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

Optimal. Leaf size=233 \[ \frac {\left (384 a^2 x^4-136 a x^2-255\right ) \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}} \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{1920 a^2}+\frac {\left (-384 a^2 x^5+568 a x^3+85 x\right ) \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{1920 a b}+\frac {17 \log \left (b \left (-\sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}\right )+\sqrt {2} \sqrt {a} \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}-a x\right )}{128 \sqrt {2} a^{3/2} b} \]

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Rubi [F]  time = 2.10, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {x^3 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {x^3 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx &=\int \frac {x^3 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx\\ \end {align*}

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Mathematica [C]  time = 12.15, size = 733, normalized size = 3.15 \begin {gather*} \frac {b x^2 \sqrt {\frac {a \left (a x^2-1\right )}{b^2}} \left (\frac {2 x^2 \sqrt {x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )} \left (-\sqrt {2} a^5 \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^2 \sqrt {x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )} \, _2F_1\left (-\frac {9}{2},-\frac {3}{2};-\frac {1}{2};-2 a x^2-2 b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}} x+1\right )-3 \left (\left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^2+a\right )^6\right )}{b}-\frac {10 \sqrt {x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )} \left (\left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^2+a\right )^2 \left (-\sqrt {2} a^3 \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^2 \sqrt {x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )} \, _2F_1\left (-\frac {5}{2},-\frac {1}{2};\frac {1}{2};-2 a x^2-2 b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}} x+1\right )-\left (\left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^2+a\right )^4\right )}{a b}-\frac {2 \sqrt {2} a^5 x^3 \sqrt {\frac {a \left (a x^2-1\right )}{b^2}} \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^2 \left (15 \left (8 a^2 x^4+8 a x^2 \left (b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}-1\right )-4 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+1\right ) \, _2F_1\left (-\frac {7}{2},-\frac {1}{2};\frac {1}{2};-2 a x^2-2 b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}} x+1\right )+3 \, _2F_1\left (-\frac {7}{2},-\frac {5}{2};-\frac {3}{2};-2 a x^2-2 b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}} x+1\right )-10 \left (2 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+2 a x^2-1\right ) \, _2F_1\left (-\frac {7}{2},-\frac {3}{2};-\frac {1}{2};-2 a x^2-2 b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}} x+1\right )\right )}{b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x^2-1}\right )}{60 a \left (b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x^2-1\right ) \left (\left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^2+a\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x^2*Sqrt[(a*(-1 + a*x^2))/b^2]*((2*x^2*Sqrt[x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])]*(-3*(a + (a*x + b*Sqrt[
(a*(-1 + a*x^2))/b^2])^2)^6 - Sqrt[2]*a^5*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])^2*Sqrt[x*(a*x + b*Sqrt[(a*(-1 +
a*x^2))/b^2])]*Hypergeometric2F1[-9/2, -3/2, -1/2, 1 - 2*a*x^2 - 2*b*x*Sqrt[(a*(-1 + a*x^2))/b^2]]))/b - (2*S
qrt[2]*a^5*x^3*Sqrt[(a*(-1 + a*x^2))/b^2]*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])^2*(3*Hypergeometric2F1[-7/2, -5
/2, -3/2, 1 - 2*a*x^2 - 2*b*x*Sqrt[(a*(-1 + a*x^2))/b^2]] - 10*(-1 + 2*a*x^2 + 2*b*x*Sqrt[(a*(-1 + a*x^2))/b^2
])*Hypergeometric2F1[-7/2, -3/2, -1/2, 1 - 2*a*x^2 - 2*b*x*Sqrt[(a*(-1 + a*x^2))/b^2]] + 15*(1 + 8*a^2*x^4 - 4
*b*x*Sqrt[(a*(-1 + a*x^2))/b^2] + 8*a*x^2*(-1 + b*x*Sqrt[(a*(-1 + a*x^2))/b^2]))*Hypergeometric2F1[-7/2, -1/2,
1/2, 1 - 2*a*x^2 - 2*b*x*Sqrt[(a*(-1 + a*x^2))/b^2]]))/(-1 + a*x^2 + b*x*Sqrt[(a*(-1 + a*x^2))/b^2]) - (10*Sq
rt[x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])]*(a + (a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])^2)^2*(-(a + (a*x + b*Sqrt
[(a*(-1 + a*x^2))/b^2])^2)^4 - Sqrt[2]*a^3*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])^2*Sqrt[x*(a*x + b*Sqrt[(a*(-1
+ a*x^2))/b^2])]*Hypergeometric2F1[-5/2, -1/2, 1/2, 1 - 2*a*x^2 - 2*b*x*Sqrt[(a*(-1 + a*x^2))/b^2]]))/(a*b)))/
(60*a*(-1 + a*x^2 + b*x*Sqrt[(a*(-1 + a*x^2))/b^2])*(a + (a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])^2)^5)

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IntegrateAlgebraic [A]  time = 3.51, size = 233, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \left (-255-136 a x^2+384 a^2 x^4\right ) \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{1920 a^2}+\frac {\left (85 x+568 a x^3-384 a^2 x^5\right ) \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{1920 a b}+\frac {17 \log \left (-a x-b \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}+\sqrt {2} \sqrt {a} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{128 \sqrt {2} a^{3/2} b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-(a/b^2) + (a^2*x^2)/b^2]*(-255 - 136*a*x^2 + 384*a^2*x^4)*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^
2]])/(1920*a^2) + ((85*x + 568*a*x^3 - 384*a^2*x^5)*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]])/(1920*a*
b) + (17*Log[-(a*x) - b*Sqrt[-(a/b^2) + (a^2*x^2)/b^2] + Sqrt[2]*Sqrt[a]*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2
*x^2)/b^2]]])/(128*Sqrt[2]*a^(3/2)*b)

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fricas [A]  time = 40.47, size = 359, normalized size = 1.54 \begin {gather*} \left [\frac {255 \, \sqrt {2} \sqrt {a} \log \left (-4 \, a^{2} x^{2} - 4 \, a b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} + 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x + \sqrt {2} \sqrt {a} b \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}\right )} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} + a\right ) - 4 \, {\left (384 \, a^{3} x^{5} - 568 \, a^{2} x^{3} - 85 \, a x - {\left (384 \, a^{2} b x^{4} - 136 \, a b x^{2} - 255 \, b\right )} \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}\right )} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}}}{7680 \, a^{2} b}, \frac {255 \, \sqrt {2} \sqrt {-a} \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} \sqrt {-a}}{2 \, a x}\right ) - 2 \, {\left (384 \, a^{3} x^{5} - 568 \, a^{2} x^{3} - 85 \, a x - {\left (384 \, a^{2} b x^{4} - 136 \, a b x^{2} - 255 \, b\right )} \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}\right )} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}}}{3840 \, a^{2} b}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/7680*(255*sqrt(2)*sqrt(a)*log(-4*a^2*x^2 - 4*a*b*x*sqrt((a^2*x^2 - a)/b^2) + 2*(sqrt(2)*a^(3/2)*x + sqrt(2)
*sqrt(a)*b*sqrt((a^2*x^2 - a)/b^2))*sqrt(a*x^2 + b*x*sqrt((a^2*x^2 - a)/b^2)) + a) - 4*(384*a^3*x^5 - 568*a^2*
x^3 - 85*a*x - (384*a^2*b*x^4 - 136*a*b*x^2 - 255*b)*sqrt((a^2*x^2 - a)/b^2))*sqrt(a*x^2 + b*x*sqrt((a^2*x^2 -
a)/b^2)))/(a^2*b), 1/3840*(255*sqrt(2)*sqrt(-a)*arctan(1/2*sqrt(2)*sqrt(a*x^2 + b*x*sqrt((a^2*x^2 - a)/b^2))*
sqrt(-a)/(a*x)) - 2*(384*a^3*x^5 - 568*a^2*x^3 - 85*a*x - (384*a^2*b*x^4 - 136*a*b*x^2 - 255*b)*sqrt((a^2*x^2
- a)/b^2))*sqrt(a*x^2 + b*x*sqrt((a^2*x^2 - a)/b^2)))/(a^2*b)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}{\sqrt {a \,x^{2}+b x \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} x^{3}}{\sqrt {a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \sqrt {\frac {a \left (a x^{2} - 1\right )}{b^{2}}}}{\sqrt {x \left (a x + b \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3*sqrt(a*(a*x**2 - 1)/b**2)/sqrt(x*(a*x + b*sqrt(a**2*x**2/b**2 - a/b**2))), x)

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