3.233 \(\int \frac{x^{9/2} (A+B x)}{(b x+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=178 \[ -\frac{32 b^2 \sqrt{b x+c x^2} (8 b B-7 A c)}{35 c^5 \sqrt{x}}+\frac{2 x^{5/2} \sqrt{b x+c x^2} (8 b B-7 A c)}{7 b c^2}-\frac{12 x^{3/2} \sqrt{b x+c x^2} (8 b B-7 A c)}{35 c^3}+\frac{16 b \sqrt{x} \sqrt{b x+c x^2} (8 b B-7 A c)}{35 c^4}-\frac{2 x^{9/2} (b B-A c)}{b c \sqrt{b x+c x^2}} \]

[Out]

(-2*(b*B - A*c)*x^(9/2))/(b*c*Sqrt[b*x + c*x^2]) - (32*b^2*(8*b*B - 7*A*c)*Sqrt[b*x + c*x^2])/(35*c^5*Sqrt[x])
+ (16*b*(8*b*B - 7*A*c)*Sqrt[x]*Sqrt[b*x + c*x^2])/(35*c^4) - (12*(8*b*B - 7*A*c)*x^(3/2)*Sqrt[b*x + c*x^2])/
(35*c^3) + (2*(8*b*B - 7*A*c)*x^(5/2)*Sqrt[b*x + c*x^2])/(7*b*c^2)

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Rubi [A]  time = 0.149217, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {788, 656, 648} \[ -\frac{32 b^2 \sqrt{b x+c x^2} (8 b B-7 A c)}{35 c^5 \sqrt{x}}+\frac{2 x^{5/2} \sqrt{b x+c x^2} (8 b B-7 A c)}{7 b c^2}-\frac{12 x^{3/2} \sqrt{b x+c x^2} (8 b B-7 A c)}{35 c^3}+\frac{16 b \sqrt{x} \sqrt{b x+c x^2} (8 b B-7 A c)}{35 c^4}-\frac{2 x^{9/2} (b B-A c)}{b c \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^(9/2)*(A + B*x))/(b*x + c*x^2)^(3/2),x]

[Out]

(-2*(b*B - A*c)*x^(9/2))/(b*c*Sqrt[b*x + c*x^2]) - (32*b^2*(8*b*B - 7*A*c)*Sqrt[b*x + c*x^2])/(35*c^5*Sqrt[x])
+ (16*b*(8*b*B - 7*A*c)*Sqrt[x]*Sqrt[b*x + c*x^2])/(35*c^4) - (12*(8*b*B - 7*A*c)*x^(3/2)*Sqrt[b*x + c*x^2])/
(35*c^3) + (2*(8*b*B - 7*A*c)*x^(5/2)*Sqrt[b*x + c*x^2])/(7*b*c^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 )^{3/2}} \, dx &=-\frac{2 (b B-A c) x^{9/2}}{b c \sqrt{b x+c x^2}}-\left (\frac{7 A}{b}-\frac{8 B}{c}\right ) \int \frac{x^{7/2}}{\sqrt{b x+c x^2}} \, dx\\ &=-\frac{2 (b B-A c) x^{9/2}}{b c \sqrt{b x+c x^2}}+\frac{2 (8 b B-7 A c) x^{5/2} \sqrt{b x+c x^2}}{7 b c^2}-\frac{(6 (8 b B-7 A c)) \int \frac{x^{5/2}}{\sqrt{b x+c x^2}} \, dx}{7 c^2}\\ &=-\frac{2 (b B-A c) x^{9/2}}{b c \sqrt{b x+c x^2}}-\frac{12 (8 b B-7 A c) x^{3/2} \sqrt{b x+c x^2}}{35 c^3}+\frac{2 (8 b B-7 A c) x^{5/2} \sqrt{b x+c x^2}}{7 b c^2}+\frac{(24 b (8 b B-7 A c)) \int \frac{x^{3/2}}{\sqrt{b x+c x^2}} \, dx}{35 c^3}\\ &=-\frac{2 (b B-A c) x^{9/2}}{b c \sqrt{b x+c x^2}}+\frac{16 b (8 b B-7 A c) \sqrt{x} \sqrt{b x+c x^2}}{35 c^4}-\frac{12 (8 b B-7 A c) x^{3/2} \sqrt{b x+c x^2}}{35 c^3}+\frac{2 (8 b B-7 A c) x^{5/2} \sqrt{b x+c x^2}}{7 b c^2}-\frac{\left (16 b^2 (8 b B-7 A c)\right ) \int \frac{\sqrt{x}}{\sqrt{b x+c x^2}} \, dx}{35 c^4}\\ &=-\frac{2 (b B-A c) x^{9/2}}{b c \sqrt{b x+c x^2}}-\frac{32 b^2 (8 b B-7 A c) \sqrt{b x+c x^2}}{35 c^5 \sqrt{x}}+\frac{16 b (8 b B-7 A c) \sqrt{x} \sqrt{b x+c x^2}}{35 c^4}-\frac{12 (8 b B-7 A c) x^{3/2} \sqrt{b x+c x^2}}{35 c^3}+\frac{2 (8 b B-7 A c) x^{5/2} \sqrt{b x+c x^2}}{7 b c^2}\\ \end{align*}

Mathematica [A]  time = 0.0663672, size = 93, normalized size = 0.52 \[ \frac{2 \sqrt{x} \left (8 b^2 c^2 x (7 A+2 B x)+16 b^3 c (7 A-4 B x)-2 b c^3 x^2 (7 A+4 B x)+c^4 x^3 (7 A+5 B x)-128 b^4 B\right )}{35 c^5 \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^(9/2)*(A + B*x))/(b*x + c*x^2)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.004, size = 107, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 5\,B{x}^{4}{c}^{4}+7\,A{c}^{4}{x}^{3}-8\,Bb{c}^{3}{x}^{3}-14\,Ab{c}^{3}{x}^{2}+16\,B{b}^{2}{c}^{2}{x}^{2}+56\,A{b}^{2}{c}^{2}x-64\,B{b}^{3}cx+112\,A{b}^{3}c-128\,{b}^{4}B \right ) }{35\,{c}^{5}}{x}^{{\frac{3}{2}}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{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)^(3/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \,{\left ({\left (15 \, B c^{5} x^{3} + 3 \, B b c^{4} x^{2} - 4 \, B b^{2} c^{3} x + 8 \, B b^{3} c^{2}\right )} x^{4} +{\left (16 \, B b^{4} c - 3 \,{\left (4 \, B b c^{4} - 7 \, A c^{5}\right )} x^{3} -{\left (8 \, B b^{2} c^{3} - 7 \, A b c^{4}\right )} x^{2} + 2 \,{\left (10 \, B b^{3} c^{2} - 7 \, A b^{2} c^{3}\right )} x\right )} x^{3} + 4 \,{\left (2 \, B b^{5} +{\left (9 \, B b^{2} c^{3} - 7 \, A b c^{4}\right )} x^{3} + 2 \,{\left (10 \, B b^{3} c^{2} - 7 \, A b^{2} c^{3}\right )} x^{2} +{\left (13 \, B b^{4} c - 7 \, A b^{3} c^{2}\right )} x\right )} x^{2}\right )} \sqrt{c x + b}}{105 \,{\left (c^{7} x^{4} + 2 \, b c^{6} x^{3} + b^{2} c^{5} x^{2}\right )}} + \int -\frac{4 \,{\left (4 \, B b^{5} - 2 \, A b^{4} c +{\left (9 \, B b^{3} c^{2} - 7 \, A b^{2} c^{3}\right )} x^{2} +{\left (13 \, B b^{4} c - 9 \, A b^{3} c^{2}\right )} x\right )} \sqrt{c x + b} x^{2}}{15 \,{\left (c^{7} x^{5} + 3 \, b c^{6} x^{4} + 3 \, b^{2} c^{5} x^{3} + b^{3} c^{4} x^{2}\right )}}\,{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)^(3/2),x, algorithm="maxima")

[Out]

2/105*((15*B*c^5*x^3 + 3*B*b*c^4*x^2 - 4*B*b^2*c^3*x + 8*B*b^3*c^2)*x^4 + (16*B*b^4*c - 3*(4*B*b*c^4 - 7*A*c^5
)*x^3 - (8*B*b^2*c^3 - 7*A*b*c^4)*x^2 + 2*(10*B*b^3*c^2 - 7*A*b^2*c^3)*x)*x^3 + 4*(2*B*b^5 + (9*B*b^2*c^3 - 7*
A*b*c^4)*x^3 + 2*(10*B*b^3*c^2 - 7*A*b^2*c^3)*x^2 + (13*B*b^4*c - 7*A*b^3*c^2)*x)*x^2)*sqrt(c*x + b)/(c^7*x^4
+ 2*b*c^6*x^3 + b^2*c^5*x^2) + integrate(-4/15*(4*B*b^5 - 2*A*b^4*c + (9*B*b^3*c^2 - 7*A*b^2*c^3)*x^2 + (13*B*
b^4*c - 9*A*b^3*c^2)*x)*sqrt(c*x + b)*x^2/(c^7*x^5 + 3*b*c^6*x^4 + 3*b^2*c^5*x^3 + b^3*c^4*x^2), x)

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Fricas [A]  time = 1.84342, size = 251, normalized size = 1.41 \begin{align*} \frac{2 \,{\left (5 \, B c^{4} x^{4} - 128 \, B b^{4} + 112 \, A b^{3} c -{\left (8 \, B b c^{3} - 7 \, A c^{4}\right )} x^{3} + 2 \,{\left (8 \, B b^{2} c^{2} - 7 \, A b c^{3}\right )} x^{2} - 8 \,{\left (8 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{35 \,{\left (c^{6} x^{2} + b c^{5} 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)^(3/2),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [A]  time = 1.17118, size = 182, normalized size = 1.02 \begin{align*} \frac{2 \,{\left (5 \,{\left (c x + b\right )}^{\frac{7}{2}} B - 28 \,{\left (c x + b\right )}^{\frac{5}{2}} B b + 70 \,{\left (c x + b\right )}^{\frac{3}{2}} B b^{2} - 140 \, \sqrt{c x + b} B b^{3} + 7 \,{\left (c x + b\right )}^{\frac{5}{2}} A c - 35 \,{\left (c x + b\right )}^{\frac{3}{2}} A b c + 105 \, \sqrt{c x + b} A b^{2} c - \frac{35 \,{\left (B b^{4} - A b^{3} c\right )}}{\sqrt{c x + b}}\right )}}{35 \, c^{5}} + \frac{32 \,{\left (8 \, B b^{4} - 7 \, A b^{3} c\right )}}{35 \, \sqrt{b} c^{5}} \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)^(3/2),x, algorithm="giac")

[Out]

2/35*(5*(c*x + b)^(7/2)*B - 28*(c*x + b)^(5/2)*B*b + 70*(c*x + b)^(3/2)*B*b^2 - 140*sqrt(c*x + b)*B*b^3 + 7*(c
*x + b)^(5/2)*A*c - 35*(c*x + b)^(3/2)*A*b*c + 105*sqrt(c*x + b)*A*b^2*c - 35*(B*b^4 - A*b^3*c)/sqrt(c*x + b))
/c^5 + 32/35*(8*B*b^4 - 7*A*b^3*c)/(sqrt(b)*c^5)