\(\int \frac {(a x+\sqrt {-b x+a^2 x^2})^{3/4}}{\sqrt {-b x+a^2 x^2}} \, dx\) [2059]

Optimal result

Integrand size = 41, antiderivative size = 147 \[ \int \frac {\left (a x+\sqrt {-b x+a^2 x^2}\right )^{3/4}}{\sqrt {-b x+a^2 x^2}} \, dx=\frac {4 \left (a x+\sqrt {-b x+a^2 x^2}\right )^{3/4}}{3 a}+\frac {\sqrt [4]{2} b^{3/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{a x+\sqrt {-b x+a^2 x^2}}}{\sqrt [4]{b}}\right )}{a^{7/4}}-\frac {\sqrt [4]{2} b^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{a x+\sqrt {-b x+a^2 x^2}}}{\sqrt [4]{b}}\right )}{a^{7/4}} \]

[Out]

4/3*(a*x+(a^2*x^2-b*x)^(1/2))^(3/4)/a+2^(1/4)*b^(3/4)*arctan(2^(1/4)*a^(1/4)*(a*x+(a^2*x^2-b*x)^(1/2))^(1/4)/b
^(1/4))/a^(7/4)-2^(1/4)*b^(3/4)*arctanh(2^(1/4)*a^(1/4)*(a*x+(a^2*x^2-b*x)^(1/2))^(1/4)/b^(1/4))/a^(7/4)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {2146, 52, 65, 304, 211, 214} \[ \int \frac {\left (a x+\sqrt {-b x+a^2 x^2}\right )^{3/4}}{\sqrt {-b x+a^2 x^2}} \, dx=\frac {4 \left (\sqrt {a^2 x^2-b x}+a x\right )^{3/4}}{3 a}+\frac {\sqrt [4]{2} b^{3/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{\sqrt {a^2 x^2-b x}+a x}}{\sqrt [4]{b}}\right )}{a^{7/4}}-\frac {\sqrt [4]{2} b^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{\sqrt {a^2 x^2-b x}+a x}}{\sqrt [4]{b}}\right )}{a^{7/4}} \]

[In]

Int[(a*x + Sqrt[-(b*x) + a^2*x^2])^(3/4)/Sqrt[-(b*x) + a^2*x^2],x]

[Out]

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

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 214

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

Rule 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 2146

Int[((g_.) + (h_.)*(x_) + (i_.)*(x_)^2)^(m_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)
^2])^(n_.), x_Symbol] :> Dist[(2/f^(2*m))*(i/c)^m, Subst[Int[x^n*((d^2*e - (b*d - a*e)*f^2 - (2*d*e - b*f^2)*x
+ e*x^2)^(2*m + 1)/(-2*d*e + b*f^2 + 2*e*x)^(2*(m + 1))), x], x, d + e*x + f*Sqrt[a + b*x + c*x^2]], x] /; Fr
eeQ[{a, b, c, d, e, f, g, h, i, n}, x] && EqQ[e^2 - c*f^2, 0] && EqQ[c*g - a*i, 0] && EqQ[c*h - b*i, 0] && Int
egerQ[2*m] && (IntegerQ[m] || GtQ[i/c, 0])

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^{3/4}}{-b+2 a x} \, dx,x,a x+\sqrt {-b x+a^2 x^2}\right ) \\ & = \frac {4 \left (a x+\sqrt {-b x+a^2 x^2}\right )^{3/4}}{3 a}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt [4]{x} (-b+2 a x)} \, dx,x,a x+\sqrt {-b x+a^2 x^2}\right )}{a} \\ & = \frac {4 \left (a x+\sqrt {-b x+a^2 x^2}\right )^{3/4}}{3 a}+\frac {(4 b) \text {Subst}\left (\int \frac {x^2}{-b+2 a x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b x+a^2 x^2}}\right )}{a} \\ & = \frac {4 \left (a x+\sqrt {-b x+a^2 x^2}\right )^{3/4}}{3 a}-\frac {\left (\sqrt {2} b\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {2} \sqrt {a} x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b x+a^2 x^2}}\right )}{a^{3/2}}+\frac {\left (\sqrt {2} b\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {2} \sqrt {a} x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b x+a^2 x^2}}\right )}{a^{3/2}} \\ & = \frac {4 \left (a x+\sqrt {-b x+a^2 x^2}\right )^{3/4}}{3 a}+\frac {\sqrt [4]{2} b^{3/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{a x+\sqrt {-b x+a^2 x^2}}}{\sqrt [4]{b}}\right )}{a^{7/4}}-\frac {\sqrt [4]{2} b^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{a x+\sqrt {-b x+a^2 x^2}}}{\sqrt [4]{b}}\right )}{a^{7/4}} \\ \end{align*}

Mathematica [F]

\[ \int \frac {\left (a x+\sqrt {-b x+a^2 x^2}\right )^{3/4}}{\sqrt {-b x+a^2 x^2}} \, dx=\int \frac {\left (a x+\sqrt {-b x+a^2 x^2}\right )^{3/4}}{\sqrt {-b x+a^2 x^2}} \, dx \]

[In]

Integrate[(a*x + Sqrt[-(b*x) + a^2*x^2])^(3/4)/Sqrt[-(b*x) + a^2*x^2],x]

[Out]

Integrate[(a*x + Sqrt[-(b*x) + a^2*x^2])^(3/4)/Sqrt[-(b*x) + a^2*x^2], x]

Maple [F]

\[\int \frac {\left (a x +\sqrt {a^{2} x^{2}-b x}\right )^{\frac {3}{4}}}{\sqrt {a^{2} x^{2}-b x}}d x\]

[In]

int((a*x+(a^2*x^2-b*x)^(1/2))^(3/4)/(a^2*x^2-b*x)^(1/2),x)

[Out]

int((a*x+(a^2*x^2-b*x)^(1/2))^(3/4)/(a^2*x^2-b*x)^(1/2),x)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.77 \[ \int \frac {\left (a x+\sqrt {-b x+a^2 x^2}\right )^{3/4}}{\sqrt {-b x+a^2 x^2}} \, dx=-\frac {3 \, \left (\frac {1}{8}\right )^{\frac {1}{4}} a \left (\frac {b^{3}}{a^{7}}\right )^{\frac {1}{4}} \log \left (4 \, \left (\frac {1}{8}\right )^{\frac {3}{4}} a^{5} \left (\frac {b^{3}}{a^{7}}\right )^{\frac {3}{4}} + {\left (a x + \sqrt {a^{2} x^{2} - b x}\right )}^{\frac {1}{4}} b^{2}\right ) - 3 i \, \left (\frac {1}{8}\right )^{\frac {1}{4}} a \left (\frac {b^{3}}{a^{7}}\right )^{\frac {1}{4}} \log \left (4 i \, \left (\frac {1}{8}\right )^{\frac {3}{4}} a^{5} \left (\frac {b^{3}}{a^{7}}\right )^{\frac {3}{4}} + {\left (a x + \sqrt {a^{2} x^{2} - b x}\right )}^{\frac {1}{4}} b^{2}\right ) + 3 i \, \left (\frac {1}{8}\right )^{\frac {1}{4}} a \left (\frac {b^{3}}{a^{7}}\right )^{\frac {1}{4}} \log \left (-4 i \, \left (\frac {1}{8}\right )^{\frac {3}{4}} a^{5} \left (\frac {b^{3}}{a^{7}}\right )^{\frac {3}{4}} + {\left (a x + \sqrt {a^{2} x^{2} - b x}\right )}^{\frac {1}{4}} b^{2}\right ) - 3 \, \left (\frac {1}{8}\right )^{\frac {1}{4}} a \left (\frac {b^{3}}{a^{7}}\right )^{\frac {1}{4}} \log \left (-4 \, \left (\frac {1}{8}\right )^{\frac {3}{4}} a^{5} \left (\frac {b^{3}}{a^{7}}\right )^{\frac {3}{4}} + {\left (a x + \sqrt {a^{2} x^{2} - b x}\right )}^{\frac {1}{4}} b^{2}\right ) - 4 \, {\left (a x + \sqrt {a^{2} x^{2} - b x}\right )}^{\frac {3}{4}}}{3 \, a} \]

[In]

integrate((a*x+(a^2*x^2-b*x)^(1/2))^(3/4)/(a^2*x^2-b*x)^(1/2),x, algorithm="fricas")

[Out]

-1/3*(3*(1/8)^(1/4)*a*(b^3/a^7)^(1/4)*log(4*(1/8)^(3/4)*a^5*(b^3/a^7)^(3/4) + (a*x + sqrt(a^2*x^2 - b*x))^(1/4
)*b^2) - 3*I*(1/8)^(1/4)*a*(b^3/a^7)^(1/4)*log(4*I*(1/8)^(3/4)*a^5*(b^3/a^7)^(3/4) + (a*x + sqrt(a^2*x^2 - b*x
))^(1/4)*b^2) + 3*I*(1/8)^(1/4)*a*(b^3/a^7)^(1/4)*log(-4*I*(1/8)^(3/4)*a^5*(b^3/a^7)^(3/4) + (a*x + sqrt(a^2*x
^2 - b*x))^(1/4)*b^2) - 3*(1/8)^(1/4)*a*(b^3/a^7)^(1/4)*log(-4*(1/8)^(3/4)*a^5*(b^3/a^7)^(3/4) + (a*x + sqrt(a
^2*x^2 - b*x))^(1/4)*b^2) - 4*(a*x + sqrt(a^2*x^2 - b*x))^(3/4))/a

Sympy [F]

\[ \int \frac {\left (a x+\sqrt {-b x+a^2 x^2}\right )^{3/4}}{\sqrt {-b x+a^2 x^2}} \, dx=\int \frac {\left (a x + \sqrt {a^{2} x^{2} - b x}\right )^{\frac {3}{4}}}{\sqrt {x \left (a^{2} x - b\right )}}\, dx \]

[In]

integrate((a*x+(a**2*x**2-b*x)**(1/2))**(3/4)/(a**2*x**2-b*x)**(1/2),x)

[Out]

Integral((a*x + sqrt(a**2*x**2 - b*x))**(3/4)/sqrt(x*(a**2*x - b)), x)

Maxima [F]

\[ \int \frac {\left (a x+\sqrt {-b x+a^2 x^2}\right )^{3/4}}{\sqrt {-b x+a^2 x^2}} \, dx=\int { \frac {{\left (a x + \sqrt {a^{2} x^{2} - b x}\right )}^{\frac {3}{4}}}{\sqrt {a^{2} x^{2} - b x}} \,d x } \]

[In]

integrate((a*x+(a^2*x^2-b*x)^(1/2))^(3/4)/(a^2*x^2-b*x)^(1/2),x, algorithm="maxima")

[Out]

integrate((a*x + sqrt(a^2*x^2 - b*x))^(3/4)/sqrt(a^2*x^2 - b*x), x)

Giac [F(-1)]

Timed out. \[ \int \frac {\left (a x+\sqrt {-b x+a^2 x^2}\right )^{3/4}}{\sqrt {-b x+a^2 x^2}} \, dx=\text {Timed out} \]

[In]

integrate((a*x+(a^2*x^2-b*x)^(1/2))^(3/4)/(a^2*x^2-b*x)^(1/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a x+\sqrt {-b x+a^2 x^2}\right )^{3/4}}{\sqrt {-b x+a^2 x^2}} \, dx=\int \frac {{\left (a\,x+\sqrt {a^2\,x^2-b\,x}\right )}^{3/4}}{\sqrt {a^2\,x^2-b\,x}} \,d x \]

[In]

int((a*x + (a^2*x^2 - b*x)^(1/2))^(3/4)/(a^2*x^2 - b*x)^(1/2),x)

[Out]

int((a*x + (a^2*x^2 - b*x)^(1/2))^(3/4)/(a^2*x^2 - b*x)^(1/2), x)