3.225 \(\int (f x)^m \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^{3/2} \, dx\)

Optimal. Leaf size=319 \[ \frac{a d (f x)^{m+1} \sqrt{a+b x^2+c x^4} F_1\left (\frac{m+1}{2};-\frac{3}{2},-\frac{3}{2};\frac{m+3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{f (m+1) \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}+\frac{a e (f x)^{m+3} \sqrt{a+b x^2+c x^4} F_1\left (\frac{m+3}{2};-\frac{3}{2},-\frac{3}{2};\frac{m+5}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{f^3 (m+3) \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}} \]

[Out]

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

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Rubi [A]  time = 1.1055, antiderivative size = 319, 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 \[ \frac{a d (f x)^{m+1} \sqrt{a+b x^2+c x^4} F_1\left (\frac{m+1}{2};-\frac{3}{2},-\frac{3}{2};\frac{m+3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{f (m+1) \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}}+\frac{a e (f x)^{m+3} \sqrt{a+b x^2+c x^4} F_1\left (\frac{m+3}{2};-\frac{3}{2},-\frac{3}{2};\frac{m+5}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{f^3 (m+3) \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1}} \]

Antiderivative was successfully verified.

[In]  Int[(f*x)^m*(d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 94.8986, size = 286, normalized size = 0.9 \[ \frac{a d \left (f x\right )^{m + 1} \sqrt{a + b x^{2} + c x^{4}} \operatorname{appellf_{1}}{\left (\frac{m}{2} + \frac{1}{2},- \frac{3}{2},- \frac{3}{2},\frac{m}{2} + \frac{3}{2},- \frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{f \left (m + 1\right ) \sqrt{\frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} + \frac{a e \left (f x\right )^{m + 3} \sqrt{a + b x^{2} + c x^{4}} \operatorname{appellf_{1}}{\left (\frac{m}{2} + \frac{3}{2},- \frac{3}{2},- \frac{3}{2},\frac{m}{2} + \frac{5}{2},- \frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{f^{3} \left (m + 3\right ) \sqrt{\frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

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)**(3/2),x)

[Out]

a*d*(f*x)**(m + 1)*sqrt(a + b*x**2 + c*x**4)*appellf1(m/2 + 1/2, -3/2, -3/2, 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))
)/(f*(m + 1)*sqrt(2*c*x**2/(b - sqrt(-4*a*c + b**2)) + 1)*sqrt(2*c*x**2/(b + sqr
t(-4*a*c + b**2)) + 1)) + a*e*(f*x)**(m + 3)*sqrt(a + b*x**2 + c*x**4)*appellf1(
m/2 + 3/2, -3/2, -3/2, 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)))/(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 = 5.37283, size = 2559, normalized size = 8.02 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(f*x)^m*(d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2),x]

[Out]

(a*(b - Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c])*d*(3 + m)*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)*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])])/(8*c^2*(1 + m)*Sqrt[a + b*x^2 + c*x^4]*(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, 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])] + (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])]))) + (b*(b -
Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c])*d*(5 + m)*x^3*(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)*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])])/(8*c^2*(3 + m)*Sqrt[a + b*x^2 + c*x^4]*(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, 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])]
+ (b - Sqrt[b^2 - 4*a*c])*AppellF1[(5 + m)/2, 1/2, -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])]))) + (a*(b - Sqrt[b
^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c])*e*(5 + m)*x^3*(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)*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])
])/(8*c^2*(3 + m)*Sqrt[a + b*x^2 + c*x^4]*(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, 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])] + (b
- Sqrt[b^2 - 4*a*c])*AppellF1[(5 + m)/2, 1/2, -1/2, (7 + m)/2, (-2*c*x^2)/(b + S
qrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]))) + ((b - Sqrt[b^2 - 4*a
*c])*(b + Sqrt[b^2 - 4*a*c])*d*(7 + m)*x^5*(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)*AppellF1[(5 + m)/2, -1/2, -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])])/(8*c*
(5 + m)*Sqrt[a + b*x^2 + c*x^4]*(2*a*(7 + m)*AppellF1[(5 + m)/2, -1/2, -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])]
+ x^2*((b + Sqrt[b^2 - 4*a*c])*AppellF1[(7 + m)/2, -1/2, 1/2, (9 + 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[(7 + m)/2, 1/2, -1/2, (9 + 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*(b - Sqrt[b^2 - 4*a*c])*(b
+ Sqrt[b^2 - 4*a*c])*e*(7 + m)*x^5*(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)*AppellF1[(5 + m)/2, -1/2, -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])])/(8*c^2*(5 + m
)*Sqrt[a + b*x^2 + c*x^4]*(2*a*(7 + m)*AppellF1[(5 + m)/2, -1/2, -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])] + x^2*
((b + Sqrt[b^2 - 4*a*c])*AppellF1[(7 + m)/2, -1/2, 1/2, (9 + 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[(7 + m)/2, 1/2, -1/2, (9 + 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])*(b + Sqrt[b
^2 - 4*a*c])*e*(9 + m)*x^7*(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)*AppellF1[(7 + m)/2, -1/2, -1/2, (9 + m)/2, (-2*c*x^2)/(b
+ Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/(8*c*(7 + m)*Sqrt[a +
b*x^2 + c*x^4]*(2*a*(9 + m)*AppellF1[(7 + m)/2, -1/2, -1/2, (9 + 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[(9 + m)/2, -1/2, 1/2, (11 + 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])*Appe
llF1[(9 + m)/2, 1/2, -1/2, (11 + 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 = 0.015, size = 0, normalized size = 0. \[ \int \left ( fx \right ) ^{m} \left ( e{x}^{2}+d \right ) \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}\, dx \]

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)^(3/2),x)

[Out]

int((f*x)^m*(e*x^2+d)*(c*x^4+b*x^2+a)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}{\left (e x^{2} + d\right )} \left (f x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^(3/2)*(e*x^2 + d)*(f*x)^m,x, algorithm="maxima")

[Out]

integrate((c*x^4 + b*x^2 + a)^(3/2)*(e*x^2 + d)*(f*x)^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c e x^{6} +{\left (c d + b e\right )} x^{4} +{\left (b d + a e\right )} x^{2} + a d\right )} \sqrt{c x^{4} + b x^{2} + a} \left (f x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^(3/2)*(e*x^2 + d)*(f*x)^m,x, algorithm="fricas")

[Out]

integral((c*e*x^6 + (c*d + b*e)*x^4 + (b*d + a*e)*x^2 + a*d)*sqrt(c*x^4 + b*x^2
+ a)*(f*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((f*x)**m*(e*x**2+d)*(c*x**4+b*x**2+a)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}{\left (e x^{2} + d\right )} \left (f x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^(3/2)*(e*x^2 + d)*(f*x)^m,x, algorithm="giac")

[Out]

integrate((c*x^4 + b*x^2 + a)^(3/2)*(e*x^2 + d)*(f*x)^m, x)