3.243 \(\int \frac{x^{9/2} (A+B x)}{(b x+c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=143 \[ \frac{2 x^{5/2} (2 b B-A c)}{3 b c^2 \sqrt{b x+c x^2}}-\frac{8 x^{3/2} (2 b B-A c)}{3 c^3 \sqrt{b x+c x^2}}-\frac{16 b \sqrt{x} (2 b B-A c)}{3 c^4 \sqrt{b x+c x^2}}-\frac{2 x^{9/2} (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]

[Out]

(-2*(b*B - A*c)*x^(9/2))/(3*b*c*(b*x + c*x^2)^(3/2)) - (16*b*(2*b*B - A*c)*Sqrt[x])/(3*c^4*Sqrt[b*x + c*x^2])
- (8*(2*b*B - A*c)*x^(3/2))/(3*c^3*Sqrt[b*x + c*x^2]) + (2*(2*b*B - A*c)*x^(5/2))/(3*b*c^2*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 0.115188, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {788, 656, 648} \[ \frac{2 x^{5/2} (2 b B-A c)}{3 b c^2 \sqrt{b x+c x^2}}-\frac{8 x^{3/2} (2 b B-A c)}{3 c^3 \sqrt{b x+c x^2}}-\frac{16 b \sqrt{x} (2 b B-A c)}{3 c^4 \sqrt{b x+c x^2}}-\frac{2 x^{9/2} (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^(9/2)*(A + B*x))/(b*x + c*x^2)^(5/2),x]

[Out]

(-2*(b*B - A*c)*x^(9/2))/(3*b*c*(b*x + c*x^2)^(3/2)) - (16*b*(2*b*B - A*c)*Sqrt[x])/(3*c^4*Sqrt[b*x + c*x^2])
- (8*(2*b*B - A*c)*x^(3/2))/(3*c^3*Sqrt[b*x + c*x^2]) + (2*(2*b*B - A*c)*x^(5/2))/(3*b*c^2*Sqrt[b*x + c*x^2])

Rule 788

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((g*(c*d - b*e) + c*e*f)*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)*(2*c*d - b*e)), x] - Dist[(e*(m*(g
*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g)))/(c*(p + 1)*(2*c*d - b*e)), Int[(d + e*x)^(m - 1)*(a + b*x +
c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2,
0] && LtQ[p, -1] && GtQ[m, 0]

Rule 656

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[(Simplify[m + p]*(2*c*d - b*e))/(c*(m + 2*p + 1)), In
t[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p], 0]

Rule 648

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c
*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0]

Rubi steps

\begin{align*} \int \frac{x^{9/2} (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 (b B-A c) x^{9/2}}{3 b c \left (b x+c x^2\right )^{3/2}}-\frac{\left (2 \left (\frac{9}{2} (-b B+A c)-\frac{3}{2} (-b B+2 A c)\right )\right ) \int \frac{x^{7/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b c}\\ &=-\frac{2 (b B-A c) x^{9/2}}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac{2 (2 b B-A c) x^{5/2}}{3 b c^2 \sqrt{b x+c x^2}}-\frac{(4 (2 b B-A c)) \int \frac{x^{5/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 c^2}\\ &=-\frac{2 (b B-A c) x^{9/2}}{3 b c \left (b x+c x^2\right )^{3/2}}-\frac{8 (2 b B-A c) x^{3/2}}{3 c^3 \sqrt{b x+c x^2}}+\frac{2 (2 b B-A c) x^{5/2}}{3 b c^2 \sqrt{b x+c x^2}}+\frac{(8 b (2 b B-A c)) \int \frac{x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 c^3}\\ &=-\frac{2 (b B-A c) x^{9/2}}{3 b c \left (b x+c x^2\right )^{3/2}}-\frac{16 b (2 b B-A c) \sqrt{x}}{3 c^4 \sqrt{b x+c x^2}}-\frac{8 (2 b B-A c) x^{3/2}}{3 c^3 \sqrt{b x+c x^2}}+\frac{2 (2 b B-A c) x^{5/2}}{3 b c^2 \sqrt{b x+c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0507561, size = 70, normalized size = 0.49 \[ \frac{2 x^{3/2} \left (8 b^2 c (A-3 B x)-6 b c^2 x (B x-2 A)+c^3 x^2 (3 A+B x)-16 b^3 B\right )}{3 c^4 (x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^(9/2)*(A + B*x))/(b*x + c*x^2)^(5/2),x]

[Out]

(2*x^(3/2)*(-16*b^3*B + 8*b^2*c*(A - 3*B*x) - 6*b*c^2*x*(-2*A + B*x) + c^3*x^2*(3*A + B*x)))/(3*c^4*(x*(b + c*
x))^(3/2))

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Maple [A]  time = 0.007, size = 82, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,cx+2\,b \right ) \left ( B{c}^{3}{x}^{3}+3\,A{x}^{2}{c}^{3}-6\,B{x}^{2}b{c}^{2}+12\,Ab{c}^{2}x-24\,B{b}^{2}cx+8\,A{b}^{2}c-16\,{b}^{3}B \right ) }{3\,{c}^{4}}{x}^{{\frac{5}{2}}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(9/2)*(B*x+A)/(c*x^2+b*x)^(5/2),x)

[Out]

2/3*(c*x+b)*(B*c^3*x^3+3*A*c^3*x^2-6*B*b*c^2*x^2+12*A*b*c^2*x-24*B*b^2*c*x+8*A*b^2*c-16*B*b^3)*x^(5/2)/c^4/(c*
x^2+b*x)^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \,{\left (B c x + B b\right )} \sqrt{c x + b} x^{3}}{3 \,{\left (c^{4} x^{3} + 3 \, b c^{3} x^{2} + 3 \, b^{2} c^{2} x + b^{3} c\right )}} + \int \frac{{\left (A b c x^{3} -{\left (2 \, B b^{2} +{\left (2 \, B b c - A c^{2}\right )} x\right )} x^{3}\right )} \sqrt{c x + b}}{c^{5} x^{5} + 4 \, b c^{4} x^{4} + 6 \, b^{2} c^{3} x^{3} + 4 \, b^{3} c^{2} x^{2} + b^{4} c x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(9/2)*(B*x+A)/(c*x^2+b*x)^(5/2),x, algorithm="maxima")

[Out]

2/3*(B*c*x + B*b)*sqrt(c*x + b)*x^3/(c^4*x^3 + 3*b*c^3*x^2 + 3*b^2*c^2*x + b^3*c) + integrate((A*b*c*x^3 - (2*
B*b^2 + (2*B*b*c - A*c^2)*x)*x^3)*sqrt(c*x + b)/(c^5*x^5 + 4*b*c^4*x^4 + 6*b^2*c^3*x^3 + 4*b^3*c^2*x^2 + b^4*c
*x), x)

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Fricas [A]  time = 1.8013, size = 215, normalized size = 1.5 \begin{align*} \frac{2 \,{\left (B c^{3} x^{3} - 16 \, B b^{3} + 8 \, A b^{2} c - 3 \,{\left (2 \, B b c^{2} - A c^{3}\right )} x^{2} - 12 \,{\left (2 \, B b^{2} c - A b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{3 \,{\left (c^{6} x^{3} + 2 \, b c^{5} x^{2} + b^{2} c^{4} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(9/2)*(B*x+A)/(c*x^2+b*x)^(5/2),x, algorithm="fricas")

[Out]

2/3*(B*c^3*x^3 - 16*B*b^3 + 8*A*b^2*c - 3*(2*B*b*c^2 - A*c^3)*x^2 - 12*(2*B*b^2*c - A*b*c^2)*x)*sqrt(c*x^2 + b
*x)*sqrt(x)/(c^6*x^3 + 2*b*c^5*x^2 + b^2*c^4*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(9/2)*(B*x+A)/(c*x**2+b*x)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.1618, size = 136, normalized size = 0.95 \begin{align*} \frac{2 \,{\left ({\left (c x + b\right )}^{\frac{3}{2}} B - 9 \, \sqrt{c x + b} B b + 3 \, \sqrt{c x + b} A c - \frac{9 \,{\left (c x + b\right )} B b^{2} - B b^{3} - 6 \,{\left (c x + b\right )} A b c + A b^{2} c}{{\left (c x + b\right )}^{\frac{3}{2}}}\right )}}{3 \, c^{4}} + \frac{16 \,{\left (2 \, B b^{2} - A b c\right )}}{3 \, \sqrt{b} c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(9/2)*(B*x+A)/(c*x^2+b*x)^(5/2),x, algorithm="giac")

[Out]

2/3*((c*x + b)^(3/2)*B - 9*sqrt(c*x + b)*B*b + 3*sqrt(c*x + b)*A*c - (9*(c*x + b)*B*b^2 - B*b^3 - 6*(c*x + b)*
A*b*c + A*b^2*c)/(c*x + b)^(3/2))/c^4 + 16/3*(2*B*b^2 - A*b*c)/(sqrt(b)*c^4)