\(\int \frac {(a x+b x^2)^{5/2}}{x^9} \, dx\) [35]

Optimal result

Integrand size = 17, antiderivative size = 74 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^9} \, dx=-\frac {2 \left (a x+b x^2\right )^{7/2}}{11 a x^9}+\frac {8 b \left (a x+b x^2\right )^{7/2}}{99 a^2 x^8}-\frac {16 b^2 \left (a x+b x^2\right )^{7/2}}{693 a^3 x^7} \]

[Out]

-2/11*(b*x^2+a*x)^(7/2)/a/x^9+8/99*b*(b*x^2+a*x)^(7/2)/a^2/x^8-16/693*b^2*(b*x^2+a*x)^(7/2)/a^3/x^7

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {672, 664} \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^9} \, dx=-\frac {16 b^2 \left (a x+b x^2\right )^{7/2}}{693 a^3 x^7}+\frac {8 b \left (a x+b x^2\right )^{7/2}}{99 a^2 x^8}-\frac {2 \left (a x+b x^2\right )^{7/2}}{11 a x^9} \]

[In]

Int[(a*x + b*x^2)^(5/2)/x^9,x]

[Out]

(-2*(a*x + b*x^2)^(7/2))/(11*a*x^9) + (8*b*(a*x + b*x^2)^(7/2))/(99*a^2*x^8) - (16*b^2*(a*x + b*x^2)^(7/2))/(6
93*a^3*x^7)

Rule 664

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]

Rule 672

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (a x+b x^2\right )^{7/2}}{11 a x^9}-\frac {(4 b) \int \frac {\left (a x+b x^2\right )^{5/2}}{x^8} \, dx}{11 a} \\ & = -\frac {2 \left (a x+b x^2\right )^{7/2}}{11 a x^9}+\frac {8 b \left (a x+b x^2\right )^{7/2}}{99 a^2 x^8}+\frac {\left (8 b^2\right ) \int \frac {\left (a x+b x^2\right )^{5/2}}{x^7} \, dx}{99 a^2} \\ & = -\frac {2 \left (a x+b x^2\right )^{7/2}}{11 a x^9}+\frac {8 b \left (a x+b x^2\right )^{7/2}}{99 a^2 x^8}-\frac {16 b^2 \left (a x+b x^2\right )^{7/2}}{693 a^3 x^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.54 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^9} \, dx=-\frac {2 (x (a+b x))^{7/2} \left (63 a^2-28 a b x+8 b^2 x^2\right )}{693 a^3 x^9} \]

[In]

Integrate[(a*x + b*x^2)^(5/2)/x^9,x]

[Out]

(-2*(x*(a + b*x))^(7/2)*(63*a^2 - 28*a*b*x + 8*b^2*x^2))/(693*a^3*x^9)

Maple [A] (verified)

Time = 2.17 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.57

method	result	size
pseudoelliptic	\(-\frac {2 \left (\frac {8}{63} b^{2} x^{2}-\frac {4}{9} a b x +a^{2}\right ) \left (b x +a \right )^{3} \sqrt {x \left (b x +a \right )}}{11 x^{6} a^{3}}\)	\(42\)
gosper	\(-\frac {2 \left (b x +a \right ) \left (8 b^{2} x^{2}-28 a b x +63 a^{2}\right ) \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{693 x^{8} a^{3}}\)	\(44\)
default	\(-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{11 a \,x^{9}}-\frac {4 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{9 a \,x^{8}}+\frac {4 b \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{63 a^{2} x^{7}}\right )}{11 a}\)	\(67\)
trager	\(-\frac {2 \left (8 b^{5} x^{5}-4 a \,b^{4} x^{4}+3 a^{2} b^{3} x^{3}+113 a^{3} b^{2} x^{2}+161 a^{4} b x +63 a^{5}\right ) \sqrt {b \,x^{2}+a x}}{693 x^{6} a^{3}}\)	\(72\)
risch	\(-\frac {2 \left (b x +a \right ) \left (8 b^{5} x^{5}-4 a \,b^{4} x^{4}+3 a^{2} b^{3} x^{3}+113 a^{3} b^{2} x^{2}+161 a^{4} b x +63 a^{5}\right )}{693 x^{5} \sqrt {x \left (b x +a \right )}\, a^{3}}\)	\(75\)

[In]

int((b*x^2+a*x)^(5/2)/x^9,x,method=_RETURNVERBOSE)

[Out]

-2/11*(8/63*b^2*x^2-4/9*a*b*x+a^2)*(b*x+a)^3*(x*(b*x+a))^(1/2)/x^6/a^3

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^9} \, dx=-\frac {2 \, {\left (8 \, b^{5} x^{5} - 4 \, a b^{4} x^{4} + 3 \, a^{2} b^{3} x^{3} + 113 \, a^{3} b^{2} x^{2} + 161 \, a^{4} b x + 63 \, a^{5}\right )} \sqrt {b x^{2} + a x}}{693 \, a^{3} x^{6}} \]

[In]

integrate((b*x^2+a*x)^(5/2)/x^9,x, algorithm="fricas")

[Out]

-2/693*(8*b^5*x^5 - 4*a*b^4*x^4 + 3*a^2*b^3*x^3 + 113*a^3*b^2*x^2 + 161*a^4*b*x + 63*a^5)*sqrt(b*x^2 + a*x)/(a
^3*x^6)

Sympy [F]

\[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^9} \, dx=\int \frac {\left (x \left (a + b x\right )\right )^{\frac {5}{2}}}{x^{9}}\, dx \]

[In]

integrate((b*x**2+a*x)**(5/2)/x**9,x)

[Out]

Integral((x*(a + b*x))**(5/2)/x**9, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (62) = 124\).

Time = 0.20 (sec) , antiderivative size = 156, normalized size of antiderivative = 2.11 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^9} \, dx=-\frac {16 \, \sqrt {b x^{2} + a x} b^{5}}{693 \, a^{3} x} + \frac {8 \, \sqrt {b x^{2} + a x} b^{4}}{693 \, a^{2} x^{2}} - \frac {2 \, \sqrt {b x^{2} + a x} b^{3}}{231 \, a x^{3}} + \frac {5 \, \sqrt {b x^{2} + a x} b^{2}}{693 \, x^{4}} - \frac {5 \, \sqrt {b x^{2} + a x} a b}{792 \, x^{5}} - \frac {5 \, \sqrt {b x^{2} + a x} a^{2}}{88 \, x^{6}} + \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a}{24 \, x^{7}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}}}{3 \, x^{8}} \]

[In]

integrate((b*x^2+a*x)^(5/2)/x^9,x, algorithm="maxima")

[Out]

-16/693*sqrt(b*x^2 + a*x)*b^5/(a^3*x) + 8/693*sqrt(b*x^2 + a*x)*b^4/(a^2*x^2) - 2/231*sqrt(b*x^2 + a*x)*b^3/(a
*x^3) + 5/693*sqrt(b*x^2 + a*x)*b^2/x^4 - 5/792*sqrt(b*x^2 + a*x)*a*b/x^5 - 5/88*sqrt(b*x^2 + a*x)*a^2/x^6 + 5
/24*(b*x^2 + a*x)^(3/2)*a/x^7 - 1/3*(b*x^2 + a*x)^(5/2)/x^8

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (62) = 124\).

Time = 0.29 (sec) , antiderivative size = 252, normalized size of antiderivative = 3.41 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^9} \, dx=\frac {2 \, {\left (924 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{8} b^{4} + 4851 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{7} a b^{\frac {7}{2}} + 11781 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{6} a^{2} b^{3} + 16863 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{5} a^{3} b^{\frac {5}{2}} + 15345 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} a^{4} b^{2} + 9009 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a^{5} b^{\frac {3}{2}} + 3311 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{6} b + 693 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a^{7} \sqrt {b} + 63 \, a^{8}\right )}}{693 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{11}} \]

[In]

integrate((b*x^2+a*x)^(5/2)/x^9,x, algorithm="giac")

[Out]

2/693*(924*(sqrt(b)*x - sqrt(b*x^2 + a*x))^8*b^4 + 4851*(sqrt(b)*x - sqrt(b*x^2 + a*x))^7*a*b^(7/2) + 11781*(s
qrt(b)*x - sqrt(b*x^2 + a*x))^6*a^2*b^3 + 16863*(sqrt(b)*x - sqrt(b*x^2 + a*x))^5*a^3*b^(5/2) + 15345*(sqrt(b)
*x - sqrt(b*x^2 + a*x))^4*a^4*b^2 + 9009*(sqrt(b)*x - sqrt(b*x^2 + a*x))^3*a^5*b^(3/2) + 3311*(sqrt(b)*x - sqr
t(b*x^2 + a*x))^2*a^6*b + 693*(sqrt(b)*x - sqrt(b*x^2 + a*x))*a^7*sqrt(b) + 63*a^8)/(sqrt(b)*x - sqrt(b*x^2 +
a*x))^11

Mupad [B] (verification not implemented)

Time = 10.10 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.66 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^9} \, dx=\frac {8\,b^4\,\sqrt {b\,x^2+a\,x}}{693\,a^2\,x^2}-\frac {226\,b^2\,\sqrt {b\,x^2+a\,x}}{693\,x^4}-\frac {2\,b^3\,\sqrt {b\,x^2+a\,x}}{231\,a\,x^3}-\frac {2\,a^2\,\sqrt {b\,x^2+a\,x}}{11\,x^6}-\frac {16\,b^5\,\sqrt {b\,x^2+a\,x}}{693\,a^3\,x}-\frac {46\,a\,b\,\sqrt {b\,x^2+a\,x}}{99\,x^5} \]

[In]

int((a*x + b*x^2)^(5/2)/x^9,x)

[Out]

(8*b^4*(a*x + b*x^2)^(1/2))/(693*a^2*x^2) - (226*b^2*(a*x + b*x^2)^(1/2))/(693*x^4) - (2*b^3*(a*x + b*x^2)^(1/
2))/(231*a*x^3) - (2*a^2*(a*x + b*x^2)^(1/2))/(11*x^6) - (16*b^5*(a*x + b*x^2)^(1/2))/(693*a^3*x) - (46*a*b*(a
*x + b*x^2)^(1/2))/(99*x^5)