### 3.36 $$\int \frac{(a x+b x^2)^{5/2}}{x^{10}} \, dx$$

Optimal. Leaf size=100 $\frac{32 b^3 \left (a x+b x^2\right )^{7/2}}{3003 a^4 x^7}-\frac{16 b^2 \left (a x+b x^2\right )^{7/2}}{429 a^3 x^8}+\frac{12 b \left (a x+b x^2\right )^{7/2}}{143 a^2 x^9}-\frac{2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}}$

[Out]

(-2*(a*x + b*x^2)^(7/2))/(13*a*x^10) + (12*b*(a*x + b*x^2)^(7/2))/(143*a^2*x^9) - (16*b^2*(a*x + b*x^2)^(7/2))
/(429*a^3*x^8) + (32*b^3*(a*x + b*x^2)^(7/2))/(3003*a^4*x^7)

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Rubi [A]  time = 0.0415272, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.118, Rules used = {658, 650} $\frac{32 b^3 \left (a x+b x^2\right )^{7/2}}{3003 a^4 x^7}-\frac{16 b^2 \left (a x+b x^2\right )^{7/2}}{429 a^3 x^8}+\frac{12 b \left (a x+b x^2\right )^{7/2}}{143 a^2 x^9}-\frac{2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a*x + b*x^2)^(5/2)/x^10,x]

[Out]

(-2*(a*x + b*x^2)^(7/2))/(13*a*x^10) + (12*b*(a*x + b*x^2)^(7/2))/(143*a^2*x^9) - (16*b^2*(a*x + b*x^2)^(7/2))
/(429*a^3*x^8) + (32*b^3*(a*x + b*x^2)^(7/2))/(3003*a^4*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^{10}} \, dx &=-\frac{2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}}-\frac{(6 b) \int \frac{\left (a x+b x^2\right )^{5/2}}{x^9} \, dx}{13 a}\\ &=-\frac{2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}}+\frac{12 b \left (a x+b x^2\right )^{7/2}}{143 a^2 x^9}+\frac{\left (24 b^2\right ) \int \frac{\left (a x+b x^2\right )^{5/2}}{x^8} \, dx}{143 a^2}\\ &=-\frac{2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}}+\frac{12 b \left (a x+b x^2\right )^{7/2}}{143 a^2 x^9}-\frac{16 b^2 \left (a x+b x^2\right )^{7/2}}{429 a^3 x^8}-\frac{\left (16 b^3\right ) \int \frac{\left (a x+b x^2\right )^{5/2}}{x^7} \, dx}{429 a^3}\\ &=-\frac{2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}}+\frac{12 b \left (a x+b x^2\right )^{7/2}}{143 a^2 x^9}-\frac{16 b^2 \left (a x+b x^2\right )^{7/2}}{429 a^3 x^8}+\frac{32 b^3 \left (a x+b x^2\right )^{7/2}}{3003 a^4 x^7}\\ \end{align*}

Mathematica [A]  time = 0.0143286, size = 58, normalized size = 0.58 $\frac{2 (a+b x)^3 \sqrt{x (a+b x)} \left (126 a^2 b x-231 a^3-56 a b^2 x^2+16 b^3 x^3\right )}{3003 a^4 x^7}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a*x + b*x^2)^(5/2)/x^10,x]

[Out]

(2*(a + b*x)^3*Sqrt[x*(a + b*x)]*(-231*a^3 + 126*a^2*b*x - 56*a*b^2*x^2 + 16*b^3*x^3))/(3003*a^4*x^7)

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Maple [A]  time = 0.043, size = 55, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,bx+2\,a \right ) \left ( -16\,{b}^{3}{x}^{3}+56\,a{b}^{2}{x}^{2}-126\,bx{a}^{2}+231\,{a}^{3} \right ) }{3003\,{x}^{9}{a}^{4}} \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^10,x)

[Out]

-2/3003*(b*x+a)*(-16*b^3*x^3+56*a*b^2*x^2-126*a^2*b*x+231*a^3)*(b*x^2+a*x)^(5/2)/x^9/a^4

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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^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.944, size = 185, normalized size = 1.85 \begin{align*} \frac{2 \,{\left (16 \, b^{6} x^{6} - 8 \, a b^{5} x^{5} + 6 \, a^{2} b^{4} x^{4} - 5 \, a^{3} b^{3} x^{3} - 371 \, a^{4} b^{2} x^{2} - 567 \, a^{5} b x - 231 \, a^{6}\right )} \sqrt{b x^{2} + a x}}{3003 \, a^{4} x^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a*x)^(5/2)/x^10,x, algorithm="fricas")

[Out]

2/3003*(16*b^6*x^6 - 8*a*b^5*x^5 + 6*a^2*b^4*x^4 - 5*a^3*b^3*x^3 - 371*a^4*b^2*x^2 - 567*a^5*b*x - 231*a^6)*sq
rt(b*x^2 + a*x)/(a^4*x^7)

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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^{10}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a*x)**(5/2)/x**10,x)

[Out]

Integral((x*(a + b*x))**(5/2)/x**10, x)

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Giac [B]  time = 1.28221, size = 379, normalized size = 3.79 \begin{align*} \frac{2 \,{\left (6006 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{9} b^{\frac{9}{2}} + 36036 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{8} a b^{4} + 99099 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{7} a^{2} b^{\frac{7}{2}} + 161733 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{6} a^{3} b^{3} + 171171 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{5} a^{4} b^{\frac{5}{2}} + 121121 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{4} a^{5} b^{2} + 57057 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{3} a^{6} b^{\frac{3}{2}} + 17199 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{2} a^{7} b + 3003 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} a^{8} \sqrt{b} + 231 \, a^{9}\right )}}{3003 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{13}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a*x)^(5/2)/x^10,x, algorithm="giac")

[Out]

2/3003*(6006*(sqrt(b)*x - sqrt(b*x^2 + a*x))^9*b^(9/2) + 36036*(sqrt(b)*x - sqrt(b*x^2 + a*x))^8*a*b^4 + 99099
*(sqrt(b)*x - sqrt(b*x^2 + a*x))^7*a^2*b^(7/2) + 161733*(sqrt(b)*x - sqrt(b*x^2 + a*x))^6*a^3*b^3 + 171171*(sq
rt(b)*x - sqrt(b*x^2 + a*x))^5*a^4*b^(5/2) + 121121*(sqrt(b)*x - sqrt(b*x^2 + a*x))^4*a^5*b^2 + 57057*(sqrt(b)
*x - sqrt(b*x^2 + a*x))^3*a^6*b^(3/2) + 17199*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2*a^7*b + 3003*(sqrt(b)*x - sqrt
(b*x^2 + a*x))*a^8*sqrt(b) + 231*a^9)/(sqrt(b)*x - sqrt(b*x^2 + a*x))^13