### 3.33 $$\int \frac{(a x+b x^2)^{5/2}}{x^7} \, dx$$

Optimal. Leaf size=23 $-\frac{2 \left (a x+b x^2\right )^{7/2}}{7 a x^7}$

[Out]

(-2*(a*x + b*x^2)^(7/2))/(7*a*x^7)

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Rubi [A]  time = 0.0074363, 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} $-\frac{2 \left (a x+b x^2\right )^{7/2}}{7 a x^7}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a*x + b*x^2)^(5/2)/x^7,x]

[Out]

(-2*(a*x + b*x^2)^(7/2))/(7*a*x^7)

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{\left (a x+b x^2\right )^{5/2}}{x^7} \, dx &=-\frac{2 \left (a x+b x^2\right )^{7/2}}{7 a x^7}\\ \end{align*}

Mathematica [A]  time = 0.0154628, size = 21, normalized size = 0.91 $-\frac{2 (x (a+b x))^{7/2}}{7 a x^7}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a*x + b*x^2)^(5/2)/x^7,x]

[Out]

(-2*(x*(a + b*x))^(7/2))/(7*a*x^7)

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Maple [A]  time = 0.044, size = 25, normalized size = 1.1 \begin{align*} -{\frac{2\,bx+2\,a}{7\,{x}^{6}a} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a*x)^(5/2)/x^7,x)

[Out]

-2/7/x^6*(b*x+a)/a*(b*x^2+a*x)^(5/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a*x)^(5/2)/x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.9822, size = 100, normalized size = 4.35 \begin{align*} -\frac{2 \,{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \sqrt{b x^{2} + a x}}{7 \, a x^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a*x)^(5/2)/x^7,x, algorithm="fricas")

[Out]

-2/7*(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)*sqrt(b*x^2 + a*x)/(a*x^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (x \left (a + b x\right )\right )^{\frac{5}{2}}}{x^{7}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a*x)**(5/2)/x**7,x)

[Out]

Integral((x*(a + b*x))**(5/2)/x**7, x)

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Giac [B]  time = 1.22441, size = 259, normalized size = 11.26 \begin{align*} \frac{2 \,{\left (7 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{6} b^{3} + 21 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{5} a b^{\frac{5}{2}} + 35 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{4} a^{2} b^{2} + 35 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{3} a^{3} b^{\frac{3}{2}} + 21 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{2} a^{4} b + 7 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} a^{5} \sqrt{b} + a^{6}\right )}}{7 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a*x)^(5/2)/x^7,x, algorithm="giac")

[Out]

2/7*(7*(sqrt(b)*x - sqrt(b*x^2 + a*x))^6*b^3 + 21*(sqrt(b)*x - sqrt(b*x^2 + a*x))^5*a*b^(5/2) + 35*(sqrt(b)*x
- sqrt(b*x^2 + a*x))^4*a^2*b^2 + 35*(sqrt(b)*x - sqrt(b*x^2 + a*x))^3*a^3*b^(3/2) + 21*(sqrt(b)*x - sqrt(b*x^2
+ a*x))^2*a^4*b + 7*(sqrt(b)*x - sqrt(b*x^2 + a*x))*a^5*sqrt(b) + a^6)/(sqrt(b)*x - sqrt(b*x^2 + a*x))^7