3.813 \(\int \left (a+b x^2\right )^{7/4} \, dx\)

Optimal. Leaf size=111 \[ -\frac{14 a^{5/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{15 \sqrt{b} \sqrt [4]{a+b x^2}}+\frac{14 a^2 x}{15 \sqrt [4]{a+b x^2}}+\frac{14}{45} a x \left (a+b x^2\right )^{3/4}+\frac{2}{9} x \left (a+b x^2\right )^{7/4} \]

[Out]

(14*a^2*x)/(15*(a + b*x^2)^(1/4)) + (14*a*x*(a + b*x^2)^(3/4))/45 + (2*x*(a + b*
x^2)^(7/4))/9 - (14*a^(5/2)*(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x)/S
qrt[a]]/2, 2])/(15*Sqrt[b]*(a + b*x^2)^(1/4))

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Rubi [A]  time = 0.0861093, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{14 a^{5/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{15 \sqrt{b} \sqrt [4]{a+b x^2}}+\frac{14 a^2 x}{15 \sqrt [4]{a+b x^2}}+\frac{14}{45} a x \left (a+b x^2\right )^{3/4}+\frac{2}{9} x \left (a+b x^2\right )^{7/4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(7/4),x]

[Out]

(14*a^2*x)/(15*(a + b*x^2)^(1/4)) + (14*a*x*(a + b*x^2)^(3/4))/45 + (2*x*(a + b*
x^2)^(7/4))/9 - (14*a^(5/2)*(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x)/S
qrt[a]]/2, 2])/(15*Sqrt[b]*(a + b*x^2)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{7 a^{3} \int \frac{1}{\left (a + b x^{2}\right )^{\frac{5}{4}}}\, dx}{15} + \frac{14 a^{2} x}{15 \sqrt [4]{a + b x^{2}}} + \frac{14 a x \left (a + b x^{2}\right )^{\frac{3}{4}}}{45} + \frac{2 x \left (a + b x^{2}\right )^{\frac{7}{4}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(7/4),x)

[Out]

-7*a**3*Integral((a + b*x**2)**(-5/4), x)/15 + 14*a**2*x/(15*(a + b*x**2)**(1/4)
) + 14*a*x*(a + b*x**2)**(3/4)/45 + 2*x*(a + b*x**2)**(7/4)/9

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Mathematica [C]  time = 0.0478442, size = 76, normalized size = 0.68 \[ \frac{21 a^2 x \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )+24 a^2 x+34 a b x^3+10 b^2 x^5}{45 \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(7/4),x]

[Out]

(24*a^2*x + 34*a*b*x^3 + 10*b^2*x^5 + 21*a^2*x*(1 + (b*x^2)/a)^(1/4)*Hypergeomet
ric2F1[1/4, 1/2, 3/2, -((b*x^2)/a)])/(45*(a + b*x^2)^(1/4))

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Maple [F]  time = 0.044, size = 0, normalized size = 0. \[ \int \left ( b{x}^{2}+a \right ) ^{{\frac{7}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(7/4),x)

[Out]

int((b*x^2+a)^(7/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{\frac{7}{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(7/4),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(7/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{2} + a\right )}^{\frac{7}{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(7/4),x, algorithm="fricas")

[Out]

integral((b*x^2 + a)^(7/4), x)

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Sympy [A]  time = 10.2539, size = 26, normalized size = 0.23 \[ a^{\frac{7}{4}} x{{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(7/4),x)

[Out]

a**(7/4)*x*hyper((-7/4, 1/2), (3/2,), b*x**2*exp_polar(I*pi)/a)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(7/4),x, algorithm="giac")

[Out]

Exception raised: TypeError