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

Optimal. Leaf size=147 \[ \frac {4 \left (\sqrt {a^2 x^2-b x}+a x\right )^{3/4}}{3 a}+\frac {\sqrt [4]{2} b^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{\sqrt {a^2 x^2-b x}+a x}}{\sqrt [4]{b}}\right )}{a^{7/4}}-\frac {\sqrt [4]{2} b^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{\sqrt {a^2 x^2-b x}+a x}}{\sqrt [4]{b}}\right )}{a^{7/4}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.23, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {2121, 50, 63, 298, 205, 208} \begin {gather*} \frac {4 \left (\sqrt {a^2 x^2-b x}+a x\right )^{3/4}}{3 a}+\frac {\sqrt [4]{2} b^{3/4} \tan ^{-1}\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} \tanh ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[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 50

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 63

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 205

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

Rule 208

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

Rule 298

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 2121

Int[((g_.) + (h_.)*(x_) + (i_.)*(x_)^2)^(m_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)
^2])^(n_.), x_Symbol] :> Dist[(2*(i/c)^m)/f^(2*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*} \int \frac {\left (a x+\sqrt {-b x+a^2 x^2}\right )^{3/4}}{\sqrt {-b x+a^2 x^2}} \, dx &=2 \operatorname {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 \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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} \tan ^{-1}\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} \tanh ^{-1}\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]  time = 1.52, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a x+\sqrt {-b x+a^2 x^2}\right )^{3/4}}{\sqrt {-b x+a^2 x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[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]

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 16.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a x+\sqrt {-b x+a^2 x^2}\right )^{3/4}}{\sqrt {-b x+a^2 x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Could not integrate

________________________________________________________________________________________

fricas [B]  time = 0.50, size = 268, normalized size = 1.82 \begin {gather*} -\frac {12 \, \left (\frac {1}{8}\right )^{\frac {1}{4}} a \left (\frac {b^{3}}{a^{7}}\right )^{\frac {1}{4}} \arctan \left (-\frac {2 \, {\left (\left (\frac {1}{8}\right )^{\frac {1}{4}} {\left (a x + \sqrt {a^{2} x^{2} - b x}\right )}^{\frac {1}{4}} a^{2} b^{2} \left (\frac {b^{3}}{a^{7}}\right )^{\frac {1}{4}} - \left (\frac {1}{8}\right )^{\frac {1}{4}} \sqrt {\sqrt {\frac {1}{2}} a^{3} b^{3} \sqrt {\frac {b^{3}}{a^{7}}} + \sqrt {a x + \sqrt {a^{2} x^{2} - b x}} b^{4}} a^{2} \left (\frac {b^{3}}{a^{7}}\right )^{\frac {1}{4}}\right )}}{b^{3}}\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 ) - 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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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*(12*(1/8)^(1/4)*a*(b^3/a^7)^(1/4)*arctan(-2*((1/8)^(1/4)*(a*x + sqrt(a^2*x^2 - b*x))^(1/4)*a^2*b^2*(b^3/a
^7)^(1/4) - (1/8)^(1/4)*sqrt(sqrt(1/2)*a^3*b^3*sqrt(b^3/a^7) + sqrt(a*x + sqrt(a^2*x^2 - b*x))*b^4)*a^2*(b^3/a
^7)^(1/4))/b^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*(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

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a x +\sqrt {a^{2} x^{2}-b x}\right )^{\frac {3}{4}}}{\sqrt {a^{2} x^{2}-b x}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a\,x+\sqrt {a^2\,x^2-b\,x}\right )}^{3/4}}{\sqrt {a^2\,x^2-b\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________