3.389 \(\int x^2 \sqrt{a+b x} (A+B x) \, dx\)

Optimal. Leaf size=95 \[ \frac{2 a^2 (a+b x)^{3/2} (A b-a B)}{3 b^4}+\frac{2 (a+b x)^{7/2} (A b-3 a B)}{7 b^4}-\frac{2 a (a+b x)^{5/2} (2 A b-3 a B)}{5 b^4}+\frac{2 B (a+b x)^{9/2}}{9 b^4} \]

[Out]

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

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Rubi [A]  time = 0.0396122, antiderivative size = 95, 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} \[ \frac{2 a^2 (a+b x)^{3/2} (A b-a B)}{3 b^4}+\frac{2 (a+b x)^{7/2} (A b-3 a B)}{7 b^4}-\frac{2 a (a+b x)^{5/2} (2 A b-3 a B)}{5 b^4}+\frac{2 B (a+b x)^{9/2}}{9 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Sqrt[a + b*x]*(A + B*x),x]

[Out]

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

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

Mathematica [A]  time = 0.0489118, size = 65, normalized size = 0.68 \[ \frac{2 (a+b x)^{3/2} \left (24 a^2 b (A+B x)-16 a^3 B-6 a b^2 x (6 A+5 B x)+5 b^3 x^2 (9 A+7 B x)\right )}{315 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sqrt[a + b*x]*(A + B*x),x]

[Out]

(2*(a + b*x)^(3/2)*(-16*a^3*B + 24*a^2*b*(A + B*x) - 6*a*b^2*x*(6*A + 5*B*x) + 5*b^3*x^2*(9*A + 7*B*x)))/(315*
b^4)

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Maple [A]  time = 0.003, size = 71, normalized size = 0.8 \begin{align*}{\frac{70\,{b}^{3}B{x}^{3}+90\,A{x}^{2}{b}^{3}-60\,B{x}^{2}a{b}^{2}-72\,a{b}^{2}Ax+48\,{a}^{2}bBx+48\,A{a}^{2}b-32\,B{a}^{3}}{315\,{b}^{4}} \left ( bx+a \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(b*x+a)^(3/2)*(35*B*b^3*x^3+45*A*b^3*x^2-30*B*a*b^2*x^2-36*A*a*b^2*x+24*B*a^2*b*x+24*A*a^2*b-16*B*a^3)/b
^4

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Maxima [A]  time = 1.09245, size = 104, normalized size = 1.09 \begin{align*} \frac{2 \,{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} B - 45 \,{\left (3 \, B a - A b\right )}{\left (b x + a\right )}^{\frac{7}{2}} + 63 \,{\left (3 \, B a^{2} - 2 \, A a b\right )}{\left (b x + a\right )}^{\frac{5}{2}} - 105 \,{\left (B a^{3} - A a^{2} b\right )}{\left (b x + a\right )}^{\frac{3}{2}}\right )}}{315 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*(b*x + a)^(9/2)*B - 45*(3*B*a - A*b)*(b*x + a)^(7/2) + 63*(3*B*a^2 - 2*A*a*b)*(b*x + a)^(5/2) - 105*
(B*a^3 - A*a^2*b)*(b*x + a)^(3/2))/b^4

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Fricas [A]  time = 2.31737, size = 213, normalized size = 2.24 \begin{align*} \frac{2 \,{\left (35 \, B b^{4} x^{4} - 16 \, B a^{4} + 24 \, A a^{3} b + 5 \,{\left (B a b^{3} + 9 \, A b^{4}\right )} x^{3} - 3 \,{\left (2 \, B a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{2} + 4 \,{\left (2 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} x\right )} \sqrt{b x + a}}{315 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*B*b^4*x^4 - 16*B*a^4 + 24*A*a^3*b + 5*(B*a*b^3 + 9*A*b^4)*x^3 - 3*(2*B*a^2*b^2 - 3*A*a*b^3)*x^2 + 4*
(2*B*a^3*b - 3*A*a^2*b^2)*x)*sqrt(b*x + a)/b^4

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Sympy [A]  time = 3.02974, size = 92, normalized size = 0.97 \begin{align*} \frac{2 \left (\frac{B \left (a + b x\right )^{\frac{9}{2}}}{9 b} + \frac{\left (a + b x\right )^{\frac{7}{2}} \left (A b - 3 B a\right )}{7 b} + \frac{\left (a + b x\right )^{\frac{5}{2}} \left (- 2 A a b + 3 B a^{2}\right )}{5 b} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (A a^{2} b - B a^{3}\right )}{3 b}\right )}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15877, size = 126, normalized size = 1.33 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2}\right )} A}{b^{2}} + \frac{{\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 )} B}{b^{3}}\right )}}{315 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(3*(15*(b*x + a)^(7/2) - 42*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2)*A/b^2 + (35*(b*x + a)^(9/2) - 13
5*(b*x + a)^(7/2)*a + 189*(b*x + a)^(5/2)*a^2 - 105*(b*x + a)^(3/2)*a^3)*B/b^3)/b