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

Optimal. Leaf size=147 \[ \frac{4 a^2 (a+b x)^{3/2} (3 A b-5 a B)}{3 b^6}-\frac{2 a^3 \sqrt{a+b x} (4 A b-5 a B)}{b^6}-\frac{2 a^4 (A b-a B)}{b^6 \sqrt{a+b x}}+\frac{2 (a+b x)^{7/2} (A b-5 a B)}{7 b^6}-\frac{4 a (a+b x)^{5/2} (2 A b-5 a B)}{5 b^6}+\frac{2 B (a+b x)^{9/2}}{9 b^6} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.060773, antiderivative size = 147, 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} \[ \frac{4 a^2 (a+b x)^{3/2} (3 A b-5 a B)}{3 b^6}-\frac{2 a^3 \sqrt{a+b x} (4 A b-5 a B)}{b^6}-\frac{2 a^4 (A b-a B)}{b^6 \sqrt{a+b x}}+\frac{2 (a+b x)^{7/2} (A b-5 a B)}{7 b^6}-\frac{4 a (a+b x)^{5/2} (2 A b-5 a B)}{5 b^6}+\frac{2 B (a+b x)^{9/2}}{9 b^6} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(A + B*x))/(a + b*x)^(3/2),x]

[Out]

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

Mathematica [A]  time = 0.0744096, size = 106, normalized size = 0.72 \[ \frac{32 a^2 b^3 x^2 (9 A+5 B x)-64 a^3 b^2 x (18 A+5 B x)-256 a^4 b (9 A-5 B x)+2560 a^5 B-4 a b^4 x^3 (36 A+25 B x)+10 b^5 x^4 (9 A+7 B x)}{315 b^6 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(A + B*x))/(a + b*x)^(3/2),x]

[Out]

(2560*a^5*B - 256*a^4*b*(9*A - 5*B*x) + 32*a^2*b^3*x^2*(9*A + 5*B*x) - 64*a^3*b^2*x*(18*A + 5*B*x) + 10*b^5*x^
4*(9*A + 7*B*x) - 4*a*b^4*x^3*(36*A + 25*B*x))/(315*b^6*Sqrt[a + b*x])

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 119, normalized size = 0.8 \begin{align*} -{\frac{-70\,{b}^{5}B{x}^{5}-90\,A{x}^{4}{b}^{5}+100\,B{x}^{4}a{b}^{4}+144\,A{x}^{3}a{b}^{4}-160\,B{x}^{3}{a}^{2}{b}^{3}-288\,A{x}^{2}{a}^{2}{b}^{3}+320\,B{x}^{2}{a}^{3}{b}^{2}+1152\,{a}^{3}{b}^{2}Ax-1280\,{a}^{4}bBx+2304\,A{a}^{4}b-2560\,B{a}^{5}}{315\,{b}^{6}}{\frac{1}{\sqrt{bx+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(B*x+A)/(b*x+a)^(3/2),x)

[Out]

-2/315/(b*x+a)^(1/2)*(-35*B*b^5*x^5-45*A*b^5*x^4+50*B*a*b^4*x^4+72*A*a*b^4*x^3-80*B*a^2*b^3*x^3-144*A*a^2*b^3*
x^2+160*B*a^3*b^2*x^2+576*A*a^3*b^2*x-640*B*a^4*b*x+1152*A*a^4*b-1280*B*a^5)/b^6

________________________________________________________________________________________

Maxima [A]  time = 1.29706, size = 177, normalized size = 1.2 \begin{align*} \frac{2 \,{\left (\frac{35 \,{\left (b x + a\right )}^{\frac{9}{2}} B - 45 \,{\left (5 \, B a - A b\right )}{\left (b x + a\right )}^{\frac{7}{2}} + 126 \,{\left (5 \, B a^{2} - 2 \, A a b\right )}{\left (b x + a\right )}^{\frac{5}{2}} - 210 \,{\left (5 \, B a^{3} - 3 \, A a^{2} b\right )}{\left (b x + a\right )}^{\frac{3}{2}} + 315 \,{\left (5 \, B a^{4} - 4 \, A a^{3} b\right )} \sqrt{b x + a}}{b} + \frac{315 \,{\left (B a^{5} - A a^{4} b\right )}}{\sqrt{b x + a} b}\right )}}{315 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)/(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

2/315*((35*(b*x + a)^(9/2)*B - 45*(5*B*a - A*b)*(b*x + a)^(7/2) + 126*(5*B*a^2 - 2*A*a*b)*(b*x + a)^(5/2) - 21
0*(5*B*a^3 - 3*A*a^2*b)*(b*x + a)^(3/2) + 315*(5*B*a^4 - 4*A*a^3*b)*sqrt(b*x + a))/b + 315*(B*a^5 - A*a^4*b)/(
sqrt(b*x + a)*b))/b^5

________________________________________________________________________________________

Fricas [A]  time = 2.33845, size = 294, normalized size = 2. \begin{align*} \frac{2 \,{\left (35 \, B b^{5} x^{5} + 1280 \, B a^{5} - 1152 \, A a^{4} b - 5 \,{\left (10 \, B a b^{4} - 9 \, A b^{5}\right )} x^{4} + 8 \,{\left (10 \, B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{3} - 16 \,{\left (10 \, B a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{2} + 64 \,{\left (10 \, B a^{4} b - 9 \, A a^{3} b^{2}\right )} x\right )} \sqrt{b x + a}}{315 \,{\left (b^{7} x + a b^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)/(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

2/315*(35*B*b^5*x^5 + 1280*B*a^5 - 1152*A*a^4*b - 5*(10*B*a*b^4 - 9*A*b^5)*x^4 + 8*(10*B*a^2*b^3 - 9*A*a*b^4)*
x^3 - 16*(10*B*a^3*b^2 - 9*A*a^2*b^3)*x^2 + 64*(10*B*a^4*b - 9*A*a^3*b^2)*x)*sqrt(b*x + a)/(b^7*x + a*b^6)

________________________________________________________________________________________

Sympy [A]  time = 15.8802, size = 146, normalized size = 0.99 \begin{align*} \frac{2 B \left (a + b x\right )^{\frac{9}{2}}}{9 b^{6}} + \frac{2 a^{4} \left (- A b + B a\right )}{b^{6} \sqrt{a + b x}} + \frac{\left (a + b x\right )^{\frac{7}{2}} \left (2 A b - 10 B a\right )}{7 b^{6}} + \frac{\left (a + b x\right )^{\frac{5}{2}} \left (- 8 A a b + 20 B a^{2}\right )}{5 b^{6}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (12 A a^{2} b - 20 B a^{3}\right )}{3 b^{6}} + \frac{\sqrt{a + b x} \left (- 8 A a^{3} b + 10 B a^{4}\right )}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(B*x+A)/(b*x+a)**(3/2),x)

[Out]

2*B*(a + b*x)**(9/2)/(9*b**6) + 2*a**4*(-A*b + B*a)/(b**6*sqrt(a + b*x)) + (a + b*x)**(7/2)*(2*A*b - 10*B*a)/(
7*b**6) + (a + b*x)**(5/2)*(-8*A*a*b + 20*B*a**2)/(5*b**6) + (a + b*x)**(3/2)*(12*A*a**2*b - 20*B*a**3)/(3*b**
6) + sqrt(a + b*x)*(-8*A*a**3*b + 10*B*a**4)/b**6

________________________________________________________________________________________

Giac [A]  time = 1.18802, size = 224, normalized size = 1.52 \begin{align*} \frac{2 \,{\left (B a^{5} - A a^{4} b\right )}}{\sqrt{b x + a} b^{6}} + \frac{2 \,{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} B b^{48} - 225 \,{\left (b x + a\right )}^{\frac{7}{2}} B a b^{48} + 630 \,{\left (b x + a\right )}^{\frac{5}{2}} B a^{2} b^{48} - 1050 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{3} b^{48} + 1575 \, \sqrt{b x + a} B a^{4} b^{48} + 45 \,{\left (b x + a\right )}^{\frac{7}{2}} A b^{49} - 252 \,{\left (b x + a\right )}^{\frac{5}{2}} A a b^{49} + 630 \,{\left (b x + a\right )}^{\frac{3}{2}} A a^{2} b^{49} - 1260 \, \sqrt{b x + a} A a^{3} b^{49}\right )}}{315 \, b^{54}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)/(b*x+a)^(3/2),x, algorithm="giac")

[Out]

2*(B*a^5 - A*a^4*b)/(sqrt(b*x + a)*b^6) + 2/315*(35*(b*x + a)^(9/2)*B*b^48 - 225*(b*x + a)^(7/2)*B*a*b^48 + 63
0*(b*x + a)^(5/2)*B*a^2*b^48 - 1050*(b*x + a)^(3/2)*B*a^3*b^48 + 1575*sqrt(b*x + a)*B*a^4*b^48 + 45*(b*x + a)^
(7/2)*A*b^49 - 252*(b*x + a)^(5/2)*A*a*b^49 + 630*(b*x + a)^(3/2)*A*a^2*b^49 - 1260*sqrt(b*x + a)*A*a^3*b^49)/
b^54