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

Optimal. Leaf size=317 \[ \frac{d (f x)^{m+1} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{m+1}{2};\frac{1}{2},\frac{1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{f (m+1) \sqrt{a+b x^2+c x^4}}+\frac{e (f x)^{m+3} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{m+3}{2};\frac{1}{2},\frac{1}{2};\frac{m+5}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{f^3 (m+3) \sqrt{a+b x^2+c x^4}} \]

[Out]

(d*(f*x)^(1 + m)*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/
(b + Sqrt[b^2 - 4*a*c])]*AppellF1[(1 + m)/2, 1/2, 1/2, (3 + m)/2, (-2*c*x^2)/(b
- Sqrt[b^2 - 4*a*c]), (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])])/(f*(1 + m)*Sqrt[a + b
*x^2 + c*x^4]) + (e*(f*x)^(3 + m)*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Sq
rt[1 + (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[(3 + m)/2, 1/2, 1/2, (5 + m)/
2, (-2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])])/(f^3
*(3 + m)*Sqrt[a + b*x^2 + c*x^4])

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Rubi [A]  time = 1.06078, antiderivative size = 317, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{d (f x)^{m+1} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{m+1}{2};\frac{1}{2},\frac{1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{f (m+1) \sqrt{a+b x^2+c x^4}}+\frac{e (f x)^{m+3} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{m+3}{2};\frac{1}{2},\frac{1}{2};\frac{m+5}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{f^3 (m+3) \sqrt{a+b x^2+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(d*(f*x)^(1 + m)*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/
(b + Sqrt[b^2 - 4*a*c])]*AppellF1[(1 + m)/2, 1/2, 1/2, (3 + m)/2, (-2*c*x^2)/(b
- Sqrt[b^2 - 4*a*c]), (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])])/(f*(1 + m)*Sqrt[a + b
*x^2 + c*x^4]) + (e*(f*x)^(3 + m)*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Sq
rt[1 + (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[(3 + m)/2, 1/2, 1/2, (5 + m)/
2, (-2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])])/(f^3
*(3 + m)*Sqrt[a + b*x^2 + c*x^4])

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Rubi in Sympy [A]  time = 86.8298, size = 279, normalized size = 0.88 \[ \frac{d \left (f x\right )^{m + 1} \sqrt{a + b x^{2} + c x^{4}} \operatorname{appellf_{1}}{\left (\frac{m}{2} + \frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{m}{2} + \frac{3}{2},- \frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{a f \left (m + 1\right ) \sqrt{\frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} + \frac{e \left (f x\right )^{m + 3} \sqrt{a + b x^{2} + c x^{4}} \operatorname{appellf_{1}}{\left (\frac{m}{2} + \frac{3}{2},\frac{1}{2},\frac{1}{2},\frac{m}{2} + \frac{5}{2},- \frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{a f^{3} \left (m + 3\right ) \sqrt{\frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*(f*x)**(m + 1)*sqrt(a + b*x**2 + c*x**4)*appellf1(m/2 + 1/2, 1/2, 1/2, m/2 + 3
/2, -2*c*x**2/(b - sqrt(-4*a*c + b**2)), -2*c*x**2/(b + sqrt(-4*a*c + b**2)))/(a
*f*(m + 1)*sqrt(2*c*x**2/(b - sqrt(-4*a*c + b**2)) + 1)*sqrt(2*c*x**2/(b + sqrt(
-4*a*c + b**2)) + 1)) + e*(f*x)**(m + 3)*sqrt(a + b*x**2 + c*x**4)*appellf1(m/2
+ 3/2, 1/2, 1/2, m/2 + 5/2, -2*c*x**2/(b - sqrt(-4*a*c + b**2)), -2*c*x**2/(b +
sqrt(-4*a*c + b**2)))/(a*f**3*(m + 3)*sqrt(2*c*x**2/(b - sqrt(-4*a*c + b**2)) +
1)*sqrt(2*c*x**2/(b + sqrt(-4*a*c + b**2)) + 1))

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Mathematica [B]  time = 2.67015, size = 728, normalized size = 2.3 \[ \frac{a x (f x)^m \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (\frac{d (m+3)^2 F_1\left (\frac{m+1}{2};\frac{1}{2},\frac{1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )}{(m+1) \left (2 a (m+3) F_1\left (\frac{m+1}{2};\frac{1}{2},\frac{1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )-x^2 \left (\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (\frac{m+3}{2};\frac{1}{2},\frac{3}{2};\frac{m+5}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{m+3}{2};\frac{3}{2},\frac{1}{2};\frac{m+5}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )\right )}-\frac{e (m+5) x^2 F_1\left (\frac{m+3}{2};\frac{1}{2},\frac{1}{2};\frac{m+5}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )}{x^2 \left (\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (\frac{m+5}{2};\frac{1}{2},\frac{3}{2};\frac{m+7}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{m+5}{2};\frac{3}{2},\frac{1}{2};\frac{m+7}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )-2 a (m+5) F_1\left (\frac{m+3}{2};\frac{1}{2},\frac{1}{2};\frac{m+5}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )}\right )}{2 c (m+3) \left (a+b x^2+c x^4\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(a*x*(f*x)^m*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)
*((d*(3 + m)^2*AppellF1[(1 + m)/2, 1/2, 1/2, (3 + m)/2, (-2*c*x^2)/(b + Sqrt[b^2
- 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/((1 + m)*(2*a*(3 + m)*AppellF1[
(1 + m)/2, 1/2, 1/2, (3 + m)/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-
b + Sqrt[b^2 - 4*a*c])] - x^2*((b + Sqrt[b^2 - 4*a*c])*AppellF1[(3 + m)/2, 1/2,
3/2, (5 + m)/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4
*a*c])] + (b - Sqrt[b^2 - 4*a*c])*AppellF1[(3 + m)/2, 3/2, 1/2, (5 + m)/2, (-2*c
*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]))) - (e*(5 +
m)*x^2*AppellF1[(3 + m)/2, 1/2, 1/2, (5 + m)/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c
]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/(-2*a*(5 + m)*AppellF1[(3 + m)/2, 1/2,
1/2, (5 + m)/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4
*a*c])] + x^2*((b + Sqrt[b^2 - 4*a*c])*AppellF1[(5 + m)/2, 1/2, 3/2, (7 + m)/2,
(-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] + (b - S
qrt[b^2 - 4*a*c])*AppellF1[(5 + m)/2, 3/2, 1/2, (7 + m)/2, (-2*c*x^2)/(b + Sqrt[
b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]))))/(2*c*(3 + m)*(a + b*x^2 +
c*x^4)^(3/2))

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Maple [F]  time = 0.019, size = 0, normalized size = 0. \[ \int{ \left ( fx \right ) ^{m} \left ( e{x}^{2}+d \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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