\(\int \frac {x^2}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx\) [1650]

Optimal result

Integrand size = 42, antiderivative size = 111 \[ \int \frac {x^2}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\frac {8 \sqrt {-b+a^2 x^2} \left (-4 b x+a^2 x^3\right )}{7 a^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}+\frac {4 \left (32 b^2-81 a^2 b x^2+18 a^4 x^4\right )}{63 a^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}} \]

[Out]

8/7*(a^2*x^2-b)^(1/2)*(a^2*x^3-4*b*x)/a^2/(a*x+(a^2*x^2-b)^(1/2))^(9/4)+4/63*(18*a^4*x^4-81*a^2*b*x^2+32*b^2)/
a^3/(a*x+(a^2*x^2-b)^(1/2))^(9/4)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.84, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2145, 276} \[ \int \frac {x^2}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=-\frac {b^2}{9 a^3 \left (\sqrt {a^2 x^2-b}+a x\right )^{9/4}}-\frac {2 b}{a^3 \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}+\frac {\left (\sqrt {a^2 x^2-b}+a x\right )^{7/4}}{7 a^3} \]

[In]

Int[x^2/(Sqrt[-b + a^2*x^2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4)),x]

[Out]

-1/9*b^2/(a^3*(a*x + Sqrt[-b + a^2*x^2])^(9/4)) - (2*b)/(a^3*(a*x + Sqrt[-b + a^2*x^2])^(1/4)) + (a*x + Sqrt[-
b + a^2*x^2])^(7/4)/(7*a^3)

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2145

Int[(x_)^(p_.)*((g_) + (i_.)*(x_)^2)^(m_.)*((e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_.), x_Symbol] :>
Dist[(1/(2^(2*m + p + 1)*e^(p + 1)*f^(2*m)))*(i/c)^m, Subst[Int[x^(n - 2*m - p - 2)*((-a)*f^2 + x^2)^p*(a*f^2
+ x^2)^(2*m + 1), x], x, e*x + f*Sqrt[a + c*x^2]], x] /; FreeQ[{a, c, e, f, g, i, n}, x] && EqQ[e^2 - c*f^2, 0
] && EqQ[c*g - a*i, 0] && IntegersQ[p, 2*m] && (IntegerQ[m] || GtQ[i/c, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (b+x^2\right )^2}{x^{13/4}} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{4 a^3} \\ & = \frac {\text {Subst}\left (\int \left (\frac {b^2}{x^{13/4}}+\frac {2 b}{x^{5/4}}+x^{3/4}\right ) \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{4 a^3} \\ & = -\frac {b^2}{9 a^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}-\frac {2 b}{a^3 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\frac {\left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4}}{7 a^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.81 \[ \int \frac {x^2}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\frac {4 \left (32 b^2+18 a^3 x^3 \left (a x+\sqrt {-b+a^2 x^2}\right )-9 a b x \left (9 a x+8 \sqrt {-b+a^2 x^2}\right )\right )}{63 a^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}} \]

[In]

Integrate[x^2/(Sqrt[-b + a^2*x^2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4)),x]

[Out]

(4*(32*b^2 + 18*a^3*x^3*(a*x + Sqrt[-b + a^2*x^2]) - 9*a*b*x*(9*a*x + 8*Sqrt[-b + a^2*x^2])))/(63*a^3*(a*x + S
qrt[-b + a^2*x^2])^(9/4))

Maple [F]

\[\int \frac {x^{2}}{\sqrt {a^{2} x^{2}-b}\, \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}}d x\]

[In]

int(x^2/(a^2*x^2-b)^(1/2)/(a*x+(a^2*x^2-b)^(1/2))^(1/4),x)

[Out]

int(x^2/(a^2*x^2-b)^(1/2)/(a*x+(a^2*x^2-b)^(1/2))^(1/4),x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.61 \[ \int \frac {x^2}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=-\frac {4 \, {\left (7 \, a^{3} x^{3} + 24 \, a b x - {\left (7 \, a^{2} x^{2} + 32 \, b\right )} \sqrt {a^{2} x^{2} - b}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}}}{63 \, a^{3} b} \]

[In]

integrate(x^2/(a^2*x^2-b)^(1/2)/(a*x+(a^2*x^2-b)^(1/2))^(1/4),x, algorithm="fricas")

[Out]

-4/63*(7*a^3*x^3 + 24*a*b*x - (7*a^2*x^2 + 32*b)*sqrt(a^2*x^2 - b))*(a*x + sqrt(a^2*x^2 - b))^(3/4)/(a^3*b)

Sympy [F]

\[ \int \frac {x^2}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\int \frac {x^{2}}{\sqrt [4]{a x + \sqrt {a^{2} x^{2} - b}} \sqrt {a^{2} x^{2} - b}}\, dx \]

[In]

integrate(x**2/(a**2*x**2-b)**(1/2)/(a*x+(a**2*x**2-b)**(1/2))**(1/4),x)

[Out]

Integral(x**2/((a*x + sqrt(a**2*x**2 - b))**(1/4)*sqrt(a**2*x**2 - b)), x)

Maxima [F]

\[ \int \frac {x^2}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\int { \frac {x^{2}}{\sqrt {a^{2} x^{2} - b} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate(x^2/(a^2*x^2-b)^(1/2)/(a*x+(a^2*x^2-b)^(1/2))^(1/4),x, algorithm="maxima")

[Out]

integrate(x^2/(sqrt(a^2*x^2 - b)*(a*x + sqrt(a^2*x^2 - b))^(1/4)), x)

Giac [F(-1)]

Timed out. \[ \int \frac {x^2}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\text {Timed out} \]

[In]

integrate(x^2/(a^2*x^2-b)^(1/2)/(a*x+(a^2*x^2-b)^(1/2))^(1/4),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\int \frac {x^2}{{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\,\sqrt {a^2\,x^2-b}} \,d x \]

[In]

int(x^2/((a*x + (a^2*x^2 - b)^(1/2))^(1/4)*(a^2*x^2 - b)^(1/2)),x)

[Out]

int(x^2/((a*x + (a^2*x^2 - b)^(1/2))^(1/4)*(a^2*x^2 - b)^(1/2)), x)