3} b^{5}\right )} x^{4} + 40 \, {\left (10 \, B a^{5} b^{3} - 17 \, A a^{4} b^{4}\right )} x^{3} - 48 \, {\left (10 \, B a^{6} b^{2} - 17 \, A a^{5} b^{3}\right )} x^{2} + 64 \, {\left (10 \, B a^{7} b - 17 \, A a^{6} b^{2}\right )} x\right )} \sqrt {b x + a}}{765765 \, b^{6}} \]

[In]

integrate(x^4*(b*x+a)^(5/2)*(B*x+A),x, algorithm="fricas")

[Out]

2/765765*(45045*B*b^8*x^8 - 1280*B*a^8 + 2176*A*a^7*b + 3003*(35*B*a*b^7 + 17*A*b^8)*x^7 + 231*(275*B*a^2*b^6
+ 527*A*a*b^7)*x^6 + 63*(5*B*a^3*b^5 + 1207*A*a^2*b^6)*x^5 - 35*(10*B*a^4*b^4 - 17*A*a^3*b^5)*x^4 + 40*(10*B*a
^5*b^3 - 17*A*a^4*b^4)*x^3 - 48*(10*B*a^6*b^2 - 17*A*a^5*b^3)*x^2 + 64*(10*B*a^7*b - 17*A*a^6*b^2)*x)*sqrt(b*x
+ a)/b^6

Sympy [A] (verification not implemented)

Time = 0.88 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.11 \[ \int x^4 (a+b x)^{5/2} (A+B x) \, dx=\begin {cases} \frac {2 \left (\frac {B \left (a + b x\right )^{\frac {17}{2}}}{17 b} + \frac {\left (a + b x\right )^{\frac {15}{2}} \left (A b - 5 B a\right )}{15 b} + \frac {\left (a + b x\right )^{\frac {13}{2}} \left (- 4 A a b + 10 B a^{2}\right )}{13 b} + \frac {\left (a + b x\right )^{\frac {11}{2}} \cdot \left (6 A a^{2} b - 10 B a^{3}\right )}{11 b} + \frac {\left (a + b x\right )^{\frac {9}{2}} \left (- 4 A a^{3} b + 5 B a^{4}\right )}{9 b} + \frac {\left (a + b x\right )^{\frac {7}{2}} \left (A a^{4} b - B a^{5}\right )}{7 b}\right )}{b^{5}} & \text {for}\: b \neq 0 \\a^{\frac {5}{2}} \left (\frac {A x^{5}}{5} + \frac {B x^{6}}{6}\right ) & \text {otherwise} \end {cases} \]

[In]

integrate(x**4*(b*x+a)**(5/2)*(B*x+A),x)

[Out]

Piecewise((2*(B*(a + b*x)**(17/2)/(17*b) + (a + b*x)**(15/2)*(A*b - 5*B*a)/(15*b) + (a + b*x)**(13/2)*(-4*A*a*
b + 10*B*a**2)/(13*b) + (a + b*x)**(11/2)*(6*A*a**2*b - 10*B*a**3)/(11*b) + (a + b*x)**(9/2)*(-4*A*a**3*b + 5*
B*a**4)/(9*b) + (a + b*x)**(7/2)*(A*a**4*b - B*a**5)/(7*b))/b**5, Ne(b, 0)), (a**(5/2)*(A*x**5/5 + B*x**6/6),
True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.81 \[ \int x^4 (a+b x)^{5/2} (A+B x) \, dx=\frac {2 \, {\left (45045 \, {\left (b x + a\right )}^{\frac {17}{2}} B - 51051 \, {\left (5 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {15}{2}} + 117810 \, {\left (5 \, B a^{2} - 2 \, A a b\right )} {\left (b x + a\right )}^{\frac {13}{2}} - 139230 \, {\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {11}{2}} + 85085 \, {\left (5 \, B a^{4} - 4 \, A a^{3} b\right )} {\left (b x + a\right )}^{\frac {9}{2}} - 109395 \, {\left (B a^{5} - A a^{4} b\right )} {\left (b x + a\right )}^{\frac {7}{2}}\right )}}{765765 \, b^{6}} \]

[In]

integrate(x^4*(b*x+a)^(5/2)*(B*x+A),x, algorithm="maxima")

[Out]

2/765765*(45045*(b*x + a)^(17/2)*B - 51051*(5*B*a - A*b)*(b*x + a)^(15/2) + 117810*(5*B*a^2 - 2*A*a*b)*(b*x +
a)^(13/2) - 139230*(5*B*a^3 - 3*A*a^2*b)*(b*x + a)^(11/2) + 85085*(5*B*a^4 - 4*A*a^3*b)*(b*x + a)^(9/2) - 1093
95*(B*a^5 - A*a^4*b)*(b*x + a)^(7/2))/b^6

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 708 vs. \(2 (128) = 256\).

Time = 0.29 (sec) , antiderivative size = 708, normalized size of antiderivative = 4.69 \[ \int x^4 (a+b x)^{5/2} (A+B x) \, dx=\text {Too large to display} \]

[In]

integrate(x^4*(b*x+a)^(5/2)*(B*x+A),x, algorithm="giac")

[Out]

2/765765*(2431*(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^3/b^4 + 1105*(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^3/b^5 + 3315*(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^2/b^4 + 765*(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^2/b^5 + 765*(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*a/b^4 + 357*
(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 + 321
75*(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*a/b
^5 + 119*(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)*A/b^4 + 7*(6435*(b*x + a)^(17/2) - 58344*(b*x + a)^(15/2)*a + 235620*(b*x + a)^(13/2)*a^2 - 556920*(b*x +
a)^(11/2)*a^3 + 850850*(b*x + a)^(9/2)*a^4 - 875160*(b*x + a)^(7/2)*a^5 + 612612*(b*x + a)^(5/2)*a^6 - 291720*
(b*x + a)^(3/2)*a^7 + 109395*sqrt(b*x + a)*a^8)*B/b^5)/b

Mupad [B] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.91 \[ \int x^4 (a+b x)^{5/2} (A+B x) \, dx=\frac {\left (20\,B\,a^2-8\,A\,a\,b\right )\,{\left (a+b\,x\right )}^{13/2}}{13\,b^6}+\frac {2\,B\,{\left (a+b\,x\right )}^{17/2}}{17\,b^6}+\frac {\left (2\,A\,b-10\,B\,a\right )\,{\left (a+b\,x\right )}^{15/2}}{15\,b^6}-\frac {\left (2\,B\,a^5-2\,A\,a^4\,b\right )\,{\left (a+b\,x\right )}^{7/2}}{7\,b^6}+\frac {\left (10\,B\,a^4-8\,A\,a^3\,b\right )\,{\left (a+b\,x\right )}^{9/2}}{9\,b^6}-\frac {\left (20\,B\,a^3-12\,A\,a^2\,b\right )\,{\left (a+b\,x\right )}^{11/2}}{11\,b^6} \]

[In]

int(x^4*(A + B*x)*(a + b*x)^(5/2),x)

[Out]

((20*B*a^2 - 8*A*a*b)*(a + b*x)^(13/2))/(13*b^6) + (2*B*(a + b*x)^(17/2))/(17*b^6) + ((2*A*b - 10*B*a)*(a + b*
x)^(15/2))/(15*b^6) - ((2*B*a^5 - 2*A*a^4*b)*(a + b*x)^(7/2))/(7*b^6) + ((10*B*a^4 - 8*A*a^3*b)*(a + b*x)^(9/2
))/(9*b^6) - ((20*B*a^3 - 12*A*a^2*b)*(a + b*x)^(11/2))/(11*b^6)