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

Optimal. Leaf size=111 \[ \frac {8 \sqrt {a^2 x^2-b} \left (a^2 x^3-4 b x\right )}{7 a^2 \left (\sqrt {a^2 x^2-b}+a x\right )^{9/4}}+\frac {4 \left (18 a^4 x^4-81 a^2 b x^2+32 b^2\right )}{63 a^3 \left (\sqrt {a^2 x^2-b}+a x\right )^{9/4}} \]

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Rubi [A]  time = 0.35, antiderivative size = 93, normalized size of antiderivative = 0.84, number of steps used = 3, number of rules used = 2, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2120, 270} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[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 270

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

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Mathematica [B]  time = 2.52, size = 629, normalized size = 5.67 \begin {gather*} \frac {4 \sqrt {a^2 x^2-b} \left (294912 a^{19} x^{19}-2506752 a^{17} b x^{17}+7745536 a^{15} b^2 x^{15}-12199936 a^{13} b^3 x^{13}[phone] a^{11} b^4 x^{11}-5854336 a^9 b^5 x^9+1812000 a^7 b^6 x^7-302768 a^5 b^7 x^5+23064 a^3 b^8 x^3-32 b^9 \sqrt {a^2 x^2-b}+4225 a^2 b^8 x^2 \sqrt {a^2 x^2-b}+294912 a^{18} x^{18} \sqrt {a^2 x^2-b}-2359296 a^{16} b x^{16} \sqrt {a^2 x^2-b}+6602752 a^{14} b^2 x^{14} \sqrt {a^2 x^2-b}-9175040 a^{12} b^3 x^{12} \sqrt {a^2 x^2-b}+7090688 a^{10} b^4 x^{10} \sqrt {a^2 x^2-b}-3127296 a^8 b^5 x^8 \sqrt {a^2 x^2-b}+760704 a^6 b^6 x^6 \sqrt {a^2 x^2-b}-91648 a^4 b^7 x^4 \sqrt {a^2 x^2-b}-520 a b^9 x\right )}{63 a^3 \left (\sqrt {a^2 x^2-b}+a x\right )^{21/4} \left (512 a^{11} x^{11}-1536 a^9 b x^9+1696 a^7 b^2 x^7-832 a^5 b^3 x^5+170 a^3 b^4 x^3-b^5 \sqrt {a^2 x^2-b}+50 a^2 b^4 x^2 \sqrt {a^2 x^2-b}+512 a^{10} x^{10} \sqrt {a^2 x^2-b}-1280 a^8 b x^8 \sqrt {a^2 x^2-b}+1120 a^6 b^2 x^6 \sqrt {a^2 x^2-b}-400 a^4 b^3 x^4 \sqrt {a^2 x^2-b}-10 a b^5 x\right ) \left (a x \left (\sqrt {a^2 x^2-b}+a x\right )-b\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/(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]*(-520*a*b^9*x + 23064*a^3*b^8*x^3 - 302768*a^5*b^7*x^5 + 1812000*a^7*b^6*x^7 - 5854336*a
^9*b^5*x^9 + 10988800*a^11*b^4*x^11 - 12199936*a^13*b^3*x^13 + 7745536*a^15*b^2*x^15 - 2506752*a^17*b*x^17 + 2
94912*a^19*x^19 - 32*b^9*Sqrt[-b + a^2*x^2] + 4225*a^2*b^8*x^2*Sqrt[-b + a^2*x^2] - 91648*a^4*b^7*x^4*Sqrt[-b
+ a^2*x^2] + 760704*a^6*b^6*x^6*Sqrt[-b + a^2*x^2] - 3127296*a^8*b^5*x^8*Sqrt[-b + a^2*x^2] + 7090688*a^10*b^4
*x^10*Sqrt[-b + a^2*x^2] - 9175040*a^12*b^3*x^12*Sqrt[-b + a^2*x^2] + 6602752*a^14*b^2*x^14*Sqrt[-b + a^2*x^2]
- 2359296*a^16*b*x^16*Sqrt[-b + a^2*x^2] + 294912*a^18*x^18*Sqrt[-b + a^2*x^2]))/(63*a^3*(a*x + Sqrt[-b + a^2
*x^2])^(21/4)*(-10*a*b^5*x + 170*a^3*b^4*x^3 - 832*a^5*b^3*x^5 + 1696*a^7*b^2*x^7 - 1536*a^9*b*x^9 + 512*a^11*
x^11 - b^5*Sqrt[-b + a^2*x^2] + 50*a^2*b^4*x^2*Sqrt[-b + a^2*x^2] - 400*a^4*b^3*x^4*Sqrt[-b + a^2*x^2] + 1120*
a^6*b^2*x^6*Sqrt[-b + a^2*x^2] - 1280*a^8*b*x^8*Sqrt[-b + a^2*x^2] + 512*a^10*x^10*Sqrt[-b + a^2*x^2])*(-b + a
*x*(a*x + Sqrt[-b + a^2*x^2])))

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IntegrateAlgebraic [A]  time = 0.17, size = 111, normalized size = 1.00 \begin {gather*} \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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.45, size = 68, normalized size = 0.61 \begin {gather*} -\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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^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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2}{{\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^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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\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**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)

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