3.37 \(\int \frac{(a x+b x^2)^{5/2}}{x^{11}} \, dx\)

Optimal. Leaf size=126 \[ -\frac{256 b^4 \left (a x+b x^2\right )^{7/2}}{45045 a^5 x^7}+\frac{128 b^3 \left (a x+b x^2\right )^{7/2}}{6435 a^4 x^8}-\frac{32 b^2 \left (a x+b x^2\right )^{7/2}}{715 a^3 x^9}+\frac{16 b \left (a x+b x^2\right )^{7/2}}{195 a^2 x^{10}}-\frac{2 \left (a x+b x^2\right )^{7/2}}{15 a x^{11}} \]

[Out]

(-2*(a*x + b*x^2)^(7/2))/(15*a*x^11) + (16*b*(a*x + b*x^2)^(7/2))/(195*a^2*x^10) - (32*b^2*(a*x + b*x^2)^(7/2)
)/(715*a^3*x^9) + (128*b^3*(a*x + b*x^2)^(7/2))/(6435*a^4*x^8) - (256*b^4*(a*x + b*x^2)^(7/2))/(45045*a^5*x^7)

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Rubi [A]  time = 0.0568929, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {658, 650} \[ -\frac{256 b^4 \left (a x+b x^2\right )^{7/2}}{45045 a^5 x^7}+\frac{128 b^3 \left (a x+b x^2\right )^{7/2}}{6435 a^4 x^8}-\frac{32 b^2 \left (a x+b x^2\right )^{7/2}}{715 a^3 x^9}+\frac{16 b \left (a x+b x^2\right )^{7/2}}{195 a^2 x^{10}}-\frac{2 \left (a x+b x^2\right )^{7/2}}{15 a x^{11}} \]

Antiderivative was successfully verified.

[In]

Int[(a*x + b*x^2)^(5/2)/x^11,x]

[Out]

(-2*(a*x + b*x^2)^(7/2))/(15*a*x^11) + (16*b*(a*x + b*x^2)^(7/2))/(195*a^2*x^10) - (32*b^2*(a*x + b*x^2)^(7/2)
)/(715*a^3*x^9) + (128*b^3*(a*x + b*x^2)^(7/2))/(6435*a^4*x^8) - (256*b^4*(a*x + b*x^2)^(7/2))/(45045*a^5*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^{11}} \, dx &=-\frac{2 \left (a x+b x^2\right )^{7/2}}{15 a x^{11}}-\frac{(8 b) \int \frac{\left (a x+b x^2\right )^{5/2}}{x^{10}} \, dx}{15 a}\\ &=-\frac{2 \left (a x+b x^2\right )^{7/2}}{15 a x^{11}}+\frac{16 b \left (a x+b x^2\right )^{7/2}}{195 a^2 x^{10}}+\frac{\left (16 b^2\right ) \int \frac{\left (a x+b x^2\right )^{5/2}}{x^9} \, dx}{65 a^2}\\ &=-\frac{2 \left (a x+b x^2\right )^{7/2}}{15 a x^{11}}+\frac{16 b \left (a x+b x^2\right )^{7/2}}{195 a^2 x^{10}}-\frac{32 b^2 \left (a x+b x^2\right )^{7/2}}{715 a^3 x^9}-\frac{\left (64 b^3\right ) \int \frac{\left (a x+b x^2\right )^{5/2}}{x^8} \, dx}{715 a^3}\\ &=-\frac{2 \left (a x+b x^2\right )^{7/2}}{15 a x^{11}}+\frac{16 b \left (a x+b x^2\right )^{7/2}}{195 a^2 x^{10}}-\frac{32 b^2 \left (a x+b x^2\right )^{7/2}}{715 a^3 x^9}+\frac{128 b^3 \left (a x+b x^2\right )^{7/2}}{6435 a^4 x^8}+\frac{\left (128 b^4\right ) \int \frac{\left (a x+b x^2\right )^{5/2}}{x^7} \, dx}{6435 a^4}\\ &=-\frac{2 \left (a x+b x^2\right )^{7/2}}{15 a x^{11}}+\frac{16 b \left (a x+b x^2\right )^{7/2}}{195 a^2 x^{10}}-\frac{32 b^2 \left (a x+b x^2\right )^{7/2}}{715 a^3 x^9}+\frac{128 b^3 \left (a x+b x^2\right )^{7/2}}{6435 a^4 x^8}-\frac{256 b^4 \left (a x+b x^2\right )^{7/2}}{45045 a^5 x^7}\\ \end{align*}

Mathematica [A]  time = 0.0166837, size = 69, normalized size = 0.55 \[ -\frac{2 (a+b x)^3 \sqrt{x (a+b x)} \left (1008 a^2 b^2 x^2-1848 a^3 b x+3003 a^4-448 a b^3 x^3+128 b^4 x^4\right )}{45045 a^5 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x + b*x^2)^(5/2)/x^11,x]

[Out]

(-2*(a + b*x)^3*Sqrt[x*(a + b*x)]*(3003*a^4 - 1848*a^3*b*x + 1008*a^2*b^2*x^2 - 448*a*b^3*x^3 + 128*b^4*x^4))/
(45045*a^5*x^8)

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Maple [A]  time = 0.047, size = 66, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,bx+2\,a \right ) \left ( 128\,{b}^{4}{x}^{4}-448\,a{b}^{3}{x}^{3}+1008\,{b}^{2}{x}^{2}{a}^{2}-1848\,x{a}^{3}b+3003\,{a}^{4} \right ) }{45045\,{x}^{10}{a}^{5}} \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^11,x)

[Out]

-2/45045*(b*x+a)*(128*b^4*x^4-448*a*b^3*x^3+1008*a^2*b^2*x^2-1848*a^3*b*x+3003*a^4)*(b*x^2+a*x)^(5/2)/x^10/a^5

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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^11,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.98007, size = 220, normalized size = 1.75 \begin{align*} -\frac{2 \,{\left (128 \, b^{7} x^{7} - 64 \, a b^{6} x^{6} + 48 \, a^{2} b^{5} x^{5} - 40 \, a^{3} b^{4} x^{4} + 35 \, a^{4} b^{3} x^{3} + 4473 \, a^{5} b^{2} x^{2} + 7161 \, a^{6} b x + 3003 \, a^{7}\right )} \sqrt{b x^{2} + a x}}{45045 \, a^{5} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a*x)^(5/2)/x^11,x, algorithm="fricas")

[Out]

-2/45045*(128*b^7*x^7 - 64*a*b^6*x^6 + 48*a^2*b^5*x^5 - 40*a^3*b^4*x^4 + 35*a^4*b^3*x^3 + 4473*a^5*b^2*x^2 + 7
161*a^6*b*x + 3003*a^7)*sqrt(b*x^2 + a*x)/(a^5*x^8)

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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^{11}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a*x)**(5/2)/x**11,x)

[Out]

Integral((x*(a + b*x))**(5/2)/x**11, x)

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Giac [B]  time = 1.1926, size = 419, normalized size = 3.33 \begin{align*} \frac{2 \,{\left (144144 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{10} b^{5} + 960960 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{9} a b^{\frac{9}{2}} + 2934360 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{8} a^{2} b^{4} + 5360355 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{7} a^{3} b^{\frac{7}{2}} + 6451445 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{6} a^{4} b^{3} + 5324319 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{5} a^{5} b^{\frac{5}{2}} + 3042585 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{4} a^{6} b^{2} + 1186185 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{3} a^{7} b^{\frac{3}{2}} + 301455 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{2} a^{8} b + 45045 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} a^{9} \sqrt{b} + 3003 \, a^{10}\right )}}{45045 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a*x)^(5/2)/x^11,x, algorithm="giac")

[Out]

2/45045*(144144*(sqrt(b)*x - sqrt(b*x^2 + a*x))^10*b^5 + 960960*(sqrt(b)*x - sqrt(b*x^2 + a*x))^9*a*b^(9/2) +
2934360*(sqrt(b)*x - sqrt(b*x^2 + a*x))^8*a^2*b^4 + 5360355*(sqrt(b)*x - sqrt(b*x^2 + a*x))^7*a^3*b^(7/2) + 64
51445*(sqrt(b)*x - sqrt(b*x^2 + a*x))^6*a^4*b^3 + 5324319*(sqrt(b)*x - sqrt(b*x^2 + a*x))^5*a^5*b^(5/2) + 3042
585*(sqrt(b)*x - sqrt(b*x^2 + a*x))^4*a^6*b^2 + 1186185*(sqrt(b)*x - sqrt(b*x^2 + a*x))^3*a^7*b^(3/2) + 301455
*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2*a^8*b + 45045*(sqrt(b)*x - sqrt(b*x^2 + a*x))*a^9*sqrt(b) + 3003*a^10)/(sqr
t(b)*x - sqrt(b*x^2 + a*x))^15