3.102 \(\int \frac{x^{13/2}}{(b x+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=164 \[ \frac{32 b^2 x^{7/2}}{63 c^3 \sqrt{b x+c x^2}}-\frac{64 b^3 x^{5/2}}{63 c^4 \sqrt{b x+c x^2}}+\frac{256 b^4 x^{3/2}}{63 c^5 \sqrt{b x+c x^2}}+\frac{512 b^5 \sqrt{x}}{63 c^6 \sqrt{b x+c x^2}}-\frac{20 b x^{9/2}}{63 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{11/2}}{9 c \sqrt{b x+c x^2}} \]

[Out]

(512*b^5*Sqrt[x])/(63*c^6*Sqrt[b*x + c*x^2]) + (256*b^4*x^(3/2))/(63*c^5*Sqrt[b*x + c*x^2]) - (64*b^3*x^(5/2))
/(63*c^4*Sqrt[b*x + c*x^2]) + (32*b^2*x^(7/2))/(63*c^3*Sqrt[b*x + c*x^2]) - (20*b*x^(9/2))/(63*c^2*Sqrt[b*x +
c*x^2]) + (2*x^(11/2))/(9*c*Sqrt[b*x + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.0748355, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {656, 648} \[ \frac{32 b^2 x^{7/2}}{63 c^3 \sqrt{b x+c x^2}}-\frac{64 b^3 x^{5/2}}{63 c^4 \sqrt{b x+c x^2}}+\frac{256 b^4 x^{3/2}}{63 c^5 \sqrt{b x+c x^2}}+\frac{512 b^5 \sqrt{x}}{63 c^6 \sqrt{b x+c x^2}}-\frac{20 b x^{9/2}}{63 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{11/2}}{9 c \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^(13/2)/(b*x + c*x^2)^(3/2),x]

[Out]

(512*b^5*Sqrt[x])/(63*c^6*Sqrt[b*x + c*x^2]) + (256*b^4*x^(3/2))/(63*c^5*Sqrt[b*x + c*x^2]) - (64*b^3*x^(5/2))
/(63*c^4*Sqrt[b*x + c*x^2]) + (32*b^2*x^(7/2))/(63*c^3*Sqrt[b*x + c*x^2]) - (20*b*x^(9/2))/(63*c^2*Sqrt[b*x +
c*x^2]) + (2*x^(11/2))/(9*c*Sqrt[b*x + c*x^2])

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^{13/2}}{\left (b x+c x^2\right )^{3/2}} \, dx &=\frac{2 x^{11/2}}{9 c \sqrt{b x+c x^2}}-\frac{(10 b) \int \frac{x^{11/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{9 c}\\ &=-\frac{20 b x^{9/2}}{63 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{11/2}}{9 c \sqrt{b x+c x^2}}+\frac{\left (80 b^2\right ) \int \frac{x^{9/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{63 c^2}\\ &=\frac{32 b^2 x^{7/2}}{63 c^3 \sqrt{b x+c x^2}}-\frac{20 b x^{9/2}}{63 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{11/2}}{9 c \sqrt{b x+c x^2}}-\frac{\left (32 b^3\right ) \int \frac{x^{7/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{21 c^3}\\ &=-\frac{64 b^3 x^{5/2}}{63 c^4 \sqrt{b x+c x^2}}+\frac{32 b^2 x^{7/2}}{63 c^3 \sqrt{b x+c x^2}}-\frac{20 b x^{9/2}}{63 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{11/2}}{9 c \sqrt{b x+c x^2}}+\frac{\left (128 b^4\right ) \int \frac{x^{5/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{63 c^4}\\ &=\frac{256 b^4 x^{3/2}}{63 c^5 \sqrt{b x+c x^2}}-\frac{64 b^3 x^{5/2}}{63 c^4 \sqrt{b x+c x^2}}+\frac{32 b^2 x^{7/2}}{63 c^3 \sqrt{b x+c x^2}}-\frac{20 b x^{9/2}}{63 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{11/2}}{9 c \sqrt{b x+c x^2}}-\frac{\left (256 b^5\right ) \int \frac{x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{63 c^5}\\ &=\frac{512 b^5 \sqrt{x}}{63 c^6 \sqrt{b x+c x^2}}+\frac{256 b^4 x^{3/2}}{63 c^5 \sqrt{b x+c x^2}}-\frac{64 b^3 x^{5/2}}{63 c^4 \sqrt{b x+c x^2}}+\frac{32 b^2 x^{7/2}}{63 c^3 \sqrt{b x+c x^2}}-\frac{20 b x^{9/2}}{63 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{11/2}}{9 c \sqrt{b x+c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0357231, size = 75, normalized size = 0.46 \[ \frac{2 \sqrt{x} \left (-32 b^3 c^2 x^2+16 b^2 c^3 x^3+128 b^4 c x+256 b^5-10 b c^4 x^4+7 c^5 x^5\right )}{63 c^6 \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(13/2)/(b*x + c*x^2)^(3/2),x]

[Out]

(2*Sqrt[x]*(256*b^5 + 128*b^4*c*x - 32*b^3*c^2*x^2 + 16*b^2*c^3*x^3 - 10*b*c^4*x^4 + 7*c^5*x^5))/(63*c^6*Sqrt[
x*(b + c*x)])

________________________________________________________________________________________

Maple [A]  time = 0.051, size = 77, normalized size = 0.5 \begin{align*}{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 7\,{x}^{5}{c}^{5}-10\,b{x}^{4}{c}^{4}+16\,{b}^{2}{x}^{3}{c}^{3}-32\,{b}^{3}{x}^{2}{c}^{2}+128\,{b}^{4}xc+256\,{b}^{5} \right ) }{63\,{c}^{6}}{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^(13/2)/(c*x^2+b*x)^(3/2),x)

[Out]

2/63*(c*x+b)*(7*c^5*x^5-10*b*c^4*x^4+16*b^2*c^3*x^3-32*b^3*c^2*x^2+128*b^4*c*x+256*b^5)*x^(3/2)/c^6/(c*x^2+b*x
)^(3/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \,{\left ({\left (35 \, c^{6} x^{5} - 5 \, b c^{5} x^{4} + 8 \, b^{2} c^{4} x^{3} - 16 \, b^{3} c^{3} x^{2} + 64 \, b^{4} c^{2} x + 128 \, b^{5} c\right )} x^{5} - 2 \,{\left (5 \, b c^{5} x^{5} - 2 \, b^{2} c^{4} x^{4} + 5 \, b^{3} c^{3} x^{3} - 28 \, b^{4} c^{2} x^{2} - 104 \, b^{5} c x - 64 \, b^{6}\right )} x^{4} + 6 \,{\left (3 \, b^{2} c^{4} x^{5} - 2 \, b^{3} c^{3} x^{4} + 11 \, b^{4} c^{2} x^{3} + 40 \, b^{5} c x^{2} + 24 \, b^{6} x\right )} x^{3} - 42 \,{\left (b^{3} c^{3} x^{5} - 2 \, b^{4} c^{2} x^{4} - 7 \, b^{5} c x^{3} - 4 \, b^{6} x^{2}\right )} x^{2} + 210 \,{\left (b^{4} c^{2} x^{5} + 2 \, b^{5} c x^{4} + b^{6} x^{3}\right )} x\right )}}{315 \,{\left (c^{7} x^{5} + b c^{6} x^{4}\right )} \sqrt{c x + b}} - \int \frac{2 \,{\left (b^{5} c x + b^{6}\right )} x}{{\left (c^{7} x^{3} + 2 \, b c^{6} x^{2} + b^{2} c^{5} x\right )} \sqrt{c x + b}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(13/2)/(c*x^2+b*x)^(3/2),x, algorithm="maxima")

[Out]

2/315*((35*c^6*x^5 - 5*b*c^5*x^4 + 8*b^2*c^4*x^3 - 16*b^3*c^3*x^2 + 64*b^4*c^2*x + 128*b^5*c)*x^5 - 2*(5*b*c^5
*x^5 - 2*b^2*c^4*x^4 + 5*b^3*c^3*x^3 - 28*b^4*c^2*x^2 - 104*b^5*c*x - 64*b^6)*x^4 + 6*(3*b^2*c^4*x^5 - 2*b^3*c
^3*x^4 + 11*b^4*c^2*x^3 + 40*b^5*c*x^2 + 24*b^6*x)*x^3 - 42*(b^3*c^3*x^5 - 2*b^4*c^2*x^4 - 7*b^5*c*x^3 - 4*b^6
*x^2)*x^2 + 210*(b^4*c^2*x^5 + 2*b^5*c*x^4 + b^6*x^3)*x)/((c^7*x^5 + b*c^6*x^4)*sqrt(c*x + b)) - integrate(2*(
b^5*c*x + b^6)*x/((c^7*x^3 + 2*b*c^6*x^2 + b^2*c^5*x)*sqrt(c*x + b)), x)

________________________________________________________________________________________

Fricas [A]  time = 1.90933, size = 185, normalized size = 1.13 \begin{align*} \frac{2 \,{\left (7 \, c^{5} x^{5} - 10 \, b c^{4} x^{4} + 16 \, b^{2} c^{3} x^{3} - 32 \, b^{3} c^{2} x^{2} + 128 \, b^{4} c x + 256 \, b^{5}\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{63 \,{\left (c^{7} x^{2} + b c^{6} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(13/2)/(c*x^2+b*x)^(3/2),x, algorithm="fricas")

[Out]

2/63*(7*c^5*x^5 - 10*b*c^4*x^4 + 16*b^2*c^3*x^3 - 32*b^3*c^2*x^2 + 128*b^4*c*x + 256*b^5)*sqrt(c*x^2 + b*x)*sq
rt(x)/(c^7*x^2 + b*c^6*x)

________________________________________________________________________________________

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**(13/2)/(c*x**2+b*x)**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.15987, size = 111, normalized size = 0.68 \begin{align*} -\frac{512 \, b^{\frac{9}{2}}}{63 \, c^{6}} + \frac{2 \,{\left (7 \,{\left (c x + b\right )}^{\frac{9}{2}} - 45 \,{\left (c x + b\right )}^{\frac{7}{2}} b + 126 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{2} - 210 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{3} + 315 \, \sqrt{c x + b} b^{4} + \frac{63 \, b^{5}}{\sqrt{c x + b}}\right )}}{63 \, c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(13/2)/(c*x^2+b*x)^(3/2),x, algorithm="giac")

[Out]

-512/63*b^(9/2)/c^6 + 2/63*(7*(c*x + b)^(9/2) - 45*(c*x + b)^(7/2)*b + 126*(c*x + b)^(5/2)*b^2 - 210*(c*x + b)
^(3/2)*b^3 + 315*sqrt(c*x + b)*b^4 + 63*b^5/sqrt(c*x + b))/c^6