\(\int \frac {\sqrt {x}}{(a x+b x^3)^{9/2}} \, dx\) [89]

Optimal result

Integrand size = 19, antiderivative size = 152 \[ \int \frac {\sqrt {x}}{\left (a x+b x^3\right )^{9/2}} \, dx=\frac {\sqrt {x}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {2}{7 a^2 \sqrt {x} \left (a x+b x^3\right )^{5/2}}+\frac {16}{21 a^3 x^{3/2} \left (a x+b x^3\right )^{3/2}}+\frac {32}{7 a^4 x^{5/2} \sqrt {a x+b x^3}}-\frac {128 \sqrt {a x+b x^3}}{21 a^5 x^{7/2}}+\frac {256 b \sqrt {a x+b x^3}}{21 a^6 x^{3/2}} \]

[Out]

16/21/a^3/x^(3/2)/(b*x^3+a*x)^(3/2)+2/7/a^2/(b*x^3+a*x)^(5/2)/x^(1/2)+1/7*x^(1/2)/a/(b*x^3+a*x)^(7/2)+32/7/a^4
/x^(5/2)/(b*x^3+a*x)^(1/2)-128/21*(b*x^3+a*x)^(1/2)/a^5/x^(7/2)+256/21*b*(b*x^3+a*x)^(1/2)/a^6/x^(3/2)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2040, 2041, 2039} \[ \int \frac {\sqrt {x}}{\left (a x+b x^3\right )^{9/2}} \, dx=\frac {256 b \sqrt {a x+b x^3}}{21 a^6 x^{3/2}}-\frac {128 \sqrt {a x+b x^3}}{21 a^5 x^{7/2}}+\frac {32}{7 a^4 x^{5/2} \sqrt {a x+b x^3}}+\frac {16}{21 a^3 x^{3/2} \left (a x+b x^3\right )^{3/2}}+\frac {2}{7 a^2 \sqrt {x} \left (a x+b x^3\right )^{5/2}}+\frac {\sqrt {x}}{7 a \left (a x+b x^3\right )^{7/2}} \]

[In]

Int[Sqrt[x]/(a*x + b*x^3)^(9/2),x]

[Out]

Sqrt[x]/(7*a*(a*x + b*x^3)^(7/2)) + 2/(7*a^2*Sqrt[x]*(a*x + b*x^3)^(5/2)) + 16/(21*a^3*x^(3/2)*(a*x + b*x^3)^(
3/2)) + 32/(7*a^4*x^(5/2)*Sqrt[a*x + b*x^3]) - (128*Sqrt[a*x + b*x^3])/(21*a^5*x^(7/2)) + (256*b*Sqrt[a*x + b*
x^3])/(21*a^6*x^(3/2))

Rule 2039

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] &&
NeQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rule 2040

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])

Rule 2041

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*(m + j*p + 1))), x] - Dist[b*((m + n*p + n - j + 1)/(a*c^(n - j)*(m + j*p + 1))
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c,
0])

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {x}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {10 \int \frac {1}{\sqrt {x} \left (a x+b x^3\right )^{7/2}} \, dx}{7 a} \\ & = \frac {\sqrt {x}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {2}{7 a^2 \sqrt {x} \left (a x+b x^3\right )^{5/2}}+\frac {16 \int \frac {1}{x^{3/2} \left (a x+b x^3\right )^{5/2}} \, dx}{7 a^2} \\ & = \frac {\sqrt {x}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {2}{7 a^2 \sqrt {x} \left (a x+b x^3\right )^{5/2}}+\frac {16}{21 a^3 x^{3/2} \left (a x+b x^3\right )^{3/2}}+\frac {32 \int \frac {1}{x^{5/2} \left (a x+b x^3\right )^{3/2}} \, dx}{7 a^3} \\ & = \frac {\sqrt {x}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {2}{7 a^2 \sqrt {x} \left (a x+b x^3\right )^{5/2}}+\frac {16}{21 a^3 x^{3/2} \left (a x+b x^3\right )^{3/2}}+\frac {32}{7 a^4 x^{5/2} \sqrt {a x+b x^3}}+\frac {128 \int \frac {1}{x^{7/2} \sqrt {a x+b x^3}} \, dx}{7 a^4} \\ & = \frac {\sqrt {x}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {2}{7 a^2 \sqrt {x} \left (a x+b x^3\right )^{5/2}}+\frac {16}{21 a^3 x^{3/2} \left (a x+b x^3\right )^{3/2}}+\frac {32}{7 a^4 x^{5/2} \sqrt {a x+b x^3}}-\frac {128 \sqrt {a x+b x^3}}{21 a^5 x^{7/2}}-\frac {(256 b) \int \frac {1}{x^{3/2} \sqrt {a x+b x^3}} \, dx}{21 a^5} \\ & = \frac {\sqrt {x}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {2}{7 a^2 \sqrt {x} \left (a x+b x^3\right )^{5/2}}+\frac {16}{21 a^3 x^{3/2} \left (a x+b x^3\right )^{3/2}}+\frac {32}{7 a^4 x^{5/2} \sqrt {a x+b x^3}}-\frac {128 \sqrt {a x+b x^3}}{21 a^5 x^{7/2}}+\frac {256 b \sqrt {a x+b x^3}}{21 a^6 x^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.52 \[ \int \frac {\sqrt {x}}{\left (a x+b x^3\right )^{9/2}} \, dx=\frac {\sqrt {x} \left (-7 a^5+70 a^4 b x^2+560 a^3 b^2 x^4+1120 a^2 b^3 x^6+896 a b^4 x^8+256 b^5 x^{10}\right )}{21 a^6 \left (x \left (a+b x^2\right )\right )^{7/2}} \]

[In]

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

[Out]

(Sqrt[x]*(-7*a^5 + 70*a^4*b*x^2 + 560*a^3*b^2*x^4 + 1120*a^2*b^3*x^6 + 896*a*b^4*x^8 + 256*b^5*x^10))/(21*a^6*
(x*(a + b*x^2))^(7/2))

Maple [A] (verified)

Time = 2.09 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.53

method	result	size
gosper	\(-\frac {x^{\frac {3}{2}} \left (b \,x^{2}+a \right ) \left (-256 b^{5} x^{10}-896 a \,b^{4} x^{8}-1120 a^{2} b^{3} x^{6}-560 a^{3} b^{2} x^{4}-70 a^{4} b \,x^{2}+7 a^{5}\right )}{21 a^{6} \left (b \,x^{3}+a x \right )^{\frac {9}{2}}}\)	\(81\)
default	\(-\frac {\sqrt {x \left (b \,x^{2}+a \right )}\, \left (-256 b^{5} x^{10}-896 a \,b^{4} x^{8}-1120 a^{2} b^{3} x^{6}-560 a^{3} b^{2} x^{4}-70 a^{4} b \,x^{2}+7 a^{5}\right )}{21 x^{\frac {7}{2}} \left (b \,x^{2}+a \right )^{4} a^{6}}\)	\(83\)
risch	\(-\frac {\left (b \,x^{2}+a \right ) \left (-14 b \,x^{2}+a \right )}{3 a^{6} x^{\frac {5}{2}} \sqrt {x \left (b \,x^{2}+a \right )}}+\frac {\left (b \,x^{2}+a \right ) x^{\frac {3}{2}} \left (158 b^{3} x^{6}+511 a \,b^{2} x^{4}+560 a^{2} b \,x^{2}+210 a^{3}\right ) b^{2}}{21 a^{6} \left (x^{8} b^{4}+4 a \,b^{3} x^{6}+6 a^{2} x^{4} b^{2}+4 a^{3} b \,x^{2}+a^{4}\right ) \sqrt {x \left (b \,x^{2}+a \right )}}\)	\(139\)

[In]

int(x^(1/2)/(b*x^3+a*x)^(9/2),x,method=_RETURNVERBOSE)

[Out]

-1/21*x^(3/2)*(b*x^2+a)*(-256*b^5*x^10-896*a*b^4*x^8-1120*a^2*b^3*x^6-560*a^3*b^2*x^4-70*a^4*b*x^2+7*a^5)/a^6/
(b*x^3+a*x)^(9/2)

Fricas [A] (verification not implemented)

none

Time = 0.67 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {x}}{\left (a x+b x^3\right )^{9/2}} \, dx=\frac {{\left (256 \, b^{5} x^{10} + 896 \, a b^{4} x^{8} + 1120 \, a^{2} b^{3} x^{6} + 560 \, a^{3} b^{2} x^{4} + 70 \, a^{4} b x^{2} - 7 \, a^{5}\right )} \sqrt {b x^{3} + a x} \sqrt {x}}{21 \, {\left (a^{6} b^{4} x^{12} + 4 \, a^{7} b^{3} x^{10} + 6 \, a^{8} b^{2} x^{8} + 4 \, a^{9} b x^{6} + a^{10} x^{4}\right )}} \]

[In]

integrate(x^(1/2)/(b*x^3+a*x)^(9/2),x, algorithm="fricas")

[Out]

1/21*(256*b^5*x^10 + 896*a*b^4*x^8 + 1120*a^2*b^3*x^6 + 560*a^3*b^2*x^4 + 70*a^4*b*x^2 - 7*a^5)*sqrt(b*x^3 + a
*x)*sqrt(x)/(a^6*b^4*x^12 + 4*a^7*b^3*x^10 + 6*a^8*b^2*x^8 + 4*a^9*b*x^6 + a^10*x^4)

Sympy [F]

\[ \int \frac {\sqrt {x}}{\left (a x+b x^3\right )^{9/2}} \, dx=\int \frac {\sqrt {x}}{\left (x \left (a + b x^{2}\right )\right )^{\frac {9}{2}}}\, dx \]

[In]

integrate(x**(1/2)/(b*x**3+a*x)**(9/2),x)

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {x}}{\left (a x+b x^3\right )^{9/2}} \, dx=\int { \frac {\sqrt {x}}{{\left (b x^{3} + a x\right )}^{\frac {9}{2}}} \,d x } \]

[In]

integrate(x^(1/2)/(b*x^3+a*x)^(9/2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {x}}{\left (a x+b x^3\right )^{9/2}} \, dx=\frac {{\left ({\left (x^{2} {\left (\frac {158 \, b^{5} x^{2}}{a^{6}} + \frac {511 \, b^{4}}{a^{5}}\right )} + \frac {560 \, b^{3}}{a^{4}}\right )} x^{2} + \frac {210 \, b^{2}}{a^{3}}\right )} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} - \frac {4 \, {\left (6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} b^{\frac {3}{2}} - 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b^{\frac {3}{2}} + 7 \, a^{2} b^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3} a^{5}} \]

[In]

integrate(x^(1/2)/(b*x^3+a*x)^(9/2),x, algorithm="giac")

[Out]

1/21*((x^2*(158*b^5*x^2/a^6 + 511*b^4/a^5) + 560*b^3/a^4)*x^2 + 210*b^2/a^3)*x/(b*x^2 + a)^(7/2) - 4/3*(6*(sqr
t(b)*x - sqrt(b*x^2 + a))^4*b^(3/2) - 15*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*b^(3/2) + 7*a^2*b^(3/2))/(((sqrt(b)
*x - sqrt(b*x^2 + a))^2 - a)^3*a^5)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {x}}{\left (a x+b x^3\right )^{9/2}} \, dx=\int \frac {\sqrt {x}}{{\left (b\,x^3+a\,x\right )}^{9/2}} \,d x \]

[In]

int(x^(1/2)/(a*x + b*x^3)^(9/2),x)

[Out]

int(x^(1/2)/(a*x + b*x^3)^(9/2), x)