3.90 \(\int \frac{(f x)^m \left (d+e x^2\right )}{\sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx\)

Optimal. Leaf size=134 \[ \frac{\left (a+b x^2\right ) (f x)^{m+1} (b d-a e) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a b f (m+1) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{e \left (a+b x^2\right ) (f x)^{m+1}}{b f (m+1) \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

(e*(f*x)^(1 + m)*(a + b*x^2))/(b*f*(1 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + ((
b*d - a*e)*(f*x)^(1 + m)*(a + b*x^2)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2,
-((b*x^2)/a)])/(a*b*f*(1 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

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Rubi [A]  time = 0.246658, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ \frac{\left (a+b x^2\right ) (f x)^{m+1} (b d-a e) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a b f (m+1) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{e \left (a+b x^2\right ) (f x)^{m+1}}{b f (m+1) \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]  Int[((f*x)^m*(d + e*x^2))/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

(e*(f*x)^(1 + m)*(a + b*x^2))/(b*f*(1 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + ((
b*d - a*e)*(f*x)^(1 + m)*(a + b*x^2)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2,
-((b*x^2)/a)])/(a*b*f*(1 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

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Rubi in Sympy [A]  time = 34.6009, size = 114, normalized size = 0.85 \[ \frac{e \left (f x\right )^{m + 1} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{b f \left (a + b x^{2}\right ) \left (m + 1\right )} - \frac{\left (f x\right )^{m + 1} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{a b f \left (a + b x^{2}\right ) \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x)**m*(e*x**2+d)/(b**2*x**4+2*a*b*x**2+a**2)**(1/2),x)

[Out]

e*(f*x)**(m + 1)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(b*f*(a + b*x**2)*(m + 1))
- (f*x)**(m + 1)*(a*e - b*d)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)*hyper((1, m/2 +
1/2), (m/2 + 3/2,), -b*x**2/a)/(a*b*f*(a + b*x**2)*(m + 1))

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Mathematica [A]  time = 0.10266, size = 78, normalized size = 0.58 \[ -\frac{x \left (a+b x^2\right ) (f x)^m \left ((a e-b d) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )-a e\right )}{a b (m+1) \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((f*x)^m*(d + e*x^2))/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

-((x*(f*x)^m*(a + b*x^2)*(-(a*e) + (-(b*d) + a*e)*Hypergeometric2F1[1, (1 + m)/2
, (3 + m)/2, -((b*x^2)/a)]))/(a*b*(1 + m)*Sqrt[(a + b*x^2)^2]))

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \[ \int{ \left ( fx \right ) ^{m} \left ( e{x}^{2}+d \right ){\frac{1}{\sqrt{{b}^{2}{x}^{4}+2\,ab{x}^{2}+{a}^{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x)^m*(e*x^2+d)/(b^2*x^4+2*a*b*x^2+a^2)^(1/2),x)

[Out]

int((f*x)^m*(e*x^2+d)/(b^2*x^4+2*a*b*x^2+a^2)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)*(f*x)^m/sqrt(b^2*x^4 + 2*a*b*x^2 + a^2),x, algorithm="maxima")

[Out]

integrate((e*x^2 + d)*(f*x)^m/sqrt(b^2*x^4 + 2*a*b*x^2 + a^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)*(f*x)^m/sqrt(b^2*x^4 + 2*a*b*x^2 + a^2),x, algorithm="fricas")

[Out]

integral((e*x^2 + d)*(f*x)^m/sqrt(b^2*x^4 + 2*a*b*x^2 + a^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (f x\right )^{m} \left (d + e x^{2}\right )}{\sqrt{\left (a + b x^{2}\right )^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x)**m*(e*x**2+d)/(b**2*x**4+2*a*b*x**2+a**2)**(1/2),x)

[Out]

Integral((f*x)**m*(d + e*x**2)/sqrt((a + b*x**2)**2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x^{2} + d\right )} \left (f x\right )^{m}}{\sqrt{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)*(f*x)^m/sqrt(b^2*x^4 + 2*a*b*x^2 + a^2),x, algorithm="giac")

[Out]

integrate((e*x^2 + d)*(f*x)^m/sqrt(b^2*x^4 + 2*a*b*x^2 + a^2), x)