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

Optimal. Leaf size=120 \[ \frac{2 a^2 (a+b x)^{3/2} (3 A b-4 a B)}{3 b^5}-\frac{2 a^3 \sqrt{a+b x} (A b-a B)}{b^5}+\frac{2 (a+b x)^{7/2} (A b-4 a B)}{7 b^5}-\frac{6 a (a+b x)^{5/2} (A b-2 a B)}{5 b^5}+\frac{2 B (a+b x)^{9/2}}{9 b^5} \]

[Out]

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

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Rubi [A]  time = 0.0482359, antiderivative size = 120, 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} \[ \frac{2 a^2 (a+b x)^{3/2} (3 A b-4 a B)}{3 b^5}-\frac{2 a^3 \sqrt{a+b x} (A b-a B)}{b^5}+\frac{2 (a+b x)^{7/2} (A b-4 a B)}{7 b^5}-\frac{6 a (a+b x)^{5/2} (A b-2 a B)}{5 b^5}+\frac{2 B (a+b x)^{9/2}}{9 b^5} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(A + B*x))/Sqrt[a + b*x],x]

[Out]

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

Mathematica [A]  time = 0.0643471, size = 87, normalized size = 0.72 \[ \frac{2 \sqrt{a+b x} \left (24 a^2 b^2 x (3 A+2 B x)-16 a^3 b (9 A+4 B x)+128 a^4 B-2 a b^3 x^2 (27 A+20 B x)+5 b^4 x^3 (9 A+7 B x)\right )}{315 b^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[a + b*x]*(128*a^4*B + 24*a^2*b^2*x*(3*A + 2*B*x) - 16*a^3*b*(9*A + 4*B*x) + 5*b^4*x^3*(9*A + 7*B*x) -
2*a*b^3*x^2*(27*A + 20*B*x)))/(315*b^5)

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Maple [A]  time = 0.006, size = 95, normalized size = 0.8 \begin{align*} -{\frac{-70\,B{x}^{4}{b}^{4}-90\,A{b}^{4}{x}^{3}+80\,Ba{b}^{3}{x}^{3}+108\,Aa{b}^{3}{x}^{2}-96\,B{a}^{2}{b}^{2}{x}^{2}-144\,A{a}^{2}{b}^{2}x+128\,B{a}^{3}bx+288\,A{a}^{3}b-256\,B{a}^{4}}{315\,{b}^{5}}\sqrt{bx+a}} \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]

-2/315*(b*x+a)^(1/2)*(-35*B*b^4*x^4-45*A*b^4*x^3+40*B*a*b^3*x^3+54*A*a*b^3*x^2-48*B*a^2*b^2*x^2-72*A*a^2*b^2*x
+64*B*a^3*b*x+144*A*a^3*b-128*B*a^4)/b^5

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Maxima [A]  time = 1.12264, size = 135, normalized size = 1.12 \begin{align*} \frac{2 \,{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} B - 45 \,{\left (4 \, B a - A b\right )}{\left (b x + a\right )}^{\frac{7}{2}} + 189 \,{\left (2 \, B a^{2} - A a b\right )}{\left (b x + a\right )}^{\frac{5}{2}} - 105 \,{\left (4 \, B a^{3} - 3 \, A a^{2} b\right )}{\left (b x + a\right )}^{\frac{3}{2}} + 315 \,{\left (B a^{4} - A a^{3} b\right )} \sqrt{b x + a}\right )}}{315 \, 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]

2/315*(35*(b*x + a)^(9/2)*B - 45*(4*B*a - A*b)*(b*x + a)^(7/2) + 189*(2*B*a^2 - A*a*b)*(b*x + a)^(5/2) - 105*(
4*B*a^3 - 3*A*a^2*b)*(b*x + a)^(3/2) + 315*(B*a^4 - A*a^3*b)*sqrt(b*x + a))/b^5

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Fricas [A]  time = 2.2787, size = 219, normalized size = 1.82 \begin{align*} \frac{2 \,{\left (35 \, B b^{4} x^{4} + 128 \, B a^{4} - 144 \, A a^{3} b - 5 \,{\left (8 \, B a b^{3} - 9 \, A b^{4}\right )} x^{3} + 6 \,{\left (8 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{2} - 8 \,{\left (8 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x\right )} \sqrt{b x + a}}{315 \, 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]

2/315*(35*B*b^4*x^4 + 128*B*a^4 - 144*A*a^3*b - 5*(8*B*a*b^3 - 9*A*b^4)*x^3 + 6*(8*B*a^2*b^2 - 9*A*a*b^3)*x^2
- 8*(8*B*a^3*b - 9*A*a^2*b^2)*x)*sqrt(b*x + a)/b^5

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Sympy [A]  time = 35.609, size = 301, normalized size = 2.51 \begin{align*} \begin{cases} - \frac{\frac{2 A a \left (- \frac{a^{3}}{\sqrt{a + b x}} - 3 a^{2} \sqrt{a + b x} + a \left (a + b x\right )^{\frac{3}{2}} - \frac{\left (a + b x\right )^{\frac{5}{2}}}{5}\right )}{b^{3}} + \frac{2 A \left (\frac{a^{4}}{\sqrt{a + b x}} + 4 a^{3} \sqrt{a + b x} - 2 a^{2} \left (a + b x\right )^{\frac{3}{2}} + \frac{4 a \left (a + b x\right )^{\frac{5}{2}}}{5} - \frac{\left (a + b x\right )^{\frac{7}{2}}}{7}\right )}{b^{3}} + \frac{2 B a \left (\frac{a^{4}}{\sqrt{a + b x}} + 4 a^{3} \sqrt{a + b x} - 2 a^{2} \left (a + b x\right )^{\frac{3}{2}} + \frac{4 a \left (a + b x\right )^{\frac{5}{2}}}{5} - \frac{\left (a + b x\right )^{\frac{7}{2}}}{7}\right )}{b^{4}} + \frac{2 B \left (- \frac{a^{5}}{\sqrt{a + b x}} - 5 a^{4} \sqrt{a + b x} + \frac{10 a^{3} \left (a + b x\right )^{\frac{3}{2}}}{3} - 2 a^{2} \left (a + b x\right )^{\frac{5}{2}} + \frac{5 a \left (a + b x\right )^{\frac{7}{2}}}{7} - \frac{\left (a + b x\right )^{\frac{9}{2}}}{9}\right )}{b^{4}}}{b} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{4}}{4} + \frac{B x^{5}}{5}}{\sqrt{a}} & \text{otherwise} \end{cases} \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]

Piecewise((-(2*A*a*(-a**3/sqrt(a + b*x) - 3*a**2*sqrt(a + b*x) + a*(a + b*x)**(3/2) - (a + b*x)**(5/2)/5)/b**3
+ 2*A*(a**4/sqrt(a + b*x) + 4*a**3*sqrt(a + b*x) - 2*a**2*(a + b*x)**(3/2) + 4*a*(a + b*x)**(5/2)/5 - (a + b*
x)**(7/2)/7)/b**3 + 2*B*a*(a**4/sqrt(a + b*x) + 4*a**3*sqrt(a + b*x) - 2*a**2*(a + b*x)**(3/2) + 4*a*(a + b*x)
**(5/2)/5 - (a + b*x)**(7/2)/7)/b**4 + 2*B*(-a**5/sqrt(a + b*x) - 5*a**4*sqrt(a + b*x) + 10*a**3*(a + b*x)**(3
/2)/3 - 2*a**2*(a + b*x)**(5/2) + 5*a*(a + b*x)**(7/2)/7 - (a + b*x)**(9/2)/9)/b**4)/b, Ne(b, 0)), ((A*x**4/4
+ B*x**5/5)/sqrt(a), True))

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Giac [A]  time = 1.16817, size = 158, normalized size = 1.32 \begin{align*} \frac{2 \,{\left (\frac{9 \,{\left (5 \,{\left (b x + a\right )}^{\frac{7}{2}} - 21 \,{\left (b x + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} - 35 \, \sqrt{b x + a} a^{3}\right )} A}{b^{3}} + \frac{{\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 )} B}{b^{4}}\right )}}{315 \, 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]

2/315*(9*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2 - 35*sqrt(b*x + a)*a^3)*A/b^3 + (3
5*(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)*B/b^4)/b