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

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

[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)

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Rubi [A]  time = 0.0737233, antiderivative size = 151, normalized size of antiderivative = 1., 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} \[ \frac{4 a^2 (a+b x)^{7/2} (3 A b-5 a B)}{7 b^6}-\frac{2 a^3 (a+b x)^{5/2} (4 A b-5 a B)}{5 b^6}+\frac{2 a^4 (a+b x)^{3/2} (A b-a B)}{3 b^6}+\frac{2 (a+b x)^{11/2} (A b-5 a B)}{11 b^6}-\frac{4 a (a+b x)^{9/2} (2 A b-5 a B)}{9 b^6}+\frac{2 B (a+b x)^{13/2}}{13 b^6} \]

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*}

Mathematica [A]  time = 0.0827237, size = 106, normalized size = 0.7 \[ \frac{2 (a+b x)^{3/2} \left (80 a^2 b^3 x^2 (39 A+35 B x)-96 a^3 b^2 x (26 A+25 B x)+128 a^4 b (13 A+15 B x)-1280 a^5 B-70 a b^4 x^3 (52 A+45 B x)+315 b^5 x^4 (13 A+11 B x)\right )}{45045 b^6} \]

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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Maple [A]  time = 0.005, size = 119, normalized size = 0.8 \begin{align*}{\frac{6930\,{b}^{5}B{x}^{5}+8190\,A{x}^{4}{b}^{5}-6300\,B{x}^{4}a{b}^{4}-7280\,A{x}^{3}a{b}^{4}+5600\,B{x}^{3}{a}^{2}{b}^{3}+6240\,A{x}^{2}{a}^{2}{b}^{3}-4800\,B{x}^{2}{a}^{3}{b}^{2}-4992\,{a}^{3}{b}^{2}Ax+3840\,{a}^{4}bBx+3328\,A{a}^{4}b-2560\,B{a}^{5}}{45045\,{b}^{6}} \left ( bx+a \right ) ^{{\frac{3}{2}}}} \end{align*}

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 = 1.03272, size = 166, normalized size = 1.1 \begin{align*} \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{align*}

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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Fricas [A]  time = 2.3075, size = 342, normalized size = 2.26 \begin{align*} \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{align*}

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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Sympy [A]  time = 3.71365, size = 150, normalized size = 0.99 \begin{align*} \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{align*}

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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Giac [A]  time = 1.19912, size = 192, normalized size = 1.27 \begin{align*} \frac{2 \,{\left (\frac{13 \,{\left (315 \,{\left (b x + a\right )}^{\frac{11}{2}} - 1540 \,{\left (b x + a\right )}^{\frac{9}{2}} a + 2970 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{2} - 2772 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{3} + 1155 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{4}\right )} A}{b^{4}} + \frac{5 \,{\left (693 \,{\left (b x + a\right )}^{\frac{13}{2}} - 4095 \,{\left (b x + a\right )}^{\frac{11}{2}} a + 10010 \,{\left (b x + a\right )}^{\frac{9}{2}} a^{2} - 12870 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{3} + 9009 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{4} - 3003 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{5}\right )} B}{b^{5}}\right )}}{45045 \, b} \end{align*}

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*(13*(315*(b*x + a)^(11/2) - 1540*(b*x + a)^(9/2)*a + 2970*(b*x + a)^(7/2)*a^2 - 2772*(b*x + a)^(5/2)*a
^3 + 1155*(b*x + a)^(3/2)*a^4)*A/b^4 + 5*(693*(b*x + a)^(13/2) - 4095*(b*x + a)^(11/2)*a + 10010*(b*x + a)^(9/
2)*a^2 - 12870*(b*x + a)^(7/2)*a^3 + 9009*(b*x + a)^(5/2)*a^4 - 3003*(b*x + a)^(3/2)*a^5)*B/b^5)/b