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

Optimal. Leaf size=116 \[ \frac{2 a^2 \sqrt{a+b x} (3 A b-4 a B)}{b^5}+\frac{2 a^3 (A b-a B)}{b^5 \sqrt{a+b x}}+\frac{2 (a+b x)^{5/2} (A b-4 a B)}{5 b^5}-\frac{2 a (a+b x)^{3/2} (A b-2 a B)}{b^5}+\frac{2 B (a+b x)^{7/2}}{7 b^5} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0429877, antiderivative size = 116, 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} \[ \frac{2 a^2 \sqrt{a+b x} (3 A b-4 a B)}{b^5}+\frac{2 a^3 (A b-a B)}{b^5 \sqrt{a+b x}}+\frac{2 (a+b x)^{5/2} (A b-4 a B)}{5 b^5}-\frac{2 a (a+b x)^{3/2} (A b-2 a B)}{b^5}+\frac{2 B (a+b x)^{7/2}}{7 b^5} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(A + B*x))/(a + b*x)^(3/2),x]

[Out]

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

Mathematica [A]  time = 0.0578529, size = 86, normalized size = 0.74 \[ \frac{2 \left (8 a^2 b^2 x (7 A+2 B x)+16 a^3 b (7 A-4 B x)-128 a^4 B-2 a b^3 x^2 (7 A+4 B x)+b^4 x^3 (7 A+5 B x)\right )}{35 b^5 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(A + B*x))/(a + b*x)^(3/2),x]

[Out]

(2*(-128*a^4*B + 16*a^3*b*(7*A - 4*B*x) + 8*a^2*b^2*x*(7*A + 2*B*x) - 2*a*b^3*x^2*(7*A + 4*B*x) + b^4*x^3*(7*A
+ 5*B*x)))/(35*b^5*Sqrt[a + b*x])

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 95, normalized size = 0.8 \begin{align*}{\frac{10\,B{x}^{4}{b}^{4}+14\,A{b}^{4}{x}^{3}-16\,Ba{b}^{3}{x}^{3}-28\,Aa{b}^{3}{x}^{2}+32\,B{a}^{2}{b}^{2}{x}^{2}+112\,A{a}^{2}{b}^{2}x-128\,B{a}^{3}bx+224\,A{a}^{3}b-256\,B{a}^{4}}{35\,{b}^{5}}{\frac{1}{\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)^(3/2),x)

[Out]

2/35/(b*x+a)^(1/2)*(5*B*b^4*x^4+7*A*b^4*x^3-8*B*a*b^3*x^3-14*A*a*b^3*x^2+16*B*a^2*b^2*x^2+56*A*a^2*b^2*x-64*B*
a^3*b*x+112*A*a^3*b-128*B*a^4)/b^5

________________________________________________________________________________________

Maxima [A]  time = 1.1304, size = 146, normalized size = 1.26 \begin{align*} \frac{2 \,{\left (\frac{5 \,{\left (b x + a\right )}^{\frac{7}{2}} B - 7 \,{\left (4 \, B a - A b\right )}{\left (b x + a\right )}^{\frac{5}{2}} + 35 \,{\left (2 \, B a^{2} - A a b\right )}{\left (b x + a\right )}^{\frac{3}{2}} - 35 \,{\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} \sqrt{b x + a}}{b} - \frac{35 \,{\left (B a^{4} - A a^{3} b\right )}}{\sqrt{b x + a} b}\right )}}{35 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.46022, size = 230, normalized size = 1.98 \begin{align*} \frac{2 \,{\left (5 \, B b^{4} x^{4} - 128 \, B a^{4} + 112 \, A a^{3} b -{\left (8 \, B a b^{3} - 7 \, A b^{4}\right )} x^{3} + 2 \,{\left (8 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{2} - 8 \,{\left (8 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x\right )} \sqrt{b x + a}}{35 \,{\left (b^{6} x + a b^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(5*B*b^4*x^4 - 128*B*a^4 + 112*A*a^3*b - (8*B*a*b^3 - 7*A*b^4)*x^3 + 2*(8*B*a^2*b^2 - 7*A*a*b^3)*x^2 - 8*
(8*B*a^3*b - 7*A*a^2*b^2)*x)*sqrt(b*x + a)/(b^6*x + a*b^5)

________________________________________________________________________________________

Sympy [A]  time = 15.9281, size = 117, normalized size = 1.01 \begin{align*} \frac{2 B \left (a + b x\right )^{\frac{7}{2}}}{7 b^{5}} - \frac{2 a^{3} \left (- A b + B a\right )}{b^{5} \sqrt{a + b x}} + \frac{\left (a + b x\right )^{\frac{5}{2}} \left (2 A b - 8 B a\right )}{5 b^{5}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (- 6 A a b + 12 B a^{2}\right )}{3 b^{5}} + \frac{\sqrt{a + b x} \left (6 A a^{2} b - 8 B a^{3}\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*B*(a + b*x)**(7/2)/(7*b**5) - 2*a**3*(-A*b + B*a)/(b**5*sqrt(a + b*x)) + (a + b*x)**(5/2)*(2*A*b - 8*B*a)/(5
*b**5) + (a + b*x)**(3/2)*(-6*A*a*b + 12*B*a**2)/(3*b**5) + sqrt(a + b*x)*(6*A*a**2*b - 8*B*a**3)/b**5

________________________________________________________________________________________

Giac [A]  time = 1.18327, size = 181, normalized size = 1.56 \begin{align*} -\frac{2 \,{\left (B a^{4} - A a^{3} b\right )}}{\sqrt{b x + a} b^{5}} + \frac{2 \,{\left (5 \,{\left (b x + a\right )}^{\frac{7}{2}} B b^{30} - 28 \,{\left (b x + a\right )}^{\frac{5}{2}} B a b^{30} + 70 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{2} b^{30} - 140 \, \sqrt{b x + a} B a^{3} b^{30} + 7 \,{\left (b x + a\right )}^{\frac{5}{2}} A b^{31} - 35 \,{\left (b x + a\right )}^{\frac{3}{2}} A a b^{31} + 105 \, \sqrt{b x + a} A a^{2} b^{31}\right )}}{35 \, b^{35}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(B*a^4 - A*a^3*b)/(sqrt(b*x + a)*b^5) + 2/35*(5*(b*x + a)^(7/2)*B*b^30 - 28*(b*x + a)^(5/2)*B*a*b^30 + 70*(
b*x + a)^(3/2)*B*a^2*b^30 - 140*sqrt(b*x + a)*B*a^3*b^30 + 7*(b*x + a)^(5/2)*A*b^31 - 35*(b*x + a)^(3/2)*A*a*b
^31 + 105*sqrt(b*x + a)*A*a^2*b^31)/b^35