3.25.18 \(\int \frac {(b^2+a x^2)^{5/2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx\)

Optimal. Leaf size=195 \[ \frac {2 x \sqrt {a x^2+b^2} \left (45 a^2 x^4+155 a b^2 x^2+247 b^4\right )}{495 \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {2 x \left (5 a^2 b x^4+31 a b^3 x^2+247 b^5\right )}{495 \sqrt {\sqrt {a x^2+b^2}+b}}+\frac {2 \sqrt {2} b^{11/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {\sqrt {\sqrt {a x^2+b^2}+b}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {a}} \]

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Rubi [F]  time = 0.21, 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 {\left (b^2+a x^2\right )^{5/2}}{\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)^(5/2)/Sqrt[b + Sqrt[b^2 + a*x^2]],x]

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 0.69, size = 388, normalized size = 1.99 \begin {gather*} \frac {2 \left (90 a^4 x^8 \sqrt {a x^2+b^2}+170 a^4 b x^8+718 a^3 b^3 x^6+470 a^3 b^2 x^6 \sqrt {a x^2+b^2}+990 a^2 b^5 x^4+990 a^2 b^4 x^4 \sqrt {a x^2+b^2}-1485 a b^7 x^2-1980 b^8 \sqrt {a x^2+b^2}-495 a b^6 x^2 \sqrt {a x^2+b^2}-1980 b^9\right )+495 \sqrt {2} b^{11/2} \sqrt {\sqrt {a x^2+b^2}-b} \left (4 b^2 \sqrt {a x^2+b^2}+a x^2 \sqrt {a x^2+b^2}+3 a b x^2+4 b^3\right ) \tan ^{-1}\left (\frac {\sqrt {\sqrt {a x^2+b^2}-b}}{\sqrt {2} \sqrt {b}}\right )+990 b^6 \left (4 b^2 \sqrt {a x^2+b^2}+a x^2 \sqrt {a x^2+b^2}+3 a b x^2+4 b^3\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b-\sqrt {b^2+a x^2}}{2 b}\right )}{990 a x \left (\sqrt {a x^2+b^2}+b\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b^2 + a*x^2)^(5/2)/Sqrt[b + Sqrt[b^2 + a*x^2]],x]

[Out]

(2*(-1980*b^9 - 1485*a*b^7*x^2 + 990*a^2*b^5*x^4 + 718*a^3*b^3*x^6 + 170*a^4*b*x^8 - 1980*b^8*Sqrt[b^2 + a*x^2
] - 495*a*b^6*x^2*Sqrt[b^2 + a*x^2] + 990*a^2*b^4*x^4*Sqrt[b^2 + a*x^2] + 470*a^3*b^2*x^6*Sqrt[b^2 + a*x^2] +
90*a^4*x^8*Sqrt[b^2 + a*x^2]) + 495*Sqrt[2]*b^(11/2)*Sqrt[-b + Sqrt[b^2 + a*x^2]]*(4*b^3 + 3*a*b*x^2 + 4*b^2*S
qrt[b^2 + a*x^2] + a*x^2*Sqrt[b^2 + a*x^2])*ArcTan[Sqrt[-b + Sqrt[b^2 + a*x^2]]/(Sqrt[2]*Sqrt[b])] + 990*b^6*(
4*b^3 + 3*a*b*x^2 + 4*b^2*Sqrt[b^2 + a*x^2] + a*x^2*Sqrt[b^2 + a*x^2])*Hypergeometric2F1[-1/2, 1, 1/2, (b - Sq
rt[b^2 + a*x^2])/(2*b)])/(990*a*x*(b + Sqrt[b^2 + a*x^2])^(5/2))

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IntegrateAlgebraic [A]  time = 0.28, size = 162, normalized size = 0.83 \begin {gather*} \frac {2 x \sqrt {b^2+a x^2} \left (247 b^4+155 a b^2 x^2+45 a^2 x^4\right )}{495 \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {2 x \left (247 b^5+31 a b^3 x^2+5 a^2 b x^4\right )}{495 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {\sqrt {2} b^{11/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(b^2 + a*x^2)^(5/2)/Sqrt[b + Sqrt[b^2 + a*x^2]],x]

[Out]

(2*x*Sqrt[b^2 + a*x^2]*(247*b^4 + 155*a*b^2*x^2 + 45*a^2*x^4))/(495*Sqrt[b + Sqrt[b^2 + a*x^2]]) - (2*x*(247*b
^5 + 31*a*b^3*x^2 + 5*a^2*b*x^4))/(495*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (Sqrt[2]*b^(11/2)*ArcTan[(Sqrt[a]*x)/(Sq
rt[2]*Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/Sqrt[a]

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fricas [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((a*x^2+b^2)^(5/2)/(b+(a*x^2+b^2)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{2} + b^{2}\right )}^{\frac {5}{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)^(5/2)/(b+(a*x^2+b^2)^(1/2))^(1/2),x, algorithm="giac")

[Out]

integrate((a*x^2 + b^2)^(5/2)/sqrt(b + sqrt(a*x^2 + b^2)), x)

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{2}+b^{2}\right )^{\frac {5}{2}}}{\sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^2+b^2)^(5/2)/(b+(a*x^2+b^2)^(1/2))^(1/2),x)

[Out]

int((a*x^2+b^2)^(5/2)/(b+(a*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 {{\left (a x^{2} + b^{2}\right )}^{\frac {5}{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)^(5/2)/(b+(a*x^2+b^2)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate((a*x^2 + b^2)^(5/2)/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 \frac {{\left (b^2+a\,x^2\right )}^{5/2}}{\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)^(5/2)/(b + (a*x^2 + b^2)^(1/2))^(1/2),x)

[Out]

int((a*x^2 + b^2)^(5/2)/(b + (a*x^2 + 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 {\left (a x^{2} + b^{2}\right )^{\frac {5}{2}}}{\sqrt {b + \sqrt {a x^{2} + b^{2}}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**2+b**2)**(5/2)/(b+(a*x**2+b**2)**(1/2))**(1/2),x)

[Out]

Integral((a*x**2 + b**2)**(5/2)/sqrt(b + sqrt(a*x**2 + b**2)), x)

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