### 3.34 $$\int \frac{(a x+b x^2)^{5/2}}{x^8} \, dx$$

Optimal. Leaf size=48 $\frac{4 b \left (a x+b x^2\right )^{7/2}}{63 a^2 x^7}-\frac{2 \left (a x+b x^2\right )^{7/2}}{9 a x^8}$

[Out]

(-2*(a*x + b*x^2)^(7/2))/(9*a*x^8) + (4*b*(a*x + b*x^2)^(7/2))/(63*a^2*x^7)

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Rubi [A]  time = 0.0173728, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.118, Rules used = {658, 650} $\frac{4 b \left (a x+b x^2\right )^{7/2}}{63 a^2 x^7}-\frac{2 \left (a x+b x^2\right )^{7/2}}{9 a x^8}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a*x + b*x^2)^(5/2)/x^8,x]

[Out]

(-2*(a*x + b*x^2)^(7/2))/(9*a*x^8) + (4*b*(a*x + b*x^2)^(7/2))/(63*a^2*x^7)

Rule 658

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))/((m + p + 1)*(2*c*d - b*e)), x] + Dist[(c*Simplify[m + 2*p + 2])/((m + p + 1)*(2*c*d -
b*e)), Int[(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] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2], 0]

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^8} \, dx &=-\frac{2 \left (a x+b x^2\right )^{7/2}}{9 a x^8}-\frac{(2 b) \int \frac{\left (a x+b x^2\right )^{5/2}}{x^7} \, dx}{9 a}\\ &=-\frac{2 \left (a x+b x^2\right )^{7/2}}{9 a x^8}+\frac{4 b \left (a x+b x^2\right )^{7/2}}{63 a^2 x^7}\\ \end{align*}

Mathematica [A]  time = 0.0107597, size = 36, normalized size = 0.75 $\frac{2 (a+b x)^3 \sqrt{x (a+b x)} (2 b x-7 a)}{63 a^2 x^5}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a*x + b*x^2)^(5/2)/x^8,x]

[Out]

(2*(a + b*x)^3*Sqrt[x*(a + b*x)]*(-7*a + 2*b*x))/(63*a^2*x^5)

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Maple [A]  time = 0.046, size = 33, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,bx+2\,a \right ) \left ( -2\,bx+7\,a \right ) }{63\,{a}^{2}{x}^{7}} \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^8,x)

[Out]

-2/63*(b*x+a)*(-2*b*x+7*a)*(b*x^2+a*x)^(5/2)/a^2/x^7

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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^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.99, size = 130, normalized size = 2.71 \begin{align*} \frac{2 \,{\left (2 \, b^{4} x^{4} - a b^{3} x^{3} - 15 \, a^{2} b^{2} x^{2} - 19 \, a^{3} b x - 7 \, a^{4}\right )} \sqrt{b x^{2} + a x}}{63 \, a^{2} x^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a*x)^(5/2)/x^8,x, algorithm="fricas")

[Out]

2/63*(2*b^4*x^4 - a*b^3*x^3 - 15*a^2*b^2*x^2 - 19*a^3*b*x - 7*a^4)*sqrt(b*x^2 + a*x)/(a^2*x^5)

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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^{8}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a*x)**(5/2)/x**8,x)

[Out]

Integral((x*(a + b*x))**(5/2)/x**8, x)

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Giac [B]  time = 1.21591, size = 301, normalized size = 6.27 \begin{align*} \frac{2 \,{\left (63 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{7} b^{\frac{7}{2}} + 273 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{6} a b^{3} + 567 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{5} a^{2} b^{\frac{5}{2}} + 693 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{4} a^{3} b^{2} + 525 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{3} a^{4} b^{\frac{3}{2}} + 243 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{2} a^{5} b + 63 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} a^{6} \sqrt{b} + 7 \, a^{7}\right )}}{63 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{9}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a*x)^(5/2)/x^8,x, algorithm="giac")

[Out]

2/63*(63*(sqrt(b)*x - sqrt(b*x^2 + a*x))^7*b^(7/2) + 273*(sqrt(b)*x - sqrt(b*x^2 + a*x))^6*a*b^3 + 567*(sqrt(b
)*x - sqrt(b*x^2 + a*x))^5*a^2*b^(5/2) + 693*(sqrt(b)*x - sqrt(b*x^2 + a*x))^4*a^3*b^2 + 525*(sqrt(b)*x - sqrt
(b*x^2 + a*x))^3*a^4*b^(3/2) + 243*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2*a^5*b + 63*(sqrt(b)*x - sqrt(b*x^2 + a*x)
)*a^6*sqrt(b) + 7*a^7)/(sqrt(b)*x - sqrt(b*x^2 + a*x))^9