### 3.106 $$\int \frac{x^{5/2}}{(b x+c x^2)^{3/2}} \, dx$$

Optimal. Leaf size=48 $\frac{4 b \sqrt{x}}{c^2 \sqrt{b x+c x^2}}+\frac{2 x^{3/2}}{c \sqrt{b x+c x^2}}$

[Out]

(4*b*Sqrt[x])/(c^2*Sqrt[b*x + c*x^2]) + (2*x^(3/2))/(c*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 0.0181612, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.105, Rules used = {656, 648} $\frac{4 b \sqrt{x}}{c^2 \sqrt{b x+c x^2}}+\frac{2 x^{3/2}}{c \sqrt{b x+c x^2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[x^(5/2)/(b*x + c*x^2)^(3/2),x]

[Out]

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

Mathematica [A]  time = 0.0129798, size = 28, normalized size = 0.58 $\frac{2 \sqrt{x} (2 b+c x)}{c^2 \sqrt{x (b+c x)}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^(5/2)/(b*x + c*x^2)^(3/2),x]

[Out]

(2*Sqrt[x]*(2*b + c*x))/(c^2*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.045, size = 32, normalized size = 0.7 \begin{align*} 2\,{\frac{ \left ( cx+b \right ) \left ( cx+2\,b \right ){x}^{3/2}}{{c}^{2} \left ( c{x}^{2}+bx \right ) ^{3/2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)/(c*x^2+b*x)^(3/2),x)

[Out]

2*(c*x+b)*(c*x+2*b)*x^(3/2)/c^2/(c*x^2+b*x)^(3/2)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)/(c*x^2+b*x)^(3/2),x, algorithm="maxima")

[Out]

2*sqrt(c*x + b)*x/(c^2*x + b*c) - integrate(2*(b*c*x + b^2)*x/((c^3*x^3 + 2*b*c^2*x^2 + b^2*c*x)*sqrt(c*x + b)
), x)

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Fricas [A]  time = 2.02542, size = 82, normalized size = 1.71 \begin{align*} \frac{2 \, \sqrt{c x^{2} + b x}{\left (c x + 2 \, b\right )} \sqrt{x}}{c^{3} x^{2} + b c^{2} x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)/(c*x^2+b*x)^(3/2),x, algorithm="fricas")

[Out]

2*sqrt(c*x^2 + b*x)*(c*x + 2*b)*sqrt(x)/(c^3*x^2 + b*c^2*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{5}{2}}}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)/(c*x**2+b*x)**(3/2),x)

[Out]

Integral(x**(5/2)/(x*(b + c*x))**(3/2), x)

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Giac [A]  time = 1.2398, size = 42, normalized size = 0.88 \begin{align*} \frac{2 \,{\left (\sqrt{c x + b} + \frac{b}{\sqrt{c x + b}}\right )}}{c^{2}} - \frac{4 \, \sqrt{b}}{c^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)/(c*x^2+b*x)^(3/2),x, algorithm="giac")

[Out]

2*(sqrt(c*x + b) + b/sqrt(c*x + b))/c^2 - 4*sqrt(b)/c^2