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

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

[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)

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Rubi [A]  time = 0.0616942, 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} \[ \frac{4 a^2 (a+b x)^{9/2} (3 A b-5 a B)}{9 b^6}-\frac{2 a^3 (a+b x)^{7/2} (4 A b-5 a B)}{7 b^6}+\frac{2 a^4 (a+b x)^{5/2} (A b-a B)}{5 b^6}+\frac{2 (a+b x)^{13/2} (A b-5 a B)}{13 b^6}-\frac{4 a (a+b x)^{11/2} (2 A b-5 a B)}{11 b^6}+\frac{2 B (a+b x)^{15/2}}{15 b^6} \]

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 = 0.0760602, size = 103, normalized size = 0.68 \[ \frac{2 (a+b x)^{5/2} \left (1680 a^2 b^3 x^2 (A+B x)-160 a^3 b^2 x (6 A+7 B x)+128 a^4 b (3 A+5 B x)-256 a^5 B-210 a b^4 x^3 (12 A+11 B x)+231 b^5 x^4 (15 A+13 B x)\right )}{45045 b^6} \]

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)

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Maple [A]  time = 0.005, size = 119, normalized size = 0.8 \begin{align*}{\frac{6006\,{b}^{5}B{x}^{5}+6930\,A{x}^{4}{b}^{5}-4620\,B{x}^{4}a{b}^{4}-5040\,A{x}^{3}a{b}^{4}+3360\,B{x}^{3}{a}^{2}{b}^{3}+3360\,A{x}^{2}{a}^{2}{b}^{3}-2240\,B{x}^{2}{a}^{3}{b}^{2}-1920\,{a}^{3}{b}^{2}Ax+1280\,{a}^{4}bBx+768\,A{a}^{4}b-512\,B{a}^{5}}{45045\,{b}^{6}} \left ( bx+a \right ) ^{{\frac{5}{2}}}} \end{align*}

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

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Maxima [A]  time = 1.03392, size = 166, normalized size = 1.1 \begin{align*} \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{align*}

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

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Fricas [A]  time = 2.29454, size = 381, normalized size = 2.52 \begin{align*} \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{align*}

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

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Sympy [B]  time = 16.9035, size = 355, normalized size = 2.35 \begin{align*} \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{align*}

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

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Giac [B]  time = 1.26445, size = 409, normalized size = 2.71 \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 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 a}{b^{5}} + \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 )} A}{b^{4}} + \frac{{\left (3003 \,{\left (b x + a\right )}^{\frac{15}{2}} - 20790 \,{\left (b x + a\right )}^{\frac{13}{2}} a + 61425 \,{\left (b x + a\right )}^{\frac{11}{2}} a^{2} - 100100 \,{\left (b x + a\right )}^{\frac{9}{2}} a^{3} + 96525 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{4} - 54054 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{5} + 15015 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{6}\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)^(3/2)*(B*x+A),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*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*a/b^5 + 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)*A/b^4 + (3003*(b*x + a)^(15/2) - 20790*(b*x + a)^(13/2)*a + 61425*(
b*x + a)^(11/2)*a^2 - 100100*(b*x + a)^(9/2)*a^3 + 96525*(b*x + a)^(7/2)*a^4 - 54054*(b*x + a)^(5/2)*a^5 + 150
15*(b*x + a)^(3/2)*a^6)*B/b^5)/b