3.1114 \(\int (d x)^m \left (a+b x^2+c x^4\right )^p \, dx\)

Optimal. Leaf size=155 \[ \frac{(d x)^{m+1} \left (\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^2+c x^4\right )^p F_1\left (\frac{m+1}{2};-p,-p;\frac{m+3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{d (m+1)} \]

[Out]

((d*x)^(1 + m)*(a + b*x^2 + c*x^4)^p*AppellF1[(1 + m)/2, -p, -p, (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])])/(d*(1 + m)*
(1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]))^p*(1 + (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])
)^p)

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Rubi [A]  time = 0.302921, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{(d x)^{m+1} \left (\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^2+c x^4\right )^p F_1\left (\frac{m+1}{2};-p,-p;\frac{m+3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{d (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^m*(a + b*x^2 + c*x^4)^p,x]

[Out]

((d*x)^(1 + m)*(a + b*x^2 + c*x^4)^p*AppellF1[(1 + m)/2, -p, -p, (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])])/(d*(1 + m)*
(1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]))^p*(1 + (2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])
)^p)

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Rubi in Sympy [A]  time = 29.6263, size = 129, normalized size = 0.83 \[ \frac{\left (d x\right )^{m + 1} \left (\frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}} + 1\right )^{- p} \left (\frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} + 1\right )^{- p} \left (a + b x^{2} + c x^{4}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{m}{2} + \frac{1}{2},- p,- p,\frac{m}{2} + \frac{3}{2},- \frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{d \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**m*(c*x**4+b*x**2+a)**p,x)

[Out]

(d*x)**(m + 1)*(2*c*x**2/(b - sqrt(-4*a*c + b**2)) + 1)**(-p)*(2*c*x**2/(b + sqr
t(-4*a*c + b**2)) + 1)**(-p)*(a + b*x**2 + c*x**4)**p*appellf1(m/2 + 1/2, -p, -p
, 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)))/(d*(m + 1))

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Mathematica [B]  time = 4.99231, size = 499, normalized size = 3.22 \[ -\frac{c (m+3) 2^{-p-2} x \left (\sqrt{b^2-4 a c}+b\right ) (d x)^m \left (x^2 \left (\sqrt{b^2-4 a c}-b\right )-2 a\right )^2 \left (\frac{b-\sqrt{b^2-4 a c}}{2 c}+x^2\right )^{-p} \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{c}\right )^{p+1} \left (a+b x^2+c x^4\right )^{p-1} F_1\left (\frac{m+1}{2};-p,-p;\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )}{(m+1) \left (\sqrt{b^2-4 a c}-b\right ) \left (\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (p x^2 \left (\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{m+3}{2};1-p,-p;\frac{m+5}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )+\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (\frac{m+3}{2};-p,1-p;\frac{m+5}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )+a (m+3) F_1\left (\frac{m+1}{2};-p,-p;\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(d*x)^m*(a + b*x^2 + c*x^4)^p,x]

[Out]

-((2^(-2 - p)*c*(b + Sqrt[b^2 - 4*a*c])*(3 + m)*x*(d*x)^m*((b - Sqrt[b^2 - 4*a*c
] + 2*c*x^2)/c)^(1 + p)*(-2*a + (-b + Sqrt[b^2 - 4*a*c])*x^2)^2*(a + b*x^2 + c*x
^4)^(-1 + p)*AppellF1[(1 + m)/2, -p, -p, (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])])/((-b + Sqrt[b^2 - 4*a*c])*(1 + m)*(
(b - Sqrt[b^2 - 4*a*c])/(2*c) + x^2)^p*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)*(a*(3 +
m)*AppellF1[(1 + m)/2, -p, -p, (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])] + p*x^2*((b - Sqrt[b^2 - 4*a*c])*AppellF1[(3
+ m)/2, 1 - p, -p, (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, -p, 1 - p, (
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])
]))))

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Maple [F]  time = 0.083, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m*(c*x^4+b*x^2+a)^p,x)

[Out]

int((d*x)^m*(c*x^4+b*x^2+a)^p,x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^4 + b*x^2 + a)^p*(d*x)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c*x^4 + b*x^2 + a)^p*(d*x)^m, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)**m*(c*x**4+b*x**2+a)**p,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^4 + b*x^2 + a)^p*(d*x)^m, x)