3.444 \(\int \frac{x^4 (A+B x)}{(a+b x)^{5/2}} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0620635, 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} \[ -\frac{2 a^4 (A b-a B)}{3 b^6 (a+b x)^{3/2}}+\frac{2 a^3 (4 A b-5 a B)}{b^6 \sqrt{a+b x}}+\frac{4 a^2 \sqrt{a+b x} (3 A b-5 a B)}{b^6}-\frac{4 a (a+b x)^{3/2} (2 A b-5 a B)}{3 b^6}+\frac{2 (a+b x)^{5/2} (A b-5 a B)}{5 b^6}+\frac{2 B (a+b x)^{7/2}}{7 b^6} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(A + B*x))/(a + b*x)^(5/2),x]

[Out]

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

Mathematica [A]  time = 0.0758364, size = 106, normalized size = 0.72 \[ \frac{32 a^2 b^3 x^2 (21 A+5 B x)+192 a^3 b^2 x (14 A-5 B x)+256 a^4 b (7 A-15 B x)-2560 a^5 B-4 a b^4 x^3 (28 A+15 B x)+6 b^5 x^4 (7 A+5 B x)}{105 b^6 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(A + B*x))/(a + b*x)^(5/2),x]

[Out]

(-2560*a^5*B + 256*a^4*b*(7*A - 15*B*x) + 192*a^3*b^2*x*(14*A - 5*B*x) + 6*b^5*x^4*(7*A + 5*B*x) + 32*a^2*b^3*
x^2*(21*A + 5*B*x) - 4*a*b^4*x^3*(28*A + 15*B*x))/(105*b^6*(a + b*x)^(3/2))

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 119, normalized size = 0.8 \begin{align*}{\frac{30\,{b}^{5}B{x}^{5}+42\,A{x}^{4}{b}^{5}-60\,B{x}^{4}a{b}^{4}-112\,A{x}^{3}a{b}^{4}+160\,B{x}^{3}{a}^{2}{b}^{3}+672\,A{x}^{2}{a}^{2}{b}^{3}-960\,B{x}^{2}{a}^{3}{b}^{2}+2688\,{a}^{3}{b}^{2}Ax-3840\,{a}^{4}bBx+1792\,A{a}^{4}b-2560\,B{a}^{5}}{105\,{b}^{6}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(B*x+A)/(b*x+a)^(5/2),x)

[Out]

2/105/(b*x+a)^(3/2)*(15*B*b^5*x^5+21*A*b^5*x^4-30*B*a*b^4*x^4-56*A*a*b^4*x^3+80*B*a^2*b^3*x^3+336*A*a^2*b^3*x^
2-480*B*a^3*b^2*x^2+1344*A*a^3*b^2*x-1920*B*a^4*b*x+896*A*a^4*b-1280*B*a^5)/b^6

________________________________________________________________________________________

Maxima [A]  time = 1.51976, size = 174, normalized size = 1.18 \begin{align*} \frac{2 \,{\left (\frac{15 \,{\left (b x + a\right )}^{\frac{7}{2}} B - 21 \,{\left (5 \, B a - A b\right )}{\left (b x + a\right )}^{\frac{5}{2}} + 70 \,{\left (5 \, B a^{2} - 2 \, A a b\right )}{\left (b x + a\right )}^{\frac{3}{2}} - 210 \,{\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} \sqrt{b x + a}}{b} + \frac{35 \,{\left (B a^{5} - A a^{4} b - 3 \,{\left (5 \, B a^{4} - 4 \, A a^{3} b\right )}{\left (b x + a\right )}\right )}}{{\left (b x + a\right )}^{\frac{3}{2}} b}\right )}}{105 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)/(b*x+a)^(5/2),x, algorithm="maxima")

[Out]

2/105*((15*(b*x + a)^(7/2)*B - 21*(5*B*a - A*b)*(b*x + a)^(5/2) + 70*(5*B*a^2 - 2*A*a*b)*(b*x + a)^(3/2) - 210
*(5*B*a^3 - 3*A*a^2*b)*sqrt(b*x + a))/b + 35*(B*a^5 - A*a^4*b - 3*(5*B*a^4 - 4*A*a^3*b)*(b*x + a))/((b*x + a)^
(3/2)*b))/b^5

________________________________________________________________________________________

Fricas [A]  time = 2.34605, size = 316, normalized size = 2.15 \begin{align*} \frac{2 \,{\left (15 \, B b^{5} x^{5} - 1280 \, B a^{5} + 896 \, A a^{4} b - 3 \,{\left (10 \, B a b^{4} - 7 \, A b^{5}\right )} x^{4} + 8 \,{\left (10 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} x^{3} - 48 \,{\left (10 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{2} - 192 \,{\left (10 \, B a^{4} b - 7 \, A a^{3} b^{2}\right )} x\right )} \sqrt{b x + a}}{105 \,{\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} 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)^(5/2),x, algorithm="fricas")

[Out]

2/105*(15*B*b^5*x^5 - 1280*B*a^5 + 896*A*a^4*b - 3*(10*B*a*b^4 - 7*A*b^5)*x^4 + 8*(10*B*a^2*b^3 - 7*A*a*b^4)*x
^3 - 48*(10*B*a^3*b^2 - 7*A*a^2*b^3)*x^2 - 192*(10*B*a^4*b - 7*A*a^3*b^2)*x)*sqrt(b*x + a)/(b^8*x^2 + 2*a*b^7*
x + a^2*b^6)

________________________________________________________________________________________

Sympy [A]  time = 18.7949, size = 146, normalized size = 0.99 \begin{align*} \frac{2 B \left (a + b x\right )^{\frac{7}{2}}}{7 b^{6}} + \frac{2 a^{4} \left (- A b + B a\right )}{3 b^{6} \left (a + b x\right )^{\frac{3}{2}}} - \frac{2 a^{3} \left (- 4 A b + 5 B a\right )}{b^{6} \sqrt{a + b x}} + \frac{\left (a + b x\right )^{\frac{5}{2}} \left (2 A b - 10 B a\right )}{5 b^{6}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (- 8 A a b + 20 B a^{2}\right )}{3 b^{6}} + \frac{\sqrt{a + b x} \left (12 A a^{2} b - 20 B a^{3}\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)**(5/2),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.22946, size = 212, normalized size = 1.44 \begin{align*} -\frac{2 \,{\left (15 \,{\left (b x + a\right )} B a^{4} - B a^{5} - 12 \,{\left (b x + a\right )} A a^{3} b + A a^{4} b\right )}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{6}} + \frac{2 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} B b^{36} - 105 \,{\left (b x + a\right )}^{\frac{5}{2}} B a b^{36} + 350 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{2} b^{36} - 1050 \, \sqrt{b x + a} B a^{3} b^{36} + 21 \,{\left (b x + a\right )}^{\frac{5}{2}} A b^{37} - 140 \,{\left (b x + a\right )}^{\frac{3}{2}} A a b^{37} + 630 \, \sqrt{b x + a} A a^{2} b^{37}\right )}}{105 \, b^{42}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)/(b*x+a)^(5/2),x, algorithm="giac")

[Out]

-2/3*(15*(b*x + a)*B*a^4 - B*a^5 - 12*(b*x + a)*A*a^3*b + A*a^4*b)/((b*x + a)^(3/2)*b^6) + 2/105*(15*(b*x + a)
^(7/2)*B*b^36 - 105*(b*x + a)^(5/2)*B*a*b^36 + 350*(b*x + a)^(3/2)*B*a^2*b^36 - 1050*sqrt(b*x + a)*B*a^3*b^36
+ 21*(b*x + a)^(5/2)*A*b^37 - 140*(b*x + a)^(3/2)*A*a*b^37 + 630*sqrt(b*x + a)*A*a^2*b^37)/b^42