\(\int \frac {1}{\sqrt {x} (a x+b x^3)^{9/2}} \, dx\) [90]

Optimal result

Integrand size = 19, antiderivative size = 189 \[ \int \frac {1}{\sqrt {x} \left (a x+b x^3\right )^{9/2}} \, dx=\frac {1}{7 a \sqrt {x} \left (a x+b x^3\right )^{7/2}}+\frac {11}{35 a^2 x^{3/2} \left (a x+b x^3\right )^{5/2}}+\frac {33}{35 a^3 x^{5/2} \left (a x+b x^3\right )^{3/2}}+\frac {33}{5 a^4 x^{7/2} \sqrt {a x+b x^3}}-\frac {33 \sqrt {a x+b x^3}}{4 a^5 x^{9/2}}+\frac {99 b \sqrt {a x+b x^3}}{8 a^6 x^{5/2}}-\frac {99 b^2 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b x^3}}\right )}{8 a^{13/2}} \]

[Out]

11/35/a^2/x^(3/2)/(b*x^3+a*x)^(5/2)+33/35/a^3/x^(5/2)/(b*x^3+a*x)^(3/2)-99/8*b^2*arctanh(a^(1/2)*x^(1/2)/(b*x^
3+a*x)^(1/2))/a^(13/2)+1/7/a/(b*x^3+a*x)^(7/2)/x^(1/2)+33/5/a^4/x^(7/2)/(b*x^3+a*x)^(1/2)-33/4*(b*x^3+a*x)^(1/
2)/a^5/x^(9/2)+99/8*b*(b*x^3+a*x)^(1/2)/a^6/x^(5/2)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2048, 2050, 2054, 212} \[ \int \frac {1}{\sqrt {x} \left (a x+b x^3\right )^{9/2}} \, dx=-\frac {99 b^2 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b x^3}}\right )}{8 a^{13/2}}+\frac {99 b \sqrt {a x+b x^3}}{8 a^6 x^{5/2}}-\frac {33 \sqrt {a x+b x^3}}{4 a^5 x^{9/2}}+\frac {33}{5 a^4 x^{7/2} \sqrt {a x+b x^3}}+\frac {33}{35 a^3 x^{5/2} \left (a x+b x^3\right )^{3/2}}+\frac {11}{35 a^2 x^{3/2} \left (a x+b x^3\right )^{5/2}}+\frac {1}{7 a \sqrt {x} \left (a x+b x^3\right )^{7/2}} \]

[In]

Int[1/(Sqrt[x]*(a*x + b*x^3)^(9/2)),x]

[Out]

1/(7*a*Sqrt[x]*(a*x + b*x^3)^(7/2)) + 11/(35*a^2*x^(3/2)*(a*x + b*x^3)^(5/2)) + 33/(35*a^3*x^(5/2)*(a*x + b*x^
3)^(3/2)) + 33/(5*a^4*x^(7/2)*Sqrt[a*x + b*x^3]) - (33*Sqrt[a*x + b*x^3])/(4*a^5*x^(9/2)) + (99*b*Sqrt[a*x + b
*x^3])/(8*a^6*x^(5/2)) - (99*b^2*ArcTanh[(Sqrt[a]*Sqrt[x])/Sqrt[a*x + b*x^3]])/(8*a^(13/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2048

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, m}, x] &&  !IntegerQ[p] && LtQ[0, j, n]
&& (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[p, -1]

Rule 2050

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, m, p}, x] &&  !IntegerQ[p] && LtQ[0, j,
n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[m + j*p + 1, 0]

Rule 2054

Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[-2/(n - j), Subst[Int[1/(1 - a*x^2
), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{7 a \sqrt {x} \left (a x+b x^3\right )^{7/2}}+\frac {11 \int \frac {1}{x^{3/2} \left (a x+b x^3\right )^{7/2}} \, dx}{7 a} \\ & = \frac {1}{7 a \sqrt {x} \left (a x+b x^3\right )^{7/2}}+\frac {11}{35 a^2 x^{3/2} \left (a x+b x^3\right )^{5/2}}+\frac {99 \int \frac {1}{x^{5/2} \left (a x+b x^3\right )^{5/2}} \, dx}{35 a^2} \\ & = \frac {1}{7 a \sqrt {x} \left (a x+b x^3\right )^{7/2}}+\frac {11}{35 a^2 x^{3/2} \left (a x+b x^3\right )^{5/2}}+\frac {33}{35 a^3 x^{5/2} \left (a x+b x^3\right )^{3/2}}+\frac {33 \int \frac {1}{x^{7/2} \left (a x+b x^3\right )^{3/2}} \, dx}{5 a^3} \\ & = \frac {1}{7 a \sqrt {x} \left (a x+b x^3\right )^{7/2}}+\frac {11}{35 a^2 x^{3/2} \left (a x+b x^3\right )^{5/2}}+\frac {33}{35 a^3 x^{5/2} \left (a x+b x^3\right )^{3/2}}+\frac {33}{5 a^4 x^{7/2} \sqrt {a x+b x^3}}+\frac {33 \int \frac {1}{x^{9/2} \sqrt {a x+b x^3}} \, dx}{a^4} \\ & = \frac {1}{7 a \sqrt {x} \left (a x+b x^3\right )^{7/2}}+\frac {11}{35 a^2 x^{3/2} \left (a x+b x^3\right )^{5/2}}+\frac {33}{35 a^3 x^{5/2} \left (a x+b x^3\right )^{3/2}}+\frac {33}{5 a^4 x^{7/2} \sqrt {a x+b x^3}}-\frac {33 \sqrt {a x+b x^3}}{4 a^5 x^{9/2}}-\frac {(99 b) \int \frac {1}{x^{5/2} \sqrt {a x+b x^3}} \, dx}{4 a^5} \\ & = \frac {1}{7 a \sqrt {x} \left (a x+b x^3\right )^{7/2}}+\frac {11}{35 a^2 x^{3/2} \left (a x+b x^3\right )^{5/2}}+\frac {33}{35 a^3 x^{5/2} \left (a x+b x^3\right )^{3/2}}+\frac {33}{5 a^4 x^{7/2} \sqrt {a x+b x^3}}-\frac {33 \sqrt {a x+b x^3}}{4 a^5 x^{9/2}}+\frac {99 b \sqrt {a x+b x^3}}{8 a^6 x^{5/2}}+\frac {\left (99 b^2\right ) \int \frac {1}{\sqrt {x} \sqrt {a x+b x^3}} \, dx}{8 a^6} \\ & = \frac {1}{7 a \sqrt {x} \left (a x+b x^3\right )^{7/2}}+\frac {11}{35 a^2 x^{3/2} \left (a x+b x^3\right )^{5/2}}+\frac {33}{35 a^3 x^{5/2} \left (a x+b x^3\right )^{3/2}}+\frac {33}{5 a^4 x^{7/2} \sqrt {a x+b x^3}}-\frac {33 \sqrt {a x+b x^3}}{4 a^5 x^{9/2}}+\frac {99 b \sqrt {a x+b x^3}}{8 a^6 x^{5/2}}-\frac {\left (99 b^2\right ) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a x+b x^3}}\right )}{8 a^6} \\ & = \frac {1}{7 a \sqrt {x} \left (a x+b x^3\right )^{7/2}}+\frac {11}{35 a^2 x^{3/2} \left (a x+b x^3\right )^{5/2}}+\frac {33}{35 a^3 x^{5/2} \left (a x+b x^3\right )^{3/2}}+\frac {33}{5 a^4 x^{7/2} \sqrt {a x+b x^3}}-\frac {33 \sqrt {a x+b x^3}}{4 a^5 x^{9/2}}+\frac {99 b \sqrt {a x+b x^3}}{8 a^6 x^{5/2}}-\frac {99 b^2 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b x^3}}\right )}{8 a^{13/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.71 \[ \int \frac {1}{\sqrt {x} \left (a x+b x^3\right )^{9/2}} \, dx=\frac {\sqrt {x \left (a+b x^2\right )} \left (\sqrt {a} \left (-70 a^5+385 a^4 b x^2+5808 a^3 b^2 x^4+13398 a^2 b^3 x^6+11550 a b^4 x^8+3465 b^5 x^{10}\right )-3465 b^2 x^4 \left (a+b x^2\right )^{7/2} \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\right )}{280 a^{13/2} x^{9/2} \left (a+b x^2\right )^4} \]

[In]

Integrate[1/(Sqrt[x]*(a*x + b*x^3)^(9/2)),x]

[Out]

(Sqrt[x*(a + b*x^2)]*(Sqrt[a]*(-70*a^5 + 385*a^4*b*x^2 + 5808*a^3*b^2*x^4 + 13398*a^2*b^3*x^6 + 11550*a*b^4*x^
8 + 3465*b^5*x^10) - 3465*b^2*x^4*(a + b*x^2)^(7/2)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]))/(280*a^(13/2)*x^(9/2)*(
a + b*x^2)^4)

Maple [A] (verified)

Time = 2.12 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.31

method	result	size
default	\(-\frac {\sqrt {x \left (b \,x^{2}+a \right )}\, \left (3465 \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right ) b^{5} x^{10} \sqrt {b \,x^{2}+a}-3465 \sqrt {a}\, b^{5} x^{10}+10395 \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right ) a \,b^{4} x^{8} \sqrt {b \,x^{2}+a}-11550 a^{\frac {3}{2}} b^{4} x^{8}+10395 \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right ) a^{2} b^{3} x^{6} \sqrt {b \,x^{2}+a}-13398 a^{\frac {5}{2}} b^{3} x^{6}+3465 \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right ) a^{3} b^{2} x^{4} \sqrt {b \,x^{2}+a}-5808 a^{\frac {7}{2}} b^{2} x^{4}-385 a^{\frac {9}{2}} b \,x^{2}+70 a^{\frac {11}{2}}\right )}{280 a^{\frac {13}{2}} x^{\frac {9}{2}} \left (b \,x^{2}+a \right )^{4}}\)	\(247\)
risch	\(-\frac {\left (b \,x^{2}+a \right ) \left (-19 b \,x^{2}+2 a \right )}{8 a^{6} x^{\frac {7}{2}} \sqrt {x \left (b \,x^{2}+a \right )}}+\frac {\left (-\frac {99 b^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{8 a^{\frac {13}{2}}}-\frac {6311 b^{2} \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{1120 a^{6} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {6311 b^{2} \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{1120 a^{6} \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {13 b \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{140 a^{5} \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )^{3}}-\frac {711 b \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{1120 a^{6} \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}}+\frac {13 b \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{140 a^{5} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )^{3}}-\frac {711 b \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{1120 a^{6} \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}}+\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{112 a^{5} \left (x -\frac {\sqrt {-a b}}{b}\right )^{4}}+\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{112 a^{5} \left (x +\frac {\sqrt {-a b}}{b}\right )^{4}}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {x}}{\sqrt {x \left (b \,x^{2}+a \right )}}\)	\(604\)

[In]

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

[Out]

-1/280*(x*(b*x^2+a))^(1/2)/a^(13/2)*(3465*ln(2*(a^(1/2)*(b*x^2+a)^(1/2)+a)/x)*b^5*x^10*(b*x^2+a)^(1/2)-3465*a^
(1/2)*b^5*x^10+10395*ln(2*(a^(1/2)*(b*x^2+a)^(1/2)+a)/x)*a*b^4*x^8*(b*x^2+a)^(1/2)-11550*a^(3/2)*b^4*x^8+10395
*ln(2*(a^(1/2)*(b*x^2+a)^(1/2)+a)/x)*a^2*b^3*x^6*(b*x^2+a)^(1/2)-13398*a^(5/2)*b^3*x^6+3465*ln(2*(a^(1/2)*(b*x
^2+a)^(1/2)+a)/x)*a^3*b^2*x^4*(b*x^2+a)^(1/2)-5808*a^(7/2)*b^2*x^4-385*a^(9/2)*b*x^2+70*a^(11/2))/x^(9/2)/(b*x
^2+a)^4

Fricas [A] (verification not implemented)

none

Time = 0.95 (sec) , antiderivative size = 422, normalized size of antiderivative = 2.23 \[ \int \frac {1}{\sqrt {x} \left (a x+b x^3\right )^{9/2}} \, dx=\left [\frac {3465 \, {\left (b^{6} x^{13} + 4 \, a b^{5} x^{11} + 6 \, a^{2} b^{4} x^{9} + 4 \, a^{3} b^{3} x^{7} + a^{4} b^{2} x^{5}\right )} \sqrt {a} \log \left (\frac {b x^{3} + 2 \, a x - 2 \, \sqrt {b x^{3} + a x} \sqrt {a} \sqrt {x}}{x^{3}}\right ) + 2 \, {\left (3465 \, a b^{5} x^{10} + 11550 \, a^{2} b^{4} x^{8} + 13398 \, a^{3} b^{3} x^{6} + 5808 \, a^{4} b^{2} x^{4} + 385 \, a^{5} b x^{2} - 70 \, a^{6}\right )} \sqrt {b x^{3} + a x} \sqrt {x}}{560 \, {\left (a^{7} b^{4} x^{13} + 4 \, a^{8} b^{3} x^{11} + 6 \, a^{9} b^{2} x^{9} + 4 \, a^{10} b x^{7} + a^{11} x^{5}\right )}}, \frac {3465 \, {\left (b^{6} x^{13} + 4 \, a b^{5} x^{11} + 6 \, a^{2} b^{4} x^{9} + 4 \, a^{3} b^{3} x^{7} + a^{4} b^{2} x^{5}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{3} + a x} \sqrt {-a}}{a \sqrt {x}}\right ) + {\left (3465 \, a b^{5} x^{10} + 11550 \, a^{2} b^{4} x^{8} + 13398 \, a^{3} b^{3} x^{6} + 5808 \, a^{4} b^{2} x^{4} + 385 \, a^{5} b x^{2} - 70 \, a^{6}\right )} \sqrt {b x^{3} + a x} \sqrt {x}}{280 \, {\left (a^{7} b^{4} x^{13} + 4 \, a^{8} b^{3} x^{11} + 6 \, a^{9} b^{2} x^{9} + 4 \, a^{10} b x^{7} + a^{11} x^{5}\right )}}\right ] \]

[In]

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

[Out]

[1/560*(3465*(b^6*x^13 + 4*a*b^5*x^11 + 6*a^2*b^4*x^9 + 4*a^3*b^3*x^7 + a^4*b^2*x^5)*sqrt(a)*log((b*x^3 + 2*a*
x - 2*sqrt(b*x^3 + a*x)*sqrt(a)*sqrt(x))/x^3) + 2*(3465*a*b^5*x^10 + 11550*a^2*b^4*x^8 + 13398*a^3*b^3*x^6 + 5
808*a^4*b^2*x^4 + 385*a^5*b*x^2 - 70*a^6)*sqrt(b*x^3 + a*x)*sqrt(x))/(a^7*b^4*x^13 + 4*a^8*b^3*x^11 + 6*a^9*b^
2*x^9 + 4*a^10*b*x^7 + a^11*x^5), 1/280*(3465*(b^6*x^13 + 4*a*b^5*x^11 + 6*a^2*b^4*x^9 + 4*a^3*b^3*x^7 + a^4*b
^2*x^5)*sqrt(-a)*arctan(sqrt(b*x^3 + a*x)*sqrt(-a)/(a*sqrt(x))) + (3465*a*b^5*x^10 + 11550*a^2*b^4*x^8 + 13398
*a^3*b^3*x^6 + 5808*a^4*b^2*x^4 + 385*a^5*b*x^2 - 70*a^6)*sqrt(b*x^3 + a*x)*sqrt(x))/(a^7*b^4*x^13 + 4*a^8*b^3
*x^11 + 6*a^9*b^2*x^9 + 4*a^10*b*x^7 + a^11*x^5)]

Sympy [F]

\[ \int \frac {1}{\sqrt {x} \left (a x+b x^3\right )^{9/2}} \, dx=\int \frac {1}{\sqrt {x} \left (x \left (a + b x^{2}\right )\right )^{\frac {9}{2}}}\, dx \]

[In]

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

[Out]

Integral(1/(sqrt(x)*(x*(a + b*x**2))**(9/2)), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {x} \left (a x+b x^3\right )^{9/2}} \, dx=\int { \frac {1}{{\left (b x^{3} + a x\right )}^{\frac {9}{2}} \sqrt {x}} \,d x } \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\sqrt {x} \left (a x+b x^3\right )^{9/2}} \, dx=\frac {99 \, b^{2} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a} a^{6}} + \frac {350 \, {\left (b x^{2} + a\right )}^{3} b^{2} + 70 \, {\left (b x^{2} + a\right )}^{2} a b^{2} + 21 \, {\left (b x^{2} + a\right )} a^{2} b^{2} + 5 \, a^{3} b^{2}}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{6}} + \frac {19 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2} - 21 \, \sqrt {b x^{2} + a} a b^{2}}{8 \, a^{6} b^{2} x^{4}} \]

[In]

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

[Out]

99/8*b^2*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^6) + 1/35*(350*(b*x^2 + a)^3*b^2 + 70*(b*x^2 + a)^2*a*b^
2 + 21*(b*x^2 + a)*a^2*b^2 + 5*a^3*b^2)/((b*x^2 + a)^(7/2)*a^6) + 1/8*(19*(b*x^2 + a)^(3/2)*b^2 - 21*sqrt(b*x^
2 + a)*a*b^2)/(a^6*b^2*x^4)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {x} \left (a x+b x^3\right )^{9/2}} \, dx=\int \frac {1}{\sqrt {x}\,{\left (b\,x^3+a\,x\right )}^{9/2}} \,d x \]

[In]

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

[Out]

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