3.388 \(\int x^3 \sqrt{a+b x} (A+B x) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0566592, antiderivative size = 122, 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} \[ \frac{2 a^2 (a+b x)^{5/2} (3 A b-4 a B)}{5 b^5}-\frac{2 a^3 (a+b x)^{3/2} (A b-a B)}{3 b^5}+\frac{2 (a+b x)^{9/2} (A b-4 a B)}{9 b^5}-\frac{6 a (a+b x)^{7/2} (A b-2 a B)}{7 b^5}+\frac{2 B (a+b x)^{11/2}}{11 b^5} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Sqrt[a + b*x]*(A + B*x),x]

[Out]

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

Mathematica [A]  time = 0.0631749, size = 87, normalized size = 0.71 \[ \frac{2 (a+b x)^{3/2} \left (24 a^2 b^2 x (11 A+10 B x)-16 a^3 b (11 A+12 B x)+128 a^4 B-10 a b^3 x^2 (33 A+28 B x)+35 b^4 x^3 (11 A+9 B x)\right )}{3465 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Sqrt[a + b*x]*(A + B*x),x]

[Out]

(2*(a + b*x)^(3/2)*(128*a^4*B + 35*b^4*x^3*(11*A + 9*B*x) + 24*a^2*b^2*x*(11*A + 10*B*x) - 16*a^3*b*(11*A + 12
*B*x) - 10*a*b^3*x^2*(33*A + 28*B*x)))/(3465*b^5)

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Maple [A]  time = 0.004, size = 95, normalized size = 0.8 \begin{align*} -{\frac{-630\,B{x}^{4}{b}^{4}-770\,A{b}^{4}{x}^{3}+560\,Ba{b}^{3}{x}^{3}+660\,Aa{b}^{3}{x}^{2}-480\,B{a}^{2}{b}^{2}{x}^{2}-528\,A{a}^{2}{b}^{2}x+384\,B{a}^{3}bx+352\,A{a}^{3}b-256\,B{a}^{4}}{3465\,{b}^{5}} \left ( bx+a \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(B*x+A)*(b*x+a)^(1/2),x)

[Out]

-2/3465*(b*x+a)^(3/2)*(-315*B*b^4*x^4-385*A*b^4*x^3+280*B*a*b^3*x^3+330*A*a*b^3*x^2-240*B*a^2*b^2*x^2-264*A*a^
2*b^2*x+192*B*a^3*b*x+176*A*a^3*b-128*B*a^4)/b^5

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Maxima [A]  time = 1.06399, size = 135, normalized size = 1.11 \begin{align*} \frac{2 \,{\left (315 \,{\left (b x + a\right )}^{\frac{11}{2}} B - 385 \,{\left (4 \, B a - A b\right )}{\left (b x + a\right )}^{\frac{9}{2}} + 1485 \,{\left (2 \, B a^{2} - A a b\right )}{\left (b x + a\right )}^{\frac{7}{2}} - 693 \,{\left (4 \, B a^{3} - 3 \, A a^{2} b\right )}{\left (b x + a\right )}^{\frac{5}{2}} + 1155 \,{\left (B a^{4} - A a^{3} b\right )}{\left (b x + a\right )}^{\frac{3}{2}}\right )}}{3465 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(B*x+A)*(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

2/3465*(315*(b*x + a)^(11/2)*B - 385*(4*B*a - A*b)*(b*x + a)^(9/2) + 1485*(2*B*a^2 - A*a*b)*(b*x + a)^(7/2) -
693*(4*B*a^3 - 3*A*a^2*b)*(b*x + a)^(5/2) + 1155*(B*a^4 - A*a^3*b)*(b*x + a)^(3/2))/b^5

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Fricas [A]  time = 2.28348, size = 274, normalized size = 2.25 \begin{align*} \frac{2 \,{\left (315 \, B b^{5} x^{5} + 128 \, B a^{5} - 176 \, A a^{4} b + 35 \,{\left (B a b^{4} + 11 \, A b^{5}\right )} x^{4} - 5 \,{\left (8 \, B a^{2} b^{3} - 11 \, A a b^{4}\right )} x^{3} + 6 \,{\left (8 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{2} - 8 \,{\left (8 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} x\right )} \sqrt{b x + a}}{3465 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(B*x+A)*(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

2/3465*(315*B*b^5*x^5 + 128*B*a^5 - 176*A*a^4*b + 35*(B*a*b^4 + 11*A*b^5)*x^4 - 5*(8*B*a^2*b^3 - 11*A*a*b^4)*x
^3 + 6*(8*B*a^3*b^2 - 11*A*a^2*b^3)*x^2 - 8*(8*B*a^4*b - 11*A*a^3*b^2)*x)*sqrt(b*x + a)/b^5

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Sympy [A]  time = 3.1667, size = 121, normalized size = 0.99 \begin{align*} \frac{2 \left (\frac{B \left (a + b x\right )^{\frac{11}{2}}}{11 b} + \frac{\left (a + b x\right )^{\frac{9}{2}} \left (A b - 4 B a\right )}{9 b} + \frac{\left (a + b x\right )^{\frac{7}{2}} \left (- 3 A a b + 6 B a^{2}\right )}{7 b} + \frac{\left (a + b x\right )^{\frac{5}{2}} \left (3 A a^{2} b - 4 B a^{3}\right )}{5 b} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (- A a^{3} b + B a^{4}\right )}{3 b}\right )}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(B*x+A)*(b*x+a)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.17867, size = 158, normalized size = 1.3 \begin{align*} \frac{2 \,{\left (\frac{11 \,{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3}\right )} A}{b^{3}} + \frac{{\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 )} B}{b^{4}}\right )}}{3465 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(B*x+A)*(b*x+a)^(1/2),x, algorithm="giac")

[Out]

2/3465*(11*(35*(b*x + a)^(9/2) - 135*(b*x + a)^(7/2)*a + 189*(b*x + a)^(5/2)*a^2 - 105*(b*x + a)^(3/2)*a^3)*A/
b^3 + (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 + 1
155*(b*x + a)^(3/2)*a^4)*B/b^4)/b