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

Optimal. Leaf size=87 \[ \frac {2 x \sqrt {a x^2+b^2} \left (5 a x^2+13 b^2\right )}{35 \sqrt {\sqrt {a x^2+b^2}+b}}+\frac {4 x \left (3 a b x^2+11 b^3\right )}{35 \sqrt {\sqrt {a x^2+b^2}+b}} \]

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Rubi [F]  time = 0.28, 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 ) \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)*Sqrt[b + Sqrt[b^2 + a*x^2]],x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.21, size = 85, normalized size = 0.98 \begin {gather*} \frac {2 x \left (5 a^2 x^4+24 a b^2 x^2+11 a b x^2 \sqrt {a x^2+b^2}+35 b^3 \sqrt {a x^2+b^2}+35 b^4\right )}{35 \left (\sqrt {a x^2+b^2}+b\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(35*b^4 + 24*a*b^2*x^2 + 5*a^2*x^4 + 35*b^3*Sqrt[b^2 + a*x^2] + 11*a*b*x^2*Sqrt[b^2 + a*x^2]))/(35*(b + S
qrt[b^2 + a*x^2])^(3/2))

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IntegrateAlgebraic [A]  time = 0.17, size = 87, normalized size = 1.00 \begin {gather*} \frac {2 x \sqrt {b^2+a x^2} \left (13 b^2+5 a x^2\right )}{35 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {4 x \left (11 b^3+3 a b x^2\right )}{35 \sqrt {b+\sqrt {b^2+a x^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*Sqrt[b^2 + a*x^2]*(13*b^2 + 5*a*x^2))/(35*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (4*x*(11*b^3 + 3*a*b*x^2))/(35*S
qrt[b + Sqrt[b^2 + a*x^2]])

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fricas [A]  time = 0.52, size = 70, normalized size = 0.80 \begin {gather*} \frac {2 \, {\left (5 \, a^{2} x^{4} + 12 \, a b^{2} x^{2} - 9 \, b^{4} + {\left (a b x^{2} + 9 \, b^{3}\right )} \sqrt {a x^{2} + b^{2}}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{35 \, a x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(5*a^2*x^4 + 12*a*b^2*x^2 - 9*b^4 + (a*b*x^2 + 9*b^3)*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 )} \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)*(b+(a*x^2+b^2)^(1/2))^(1/2),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.05, size = 153, normalized size = 1.76

method	result	size
meijerg	\(\frac {\left (b^{2}\right )^{\frac {1}{4}} a \sqrt {2}\, x^{3} \hypergeom \left (\left [-\frac {1}{4}, \frac {1}{4}, \frac {3}{2}\right ], \left [\frac {1}{2}, \frac {5}{2}\right ], -\frac {x^{2} a}{b^{2}}\right )}{3}-\frac {b^{2} \left (b^{2}\right )^{\frac {1}{4}} \left (-\frac {32 \sqrt {\pi }\, \sqrt {2}\, x^{3} \sqrt {\frac {a}{b^{2}}}\, a \cosh \left (\frac {3 \arcsinh \left (\frac {x \sqrt {a}}{b}\right )}{2}\right )}{3 b^{2}}-\frac {8 \sqrt {\pi }\, \sqrt {2}\, \sqrt {\frac {a}{b^{2}}}\, \left (-\frac {4 x^{4} a^{2}}{3 b^{4}}-\frac {2 x^{2} a}{3 b^{2}}+\frac {2}{3}\right ) \sinh \left (\frac {3 \arcsinh \left (\frac {x \sqrt {a}}{b}\right )}{2}\right ) b}{\sqrt {a}\, \sqrt {\frac {x^{2} a}{b^{2}}+1}}\right )}{8 \sqrt {\pi }\, \sqrt {\frac {a}{b^{2}}}}\)	\(153\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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sympy [B]  time = 3.71, size = 581, normalized size = 6.68 \begin {gather*} - \frac {15 \sqrt {2} a^{2} \sqrt {b} x^{5} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{420 \pi b^{2} \sqrt {\frac {a x^{2}}{b^{2}} + 1} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1} + 420 \pi b^{2} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1}} - \frac {33 \sqrt {2} a b^{\frac {5}{2}} x^{3} \sqrt {\frac {a x^{2}}{b^{2}} + 1} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{420 \pi b^{2} \sqrt {\frac {a x^{2}}{b^{2}} + 1} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1} + 420 \pi b^{2} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1}} - \frac {37 \sqrt {2} a b^{\frac {5}{2}} x^{3} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{420 \pi b^{2} \sqrt {\frac {a x^{2}}{b^{2}} + 1} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1} + 420 \pi b^{2} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1}} - \frac {\sqrt {2} a b^{\frac {5}{2}} x^{3} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{12 \pi b^{2} \sqrt {\frac {a x^{2}}{b^{2}} + 1} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1} + 12 \pi b^{2} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1}} - \frac {3 \sqrt {2} b^{\frac {9}{2}} x \sqrt {\frac {a x^{2}}{b^{2}} + 1} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{12 \pi b^{2} \sqrt {\frac {a x^{2}}{b^{2}} + 1} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1} + 12 \pi b^{2} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1}} - \frac {3 \sqrt {2} b^{\frac {9}{2}} x \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{12 \pi b^{2} \sqrt {\frac {a x^{2}}{b^{2}} + 1} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1} + 12 \pi b^{2} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-15*sqrt(2)*a**2*sqrt(b)*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)) - 33*sqrt(2)*a*b**(5/2)*x**3*sqrt(a*x**2/b**2 + 1)*ga
mma(-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(sqr
t(a*x**2/b**2 + 1) + 1)) - 37*sqrt(2)*a*b**(5/2)*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**(5/2)*x**3*gam
ma(-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**(9/2)*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**(9/
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*sqr
t(sqrt(a*x**2/b**2 + 1) + 1))

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