3.17.5 \(\int (b^2+a x^2)^2 \sqrt {b+\sqrt {b^2+a x^2}} \, dx\)

Optimal. Leaf size=109 \[ \frac {2 x \sqrt {a x^2+b^2} \left (63 a^2 x^4+206 a b^2 x^2+271 b^4\right )}{693 \sqrt {\sqrt {a x^2+b^2}+b}}+\frac {4 x \left (35 a^2 b x^4+118 a b^3 x^2+211 b^5\right )}{693 \sqrt {\sqrt {a x^2+b^2}+b}} \]

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Rubi [F]  time = 0.48, 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 \left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}} \, dx &=\int \left (b^4 \sqrt {b+\sqrt {b^2+a x^2}}+2 a b^2 x^2 \sqrt {b+\sqrt {b^2+a x^2}}+a^2 x^4 \sqrt {b+\sqrt {b^2+a x^2}}\right ) \, dx\\ &=a^2 \int x^4 \sqrt {b+\sqrt {b^2+a x^2}} \, dx+\left (2 a b^2\right ) \int x^2 \sqrt {b+\sqrt {b^2+a x^2}} \, dx+b^4 \int \sqrt {b+\sqrt {b^2+a x^2}} \, dx\\ &=\frac {2 a b^4 x^3}{3 \left (b+\sqrt {b^2+a x^2}\right )^{3/2}}+\frac {2 b^5 x}{\sqrt {b+\sqrt {b^2+a x^2}}}+a^2 \int x^4 \sqrt {b+\sqrt {b^2+a x^2}} \, dx+\left (2 a b^2\right ) \int x^2 \sqrt {b+\sqrt {b^2+a x^2}} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.29, size = 144, normalized size = 1.32 \begin {gather*} \frac {2 x \left (63 a^3 x^6 \sqrt {a x^2+b^2}+196 a^3 b x^6+914 a^2 b^3 x^4+472 a^2 b^2 x^4 \sqrt {a x^2+b^2}+1848 a b^5 x^2+1386 b^6 \sqrt {a x^2+b^2}+1155 a b^4 x^2 \sqrt {a x^2+b^2}+1386 b^7\right )}{693 \left (\sqrt {a x^2+b^2}+b\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(1386*b^7 + 1848*a*b^5*x^2 + 914*a^2*b^3*x^4 + 196*a^3*b*x^6 + 1386*b^6*Sqrt[b^2 + a*x^2] + 1155*a*b^4*x^
2*Sqrt[b^2 + a*x^2] + 472*a^2*b^2*x^4*Sqrt[b^2 + a*x^2] + 63*a^3*x^6*Sqrt[b^2 + a*x^2]))/(693*(b + Sqrt[b^2 +
a*x^2])^(5/2))

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IntegrateAlgebraic [A]  time = 0.18, size = 109, normalized size = 1.00 \begin {gather*} \frac {2 x \sqrt {b^2+a x^2} \left (271 b^4+206 a b^2 x^2+63 a^2 x^4\right )}{693 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {4 x \left (211 b^5+118 a b^3 x^2+35 a^2 b x^4\right )}{693 \sqrt {b+\sqrt {b^2+a x^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*Sqrt[b^2 + a*x^2]*(271*b^4 + 206*a*b^2*x^2 + 63*a^2*x^4))/(693*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (4*x*(211*b
^5 + 118*a*b^3*x^2 + 35*a^2*b*x^4))/(693*Sqrt[b + Sqrt[b^2 + a*x^2]])

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fricas [A]  time = 0.54, size = 93, normalized size = 0.85 \begin {gather*} \frac {2 \, {\left (63 \, a^{3} x^{6} + 199 \, a^{2} b^{2} x^{4} + 241 \, a b^{4} x^{2} - 151 \, b^{6} + {\left (7 \, a^{2} b x^{4} + 30 \, a b^{3} x^{2} + 151 \, b^{5}\right )} \sqrt {a x^{2} + b^{2}}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{693 \, a x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/693*(63*a^3*x^6 + 199*a^2*b^2*x^4 + 241*a*b^4*x^2 - 151*b^6 + (7*a^2*b*x^4 + 30*a*b^3*x^2 + 151*b^5)*sqrt(a*
x^2 + b^2))*sqrt(b + sqrt(a*x^2 + b^2))/(a*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.12, size = 189, normalized size = 1.73

method	result	size
meijerg	\(\frac {\left (b^{2}\right )^{\frac {1}{4}} a^{2} \sqrt {2}\, x^{5} \hypergeom \left (\left [-\frac {1}{4}, \frac {1}{4}, \frac {5}{2}\right ], \left [\frac {1}{2}, \frac {7}{2}\right ], -\frac {x^{2} a}{b^{2}}\right )}{5}+\frac {2 b^{2} \left (b^{2}\right )^{\frac {1}{4}} a \sqrt {2}\, x^{3} \hypergeom \left (\left [-\frac {1}{4}, \frac {1}{4}, \frac {3}{2}\right ], \left [\frac {1}{2}, \frac {5}{2}\right ], -\frac {x^{2} a}{b^{2}}\right )}{3}-\frac {b^{4} \left (b^{2}\right )^{\frac {1}{4}} \left (-\frac {32 \sqrt {\pi }\, \sqrt {2}\, x^{3} \sqrt {\frac {a}{b^{2}}}\, a \cosh \left (\frac {3 \arcsinh \left (\frac {x \sqrt {a}}{b}\right )}{2}\right )}{3 b^{2}}-\frac {8 \sqrt {\pi }\, \sqrt {2}\, \sqrt {\frac {a}{b^{2}}}\, \left (-\frac {4 x^{4} a^{2}}{3 b^{4}}-\frac {2 x^{2} a}{3 b^{2}}+\frac {2}{3}\right ) \sinh \left (\frac {3 \arcsinh \left (\frac {x \sqrt {a}}{b}\right )}{2}\right ) b}{\sqrt {a}\, \sqrt {\frac {x^{2} a}{b^{2}}+1}}\right )}{8 \sqrt {\pi }\, \sqrt {\frac {a}{b^{2}}}}\)	\(189\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^2+b^2)^2*(b+(a*x^2+b^2)^(1/2))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/5*(b^2)^(1/4)*a^2*2^(1/2)*x^5*hypergeom([-1/4,1/4,5/2],[1/2,7/2],-x^2*a/b^2)+2/3*b^2*(b^2)^(1/4)*a*2^(1/2)*x
^3*hypergeom([-1/4,1/4,3/2],[1/2,5/2],-x^2*a/b^2)-1/8*b^4*(b^2)^(1/4)/Pi^(1/2)/(a/b^2)^(1/2)*(-32/3*Pi^(1/2)*2
^(1/2)*x^3*(a/b^2)^(1/2)*a/b^2*cosh(3/2*arcsinh(x*a^(1/2)/b))-8*Pi^(1/2)*2^(1/2)*(a/b^2)^(1/2)*(-4/3*x^4*a^2/b
^4-2/3*x^2*a/b^2+2/3)*sinh(3/2*arcsinh(x*a^(1/2)/b))/a^(1/2)*b/(x^2*a/b^2+1)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 7.33, size = 1100, normalized size = 10.09

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-315*sqrt(2)*a**4*b**(17/2)*x**7*gamma(-1/4)*gamma(1/4)/(13860*pi*a*b**10*sqrt(a*x**2/b**2 + 1)*sqrt(sqrt(a*x*
*2/b**2 + 1) + 1) + 13860*pi*a*b**10*sqrt(sqrt(a*x**2/b**2 + 1) + 1)) - 665*sqrt(2)*a**3*b**(21/2)*x**5*sqrt(a
*x**2/b**2 + 1)*gamma(-1/4)*gamma(1/4)/(13860*pi*a*b**10*sqrt(a*x**2/b**2 + 1)*sqrt(sqrt(a*x**2/b**2 + 1) + 1)
+ 13860*pi*a*b**10*sqrt(sqrt(a*x**2/b**2 + 1) + 1)) - 705*sqrt(2)*a**3*b**(21/2)*x**5*gamma(-1/4)*gamma(1/4)/
(13860*pi*a*b**10*sqrt(a*x**2/b**2 + 1)*sqrt(sqrt(a*x**2/b**2 + 1) + 1) + 13860*pi*a*b**10*sqrt(sqrt(a*x**2/b*
*2 + 1) + 1)) - 32*sqrt(2)*a**2*b**(25/2)*x**3*sqrt(a*x**2/b**2 + 1)*gamma(-1/4)*gamma(1/4)/(13860*pi*a*b**10*
sqrt(a*x**2/b**2 + 1)*sqrt(sqrt(a*x**2/b**2 + 1) + 1) + 13860*pi*a*b**10*sqrt(sqrt(a*x**2/b**2 + 1) + 1)) + 32
*sqrt(2)*a**2*b**(25/2)*x**3*gamma(-1/4)*gamma(1/4)/(13860*pi*a*b**10*sqrt(a*x**2/b**2 + 1)*sqrt(sqrt(a*x**2/b
**2 + 1) + 1) + 13860*pi*a*b**10*sqrt(sqrt(a*x**2/b**2 + 1) + 1)) - 30*sqrt(2)*a**2*b**(5/2)*x**5*gamma(-1/4)*
gamma(1/4)/(420*pi*b**2*sqrt(a*x**2/b**2 + 1)*sqrt(sqrt(a*x**2/b**2 + 1) + 1) + 420*pi*b**2*sqrt(sqrt(a*x**2/b
**2 + 1) + 1)) - 66*sqrt(2)*a*b**(9/2)*x**3*sqrt(a*x**2/b**2 + 1)*gamma(-1/4)*gamma(1/4)/(420*pi*b**2*sqrt(a*x
**2/b**2 + 1)*sqrt(sqrt(a*x**2/b**2 + 1) + 1) + 420*pi*b**2*sqrt(sqrt(a*x**2/b**2 + 1) + 1)) - 74*sqrt(2)*a*b*
*(9/2)*x**3*gamma(-1/4)*gamma(1/4)/(420*pi*b**2*sqrt(a*x**2/b**2 + 1)*sqrt(sqrt(a*x**2/b**2 + 1) + 1) + 420*pi
*b**2*sqrt(sqrt(a*x**2/b**2 + 1) + 1)) - sqrt(2)*a*b**(9/2)*x**3*gamma(-1/4)*gamma(1/4)/(12*pi*b**2*sqrt(a*x**
2/b**2 + 1)*sqrt(sqrt(a*x**2/b**2 + 1) + 1) + 12*pi*b**2*sqrt(sqrt(a*x**2/b**2 + 1) + 1)) - 3*sqrt(2)*b**(13/2
)*x*sqrt(a*x**2/b**2 + 1)*gamma(-1/4)*gamma(1/4)/(12*pi*b**2*sqrt(a*x**2/b**2 + 1)*sqrt(sqrt(a*x**2/b**2 + 1)
+ 1) + 12*pi*b**2*sqrt(sqrt(a*x**2/b**2 + 1) + 1)) - 3*sqrt(2)*b**(13/2)*x*gamma(-1/4)*gamma(1/4)/(12*pi*b**2*
sqrt(a*x**2/b**2 + 1)*sqrt(sqrt(a*x**2/b**2 + 1) + 1) + 12*pi*b**2*sqrt(sqrt(a*x**2/b**2 + 1) + 1))

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