3.4.62 \(\int x^4 \sqrt {a+b x} (A+B x) \, dx\)

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

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Rubi [A]  time = 0.07, 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)^{7/2} (3 A b-5 a B)}{7 b^6}-\frac {2 a^3 (a+b x)^{5/2} (4 A b-5 a B)}{5 b^6}+\frac {2 a^4 (a+b x)^{3/2} (A b-a B)}{3 b^6}+\frac {2 (a+b x)^{11/2} (A b-5 a B)}{11 b^6}-\frac {4 a (a+b x)^{9/2} (2 A b-5 a B)}{9 b^6}+\frac {2 B (a+b x)^{13/2}}{13 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*Sqrt[a + b*x]*(A + B*x),x]

[Out]

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

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Mathematica [A]  time = 0.12, size = 106, normalized size = 0.70 \begin {gather*} \frac {2 (a+b x)^{3/2} \left (-1280 a^5 B+128 a^4 b (13 A+15 B x)-96 a^3 b^2 x (26 A+25 B x)+80 a^2 b^3 x^2 (39 A+35 B x)-70 a b^4 x^3 (52 A+45 B x)+315 b^5 x^4 (13 A+11 B x)\right )}{45045 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4*Sqrt[a + b*x]*(A + B*x),x]

[Out]

(2*(a + b*x)^(3/2)*(-1280*a^5*B + 315*b^5*x^4*(13*A + 11*B*x) + 128*a^4*b*(13*A + 15*B*x) - 96*a^3*b^2*x*(26*A
+ 25*B*x) + 80*a^2*b^3*x^2*(39*A + 35*B*x) - 70*a*b^4*x^3*(52*A + 45*B*x)))/(45045*b^6)

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IntegrateAlgebraic [A]  time = 0.06, size = 137, normalized size = 0.91 \begin {gather*} \frac {2 (a+b x)^{3/2} \left (-15015 a^5 B+15015 a^4 A b+45045 a^4 B (a+b x)-36036 a^3 A b (a+b x)-64350 a^3 B (a+b x)^2+38610 a^2 A b (a+b x)^2+50050 a^2 B (a+b x)^3-20020 a A b (a+b x)^3+4095 A b (a+b x)^4-20475 a B (a+b x)^4+3465 B (a+b x)^5\right )}{45045 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^4*Sqrt[a + b*x]*(A + B*x),x]

[Out]

(2*(a + b*x)^(3/2)*(15015*a^4*A*b - 15015*a^5*B - 36036*a^3*A*b*(a + b*x) + 45045*a^4*B*(a + b*x) + 38610*a^2*
A*b*(a + b*x)^2 - 64350*a^3*B*(a + b*x)^2 - 20020*a*A*b*(a + b*x)^3 + 50050*a^2*B*(a + b*x)^3 + 4095*A*b*(a +
b*x)^4 - 20475*a*B*(a + b*x)^4 + 3465*B*(a + b*x)^5))/(45045*b^6)

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fricas [A]  time = 1.29, size = 143, normalized size = 0.95 \begin {gather*} \frac {2 \, {\left (3465 \, B b^{6} x^{6} - 1280 \, B a^{6} + 1664 \, A a^{5} b + 315 \, {\left (B a b^{5} + 13 \, A b^{6}\right )} x^{5} - 35 \, {\left (10 \, B a^{2} b^{4} - 13 \, A a b^{5}\right )} x^{4} + 40 \, {\left (10 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{3} - 48 \, {\left (10 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{2} + 64 \, {\left (10 \, B a^{5} b - 13 \, A a^{4} 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)*(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

2/45045*(3465*B*b^6*x^6 - 1280*B*a^6 + 1664*A*a^5*b + 315*(B*a*b^5 + 13*A*b^6)*x^5 - 35*(10*B*a^2*b^4 - 13*A*a
*b^5)*x^4 + 40*(10*B*a^3*b^3 - 13*A*a^2*b^4)*x^3 - 48*(10*B*a^4*b^2 - 13*A*a^3*b^3)*x^2 + 64*(10*B*a^5*b - 13*
A*a^4*b^2)*x)*sqrt(b*x + a)/b^6

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giac [B]  time = 1.18, size = 304, normalized size = 2.01 \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}{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}{b^{5}} + \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^{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 )} 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)*(b*x+a)^(1/2),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/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/b^5 + 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^4 + 15*(231*(b*x + a)^(13/2) - 1638*(b*x + a)^(11/2)*a + 5005*(b*x + a)^(9/2)*a^2 - 85
80*(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^5)/
b

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maple [A]  time = 0.01, size = 119, normalized size = 0.79 \begin {gather*} \frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (3465 B \,b^{5} x^{5}+4095 A \,b^{5} x^{4}-3150 B a \,b^{4} x^{4}-3640 A a \,b^{4} x^{3}+2800 B \,a^{2} b^{3} x^{3}+3120 A \,a^{2} b^{3} x^{2}-2400 B \,a^{3} b^{2} x^{2}-2496 A \,a^{3} b^{2} x +1920 B \,a^{4} b x +1664 A \,a^{4} b -1280 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)*(b*x+a)^(1/2),x)

[Out]

2/45045*(b*x+a)^(3/2)*(3465*B*b^5*x^5+4095*A*b^5*x^4-3150*B*a*b^4*x^4-3640*A*a*b^4*x^3+2800*B*a^2*b^3*x^3+3120
*A*a^2*b^3*x^2-2400*B*a^3*b^2*x^2-2496*A*a^3*b^2*x+1920*B*a^4*b*x+1664*A*a^4*b-1280*B*a^5)/b^6

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maxima [A]  time = 0.84, size = 123, normalized size = 0.81 \begin {gather*} \frac {2 \, {\left (3465 \, {\left (b x + a\right )}^{\frac {13}{2}} B - 4095 \, {\left (5 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {11}{2}} + 10010 \, {\left (5 \, B a^{2} - 2 \, A a b\right )} {\left (b x + a\right )}^{\frac {9}{2}} - 12870 \, {\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {7}{2}} + 9009 \, {\left (5 \, B a^{4} - 4 \, A a^{3} b\right )} {\left (b x + a\right )}^{\frac {5}{2}} - 15015 \, {\left (B a^{5} - A a^{4} b\right )} {\left (b x + a\right )}^{\frac {3}{2}}\right )}}{45045 \, b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)*(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

2/45045*(3465*(b*x + a)^(13/2)*B - 4095*(5*B*a - A*b)*(b*x + a)^(11/2) + 10010*(5*B*a^2 - 2*A*a*b)*(b*x + a)^(
9/2) - 12870*(5*B*a^3 - 3*A*a^2*b)*(b*x + a)^(7/2) + 9009*(5*B*a^4 - 4*A*a^3*b)*(b*x + a)^(5/2) - 15015*(B*a^5
- A*a^4*b)*(b*x + a)^(3/2))/b^6

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mupad [B]  time = 0.08, size = 137, normalized size = 0.91 \begin {gather*} \frac {\left (20\,B\,a^2-8\,A\,a\,b\right )\,{\left (a+b\,x\right )}^{9/2}}{9\,b^6}+\frac {2\,B\,{\left (a+b\,x\right )}^{13/2}}{13\,b^6}+\frac {\left (2\,A\,b-10\,B\,a\right )\,{\left (a+b\,x\right )}^{11/2}}{11\,b^6}-\frac {\left (2\,B\,a^5-2\,A\,a^4\,b\right )\,{\left (a+b\,x\right )}^{3/2}}{3\,b^6}+\frac {\left (10\,B\,a^4-8\,A\,a^3\,b\right )\,{\left (a+b\,x\right )}^{5/2}}{5\,b^6}-\frac {\left (20\,B\,a^3-12\,A\,a^2\,b\right )\,{\left (a+b\,x\right )}^{7/2}}{7\,b^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(A + B*x)*(a + b*x)^(1/2),x)

[Out]

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

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sympy [A]  time = 3.73, size = 150, normalized size = 0.99 \begin {gather*} \frac {2 \left (\frac {B \left (a + b x\right )^{\frac {13}{2}}}{13 b} + \frac {\left (a + b x\right )^{\frac {11}{2}} \left (A b - 5 B a\right )}{11 b} + \frac {\left (a + b x\right )^{\frac {9}{2}} \left (- 4 A a b + 10 B a^{2}\right )}{9 b} + \frac {\left (a + b x\right )^{\frac {7}{2}} \left (6 A a^{2} b - 10 B a^{3}\right )}{7 b} + \frac {\left (a + b x\right )^{\frac {5}{2}} \left (- 4 A a^{3} b + 5 B a^{4}\right )}{5 b} + \frac {\left (a + b x\right )^{\frac {3}{2}} \left (A a^{4} b - B a^{5}\right )}{3 b}\right )}{b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(B*x+A)*(b*x+a)**(1/2),x)

[Out]

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

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