3.1.50 \(\int \frac {x^{5/2}}{(a x+b x^3)^{9/2}} \, dx\)

Optimal. Leaf size=126 \[ -\frac {128 \sqrt {a x+b x^3}}{35 a^5 x^{3/2}}+\frac {64}{35 a^4 \sqrt {x} \sqrt {a x+b x^3}}+\frac {16 \sqrt {x}}{35 a^3 \left (a x+b x^3\right )^{3/2}}+\frac {8 x^{3/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {x^{5/2}}{7 a \left (a x+b x^3\right )^{7/2}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.19, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2015, 2014} \begin {gather*} \frac {8 x^{3/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {16 \sqrt {x}}{35 a^3 \left (a x+b x^3\right )^{3/2}}+\frac {64}{35 a^4 \sqrt {x} \sqrt {a x+b x^3}}-\frac {128 \sqrt {a x+b x^3}}{35 a^5 x^{3/2}}+\frac {x^{5/2}}{7 a \left (a x+b x^3\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(5/2)/(a*x + b*x^3)^(9/2),x]

[Out]

x^(5/2)/(7*a*(a*x + b*x^3)^(7/2)) + (8*x^(3/2))/(35*a^2*(a*x + b*x^3)^(5/2)) + (16*Sqrt[x])/(35*a^3*(a*x + b*x
^3)^(3/2)) + 64/(35*a^4*Sqrt[x]*Sqrt[a*x + b*x^3]) - (128*Sqrt[a*x + b*x^3])/(35*a^5*x^(3/2))

Rule 2014

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(c^(j - 1)*(c*x)^(m - j
+ 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rule 2015

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(c^(j - 1)*(c*x)^(m - j
+ 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] + Dist[(c^j*(m + n*p + n - j + 1))/(a*(n - j)*(p + 1)),
Int[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, j, m, n}, x] &&  !IntegerQ[p] && NeQ[n, j
] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && LtQ[p, -1] && (IntegerQ[j] || GtQ[c, 0])

Rubi steps

\begin {align*} \int \frac {x^{5/2}}{\left (a x+b x^3\right )^{9/2}} \, dx &=\frac {x^{5/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {8 \int \frac {x^{3/2}}{\left (a x+b x^3\right )^{7/2}} \, dx}{7 a}\\ &=\frac {x^{5/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {8 x^{3/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {48 \int \frac {\sqrt {x}}{\left (a x+b x^3\right )^{5/2}} \, dx}{35 a^2}\\ &=\frac {x^{5/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {8 x^{3/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {16 \sqrt {x}}{35 a^3 \left (a x+b x^3\right )^{3/2}}+\frac {64 \int \frac {1}{\sqrt {x} \left (a x+b x^3\right )^{3/2}} \, dx}{35 a^3}\\ &=\frac {x^{5/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {8 x^{3/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {16 \sqrt {x}}{35 a^3 \left (a x+b x^3\right )^{3/2}}+\frac {64}{35 a^4 \sqrt {x} \sqrt {a x+b x^3}}+\frac {128 \int \frac {1}{x^{3/2} \sqrt {a x+b x^3}} \, dx}{35 a^4}\\ &=\frac {x^{5/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {8 x^{3/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {16 \sqrt {x}}{35 a^3 \left (a x+b x^3\right )^{3/2}}+\frac {64}{35 a^4 \sqrt {x} \sqrt {a x+b x^3}}-\frac {128 \sqrt {a x+b x^3}}{35 a^5 x^{3/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 77, normalized size = 0.61 \begin {gather*} -\frac {\sqrt {x \left (a+b x^2\right )} \left (35 a^4+280 a^3 b x^2+560 a^2 b^2 x^4+448 a b^3 x^6+128 b^4 x^8\right )}{35 a^5 x^{3/2} \left (a+b x^2\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)/(a*x + b*x^3)^(9/2),x]

[Out]

-1/35*(Sqrt[x*(a + b*x^2)]*(35*a^4 + 280*a^3*b*x^2 + 560*a^2*b^2*x^4 + 448*a*b^3*x^6 + 128*b^4*x^8))/(a^5*x^(3
/2)*(a + b*x^2)^4)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 1.35, size = 68, normalized size = 0.54 \begin {gather*} -\frac {x^{5/2} \left (35 a^4+280 a^3 b x^2+560 a^2 b^2 x^4+448 a b^3 x^6+128 b^4 x^8\right )}{35 a^5 \left (a x+b x^3\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^(5/2)/(a*x + b*x^3)^(9/2),x]

[Out]

-1/35*(x^(5/2)*(35*a^4 + 280*a^3*b*x^2 + 560*a^2*b^2*x^4 + 448*a*b^3*x^6 + 128*b^4*x^8))/(a^5*(a*x + b*x^3)^(7
/2))

________________________________________________________________________________________

fricas [A]  time = 0.47, size = 110, normalized size = 0.87 \begin {gather*} -\frac {{\left (128 \, b^{4} x^{8} + 448 \, a b^{3} x^{6} + 560 \, a^{2} b^{2} x^{4} + 280 \, a^{3} b x^{2} + 35 \, a^{4}\right )} \sqrt {b x^{3} + a x} \sqrt {x}}{35 \, {\left (a^{5} b^{4} x^{10} + 4 \, a^{6} b^{3} x^{8} + 6 \, a^{7} b^{2} x^{6} + 4 \, a^{8} b x^{4} + a^{9} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)/(b*x^3+a*x)^(9/2),x, algorithm="fricas")

[Out]

-1/35*(128*b^4*x^8 + 448*a*b^3*x^6 + 560*a^2*b^2*x^4 + 280*a^3*b*x^2 + 35*a^4)*sqrt(b*x^3 + a*x)*sqrt(x)/(a^5*
b^4*x^10 + 4*a^6*b^3*x^8 + 6*a^7*b^2*x^6 + 4*a^8*b*x^4 + a^9*x^2)

________________________________________________________________________________________

giac [A]  time = 0.27, size = 90, normalized size = 0.71 \begin {gather*} -\frac {{\left ({\left (x^{2} {\left (\frac {93 \, b^{4} x^{2}}{a^{5}} + \frac {308 \, b^{3}}{a^{4}}\right )} + \frac {350 \, b^{2}}{a^{3}}\right )} x^{2} + \frac {140 \, b}{a^{2}}\right )} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {2 \, \sqrt {b}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )} a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)/(b*x^3+a*x)^(9/2),x, algorithm="giac")

[Out]

-1/35*((x^2*(93*b^4*x^2/a^5 + 308*b^3/a^4) + 350*b^2/a^3)*x^2 + 140*b/a^2)*x/(b*x^2 + a)^(7/2) + 2*sqrt(b)/(((
sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)*a^4)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 70, normalized size = 0.56 \begin {gather*} -\frac {\left (b \,x^{2}+a \right ) \left (128 x^{8} b^{4}+448 a \,x^{6} b^{3}+560 a^{2} x^{4} b^{2}+280 a^{3} x^{2} b +35 a^{4}\right ) x^{\frac {7}{2}}}{35 \left (b \,x^{3}+a x \right )^{\frac {9}{2}} a^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)/(b*x^3+a*x)^(9/2),x)

[Out]

-1/35*(b*x^2+a)*x^(7/2)*(128*b^4*x^8+448*a*b^3*x^6+560*a^2*b^2*x^4+280*a^3*b*x^2+35*a^4)/a^5/(b*x^3+a*x)^(9/2)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{\frac {5}{2}}}{{\left (b x^{3} + a x\right )}^{\frac {9}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)/(b*x^3+a*x)^(9/2),x, algorithm="maxima")

[Out]

integrate(x^(5/2)/(b*x^3 + a*x)^(9/2), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^{5/2}}{{\left (b\,x^3+a\,x\right )}^{9/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)/(a*x + b*x^3)^(9/2),x)

[Out]

int(x^(5/2)/(a*x + b*x^3)^(9/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{\frac {5}{2}}}{\left (x \left (a + b x^{2}\right )\right )^{\frac {9}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)/(b*x**3+a*x)**(9/2),x)

[Out]

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

________________________________________________________________________________________