### 3.63 $$\int \frac{x^3}{(a x+b x^2)^{5/2}} \, dx$$

Optimal. Leaf size=23 $\frac{2 x^3}{3 a \left (a x+b x^2\right )^{3/2}}$

[Out]

(2*x^3)/(3*a*(a*x + b*x^2)^(3/2))

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Rubi [A]  time = 0.0076737, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.059, Rules used = {650} $\frac{2 x^3}{3 a \left (a x+b x^2\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[x^3/(a*x + b*x^2)^(5/2),x]

[Out]

(2*x^3)/(3*a*(a*x + b*x^2)^(3/2))

Rule 650

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((p + 1)*(2*c*d - b*e)), 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 + 2*p + 2, 0]

Rubi steps

\begin{align*} \int \frac{x^3}{\left (a x+b x^2\right )^{5/2}} \, dx &=\frac{2 x^3}{3 a \left (a x+b x^2\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0098519, size = 21, normalized size = 0.91 $\frac{2 x^3}{3 a (x (a+b x))^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^3/(a*x + b*x^2)^(5/2),x]

[Out]

(2*x^3)/(3*a*(x*(a + b*x))^(3/2))

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Maple [A]  time = 0.043, size = 25, normalized size = 1.1 \begin{align*}{\frac{2\,{x}^{4} \left ( bx+a \right ) }{3\,a} \left ( b{x}^{2}+ax \right ) ^{-{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x^4*(b*x+a)/a/(b*x^2+a*x)^(5/2)

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Maxima [B]  time = 1.13721, size = 100, normalized size = 4.35 \begin{align*} -\frac{x^{2}}{{\left (b x^{2} + a x\right )}^{\frac{3}{2}} b} - \frac{a x}{3 \,{\left (b x^{2} + a x\right )}^{\frac{3}{2}} b^{2}} + \frac{2 \, x}{3 \, \sqrt{b x^{2} + a x} a b} + \frac{1}{3 \, \sqrt{b x^{2} + a x} b^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2/((b*x^2 + a*x)^(3/2)*b) - 1/3*a*x/((b*x^2 + a*x)^(3/2)*b^2) + 2/3*x/(sqrt(b*x^2 + a*x)*a*b) + 1/3/(sqrt(b
*x^2 + a*x)*b^2)

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Fricas [A]  time = 1.84234, size = 74, normalized size = 3.22 \begin{align*} \frac{2 \, \sqrt{b x^{2} + a x} x}{3 \,{\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.15981, size = 120, normalized size = 5.22 \begin{align*} \frac{2 \,{\left (3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{2} b^{\frac{3}{2}} + 3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} a b + a^{2} \sqrt{b}\right )}}{3 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \sqrt{b} + a\right )}^{3} b^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(3*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2*b^(3/2) + 3*(sqrt(b)*x - sqrt(b*x^2 + a*x))*a*b + a^2*sqrt(b))/(((sqr
t(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) + a)^3*b^2)