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

Optimal. Leaf size=62 \[ \frac {2 \left (\sqrt {a^2 x^2-b}+a x\right )^{3/4}}{3 a^2}-\frac {2 b}{5 a^2 \left (\sqrt {a^2 x^2-b}+a x\right )^{5/4}} \]

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Rubi [A]  time = 0.25, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {2120, 14} \begin {gather*} \frac {2 \left (\sqrt {a^2 x^2-b}+a x\right )^{3/4}}{3 a^2}-\frac {2 b}{5 a^2 \left (\sqrt {a^2 x^2-b}+a x\right )^{5/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b)/(5*a^2*(a*x + Sqrt[-b + a^2*x^2])^(5/4)) + (2*(a*x + Sqrt[-b + a^2*x^2])^(3/4))/(3*a^2)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2120

Int[(x_)^(p_.)*((g_) + (i_.)*(x_)^2)^(m_.)*((e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_.), x_Symbol] :>
Dist[(1*(i/c)^m)/(2^(2*m + p + 1)*e^(p + 1)*f^(2*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*} \int \frac {x}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {b+x^2}{x^{9/4}} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{2 a^2}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {b}{x^{9/4}}+\frac {1}{\sqrt [4]{x}}\right ) \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{2 a^2}\\ &=-\frac {2 b}{5 a^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}+\frac {2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{3 a^2}\\ \end {align*}

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Mathematica [B]  time = 2.44, size = 510, normalized size = 8.23 \begin {gather*} \frac {4 \sqrt {a^2 x^2-b} \left (20480 a^{15} x^{15}-84992 a^{13} b x^{13}+142592 a^{11} b^2 x^{11}-123392 a^9 b^3 x^9+58080 a^7 b^4 x^7-14308 a^5 b^5 x^5+1593 a^3 b^6 x^3-4 b^7 \sqrt {a^2 x^2-b}+353 a^2 b^6 x^2 \sqrt {a^2 x^2-b}+20480 a^{14} x^{14} \sqrt {a^2 x^2-b}-74752 a^{12} b x^{12} \sqrt {a^2 x^2-b}+107776 a^{10} b^2 x^{10} \sqrt {a^2 x^2-b}-77568 a^8 b^3 x^8 \sqrt {a^2 x^2-b}+28896 a^6 b^4 x^6 \sqrt {a^2 x^2-b}-5180 a^4 b^5 x^4 \sqrt {a^2 x^2-b}-53 a b^7 x\right )}{15 a^2 \left (\sqrt {a^2 x^2-b}+a x\right )^{13/4} \left (a x \left (\sqrt {a^2 x^2-b}+a x\right )-b\right ) \left (a b^4 x \left (9 \sqrt {a^2 x^2-b}+41 a x\right )+256 a^9 x^9 \left (\sqrt {a^2 x^2-b}+a x\right )-64 a^7 b x^7 \left (9 \sqrt {a^2 x^2-b}+11 a x\right )+16 a^5 b^2 x^5 \left (27 \sqrt {a^2 x^2-b}+43 a x\right )-40 a^3 b^3 x^3 \left (3 \sqrt {a^2 x^2-b}+7 a x\right )-b^5\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[-b + a^2*x^2]*(-53*a*b^7*x + 1593*a^3*b^6*x^3 - 14308*a^5*b^5*x^5 + 58080*a^7*b^4*x^7 - 123392*a^9*b^3
*x^9 + 142592*a^11*b^2*x^11 - 84992*a^13*b*x^13 + 20480*a^15*x^15 - 4*b^7*Sqrt[-b + a^2*x^2] + 353*a^2*b^6*x^2
*Sqrt[-b + a^2*x^2] - 5180*a^4*b^5*x^4*Sqrt[-b + a^2*x^2] + 28896*a^6*b^4*x^6*Sqrt[-b + a^2*x^2] - 77568*a^8*b
^3*x^8*Sqrt[-b + a^2*x^2] + 107776*a^10*b^2*x^10*Sqrt[-b + a^2*x^2] - 74752*a^12*b*x^12*Sqrt[-b + a^2*x^2] + 2
0480*a^14*x^14*Sqrt[-b + a^2*x^2]))/(15*a^2*(a*x + Sqrt[-b + a^2*x^2])^(13/4)*(-b + a*x*(a*x + Sqrt[-b + a^2*x
^2]))*(-b^5 + 256*a^9*x^9*(a*x + Sqrt[-b + a^2*x^2]) - 40*a^3*b^3*x^3*(7*a*x + 3*Sqrt[-b + a^2*x^2]) - 64*a^7*
b*x^7*(11*a*x + 9*Sqrt[-b + a^2*x^2]) + a*b^4*x*(41*a*x + 9*Sqrt[-b + a^2*x^2]) + 16*a^5*b^2*x^5*(43*a*x + 27*
Sqrt[-b + a^2*x^2])))

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IntegrateAlgebraic [A]  time = 0.12, size = 62, normalized size = 1.00 \begin {gather*} -\frac {2 b}{5 a^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}+\frac {2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{3 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b)/(5*a^2*(a*x + Sqrt[-b + a^2*x^2])^(5/4)) + (2*(a*x + Sqrt[-b + a^2*x^2])^(3/4))/(3*a^2)

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fricas [A]  time = 0.47, size = 56, normalized size = 0.90 \begin {gather*} -\frac {4 \, {\left (3 \, a^{2} x^{2} - 3 \, \sqrt {a^{2} x^{2} - b} a x - 4 \, b\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}}}{15 \, a^{2} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/15*(3*a^2*x^2 - 3*sqrt(a^2*x^2 - b)*a*x - 4*b)*(a*x + sqrt(a^2*x^2 - b))^(3/4)/(a^2*b)

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

[Out]

Timed out

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x}{\sqrt {a^{2} x^{2}-b}\, \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {a^{2} x^{2} - b} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x}{{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\,\sqrt {a^2\,x^2-b}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt [4]{a x + \sqrt {a^{2} x^{2} - b}} \sqrt {a^{2} x^{2} - b}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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