3.605 \(\int \frac{(d x)^m}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx\)
Optimal. Leaf size=163 \[ \frac{(d x)^{m+1} \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{m+1}{n};\frac{3}{2},\frac{3}{2};\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{a d (m+1) \sqrt{a+b x^n+c x^{2 n}}} \]
[Out]
((d*x)^(1 + m)*Sqrt[1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^n)/(b
+ Sqrt[b^2 - 4*a*c])]*AppellF1[(1 + m)/n, 3/2, 3/2, (1 + m + n)/n, (-2*c*x^n)/(
b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*d*(1 + m)*Sqrt[a
+ b*x^n + c*x^(2*n)])
_______________________________________________________________________________________
Rubi [A] time = 0.4882, antiderivative size = 163, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083
\[ \frac{(d x)^{m+1} \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{m+1}{n};\frac{3}{2},\frac{3}{2};\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{a d (m+1) \sqrt{a+b x^n+c x^{2 n}}} \]
Antiderivative was successfully verified.
[In] Int[(d*x)^m/(a + b*x^n + c*x^(2*n))^(3/2),x]
[Out]
((d*x)^(1 + m)*Sqrt[1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^n)/(b
+ Sqrt[b^2 - 4*a*c])]*AppellF1[(1 + m)/n, 3/2, 3/2, (1 + m + n)/n, (-2*c*x^n)/(
b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(a*d*(1 + m)*Sqrt[a
+ b*x^n + c*x^(2*n)])
_______________________________________________________________________________________
Rubi in Sympy [A] time = 45.2513, size = 138, normalized size = 0.85 \[ \frac{\left (d x\right )^{m + 1} \sqrt{a + b x^{n} + c x^{2 n}} \operatorname{appellf_{1}}{\left (\frac{m + 1}{n},\frac{3}{2},\frac{3}{2},\frac{m + n + 1}{n},- \frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{a^{2} d \left (m + 1\right ) \sqrt{\frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x)**m/(a+b*x**n+c*x**(2*n))**(3/2),x)
[Out]
(d*x)**(m + 1)*sqrt(a + b*x**n + c*x**(2*n))*appellf1((m + 1)/n, 3/2, 3/2, (m +
n + 1)/n, -2*c*x**n/(b - sqrt(-4*a*c + b**2)), -2*c*x**n/(b + sqrt(-4*a*c + b**2
)))/(a**2*d*(m + 1)*sqrt(2*c*x**n/(b - sqrt(-4*a*c + b**2)) + 1)*sqrt(2*c*x**n/(
b + sqrt(-4*a*c + b**2)) + 1))
_______________________________________________________________________________________
Mathematica [B] time = 6.2686, size = 3743, normalized size = 22.96 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(d*x)^m/(a + b*x^n + c*x^(2*n))^(3/2),x]
[Out]
(2*x*(d*x)^m*(-b^2 + 2*a*c - b*c*x^n))/(a*(-b^2 + 4*a*c)*n*Sqrt[a + b*x^n + c*x^
(2*n)]) - (4*a*b^2*(1 + m + n)*x*(d*x)^m*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)*(b +
Sqrt[b^2 - 4*a*c] + 2*c*x^n)*AppellF1[(1 + m)/n, 1/2, 1/2, (1 + m + n)/n, (-2*c*
x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/((-b^2 + 4*a*
c)*(b - Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c])*(1 + m)*(a + x^n*(b + c*x^n))
^(3/2)*(4*a*(1 + m + n)*AppellF1[(1 + m)/n, 1/2, 1/2, (1 + m + n)/n, (-2*c*x^n)/
(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] - n*x^n*((b + Sqrt[
b^2 - 4*a*c])*AppellF1[(1 + m + n)/n, 1/2, 3/2, (1 + m + 2*n)/n, (-2*c*x^n)/(b +
Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + (b - Sqrt[b^2 - 4*a*c
])*AppellF1[(1 + m + n)/n, 3/2, 1/2, (1 + m + 2*n)/n, (-2*c*x^n)/(b + Sqrt[b^2 -
4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]))) + (16*a^2*c*(1 + m + n)*x*(d*x)
^m*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)*AppellF1[
(1 + m)/n, 1/2, 1/2, (1 + m + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n
)/(-b + Sqrt[b^2 - 4*a*c])])/((-b^2 + 4*a*c)*(b - Sqrt[b^2 - 4*a*c])*(b + Sqrt[b
^2 - 4*a*c])*(1 + m)*(a + x^n*(b + c*x^n))^(3/2)*(4*a*(1 + m + n)*AppellF1[(1 +
m)/n, 1/2, 1/2, (1 + m + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b
+ Sqrt[b^2 - 4*a*c])] - n*x^n*((b + Sqrt[b^2 - 4*a*c])*AppellF1[(1 + m + n)/n,
1/2, 3/2, (1 + m + 2*n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + S
qrt[b^2 - 4*a*c])] + (b - Sqrt[b^2 - 4*a*c])*AppellF1[(1 + m + n)/n, 3/2, 1/2, (
1 + m + 2*n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4
*a*c])]))) + (8*a*b^2*(1 + m + n)*x*(d*x)^m*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)*(b
+ Sqrt[b^2 - 4*a*c] + 2*c*x^n)*AppellF1[(1 + m)/n, 1/2, 1/2, (1 + m + n)/n, (-2
*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/((-b^2 + 4
*a*c)*(b - Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c])*(1 + m)*n*(a + x^n*(b + c*
x^n))^(3/2)*(4*a*(1 + m + n)*AppellF1[(1 + m)/n, 1/2, 1/2, (1 + m + n)/n, (-2*c*
x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] - n*x^n*((b +
Sqrt[b^2 - 4*a*c])*AppellF1[(1 + m + n)/n, 1/2, 3/2, (1 + m + 2*n)/n, (-2*c*x^n)
/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + (b - Sqrt[b^2 -
4*a*c])*AppellF1[(1 + m + n)/n, 3/2, 1/2, (1 + m + 2*n)/n, (-2*c*x^n)/(b + Sqrt[
b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]))) - (16*a^2*c*(1 + m + n)*x*
(d*x)^m*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)*Appe
llF1[(1 + m)/n, 1/2, 1/2, (1 + m + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*
c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/((-b^2 + 4*a*c)*(b - Sqrt[b^2 - 4*a*c])*(b + S
qrt[b^2 - 4*a*c])*(1 + m)*n*(a + x^n*(b + c*x^n))^(3/2)*(4*a*(1 + m + n)*AppellF
1[(1 + m)/n, 1/2, 1/2, (1 + m + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x
^n)/(-b + Sqrt[b^2 - 4*a*c])] - n*x^n*((b + Sqrt[b^2 - 4*a*c])*AppellF1[(1 + m +
n)/n, 1/2, 3/2, (1 + m + 2*n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/
(-b + Sqrt[b^2 - 4*a*c])] + (b - Sqrt[b^2 - 4*a*c])*AppellF1[(1 + m + n)/n, 3/2,
1/2, (1 + m + 2*n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[
b^2 - 4*a*c])]))) + (8*a*b^2*m*(1 + m + n)*x*(d*x)^m*(b - Sqrt[b^2 - 4*a*c] + 2*
c*x^n)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)*AppellF1[(1 + m)/n, 1/2, 1/2, (1 + m +
n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/(
(-b^2 + 4*a*c)*(b - Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c])*(1 + m)*n*(a + x^
n*(b + c*x^n))^(3/2)*(4*a*(1 + m + n)*AppellF1[(1 + m)/n, 1/2, 1/2, (1 + m + n)/
n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] - n*x
^n*((b + Sqrt[b^2 - 4*a*c])*AppellF1[(1 + m + n)/n, 1/2, 3/2, (1 + m + 2*n)/n, (
-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + (b - Sq
rt[b^2 - 4*a*c])*AppellF1[(1 + m + n)/n, 3/2, 1/2, (1 + m + 2*n)/n, (-2*c*x^n)/(
b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]))) - (16*a^2*c*m*(1
+ m + n)*x*(d*x)^m*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)*(b + Sqrt[b^2 - 4*a*c] + 2*
c*x^n)*AppellF1[(1 + m)/n, 1/2, 1/2, (1 + m + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4
*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/((-b^2 + 4*a*c)*(b - Sqrt[b^2 - 4*a
*c])*(b + Sqrt[b^2 - 4*a*c])*(1 + m)*n*(a + x^n*(b + c*x^n))^(3/2)*(4*a*(1 + m +
n)*AppellF1[(1 + m)/n, 1/2, 1/2, (1 + m + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*
c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] - n*x^n*((b + Sqrt[b^2 - 4*a*c])*Appell
F1[(1 + m + n)/n, 1/2, 3/2, (1 + m + 2*n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]),
(2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + (b - Sqrt[b^2 - 4*a*c])*AppellF1[(1 + m +
n)/n, 3/2, 1/2, (1 + m + 2*n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/
(-b + Sqrt[b^2 - 4*a*c])]))) + (8*a*b*c*(1 + m + 2*n)*x^(1 + n)*(d*x)^m*(b - Sqr
t[b^2 - 4*a*c] + 2*c*x^n)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)*AppellF1[(1 + m + n)
/n, 1/2, 1/2, (1 + m + 2*n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b
+ Sqrt[b^2 - 4*a*c])])/((-b^2 + 4*a*c)*(b - Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 -
4*a*c])*n*(1 + m + n)*(a + x^n*(b + c*x^n))^(3/2)*(4*a*(1 + m + 2*n)*AppellF1[(1
+ m + n)/n, 1/2, 1/2, (1 + m + 2*n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c
*x^n)/(-b + Sqrt[b^2 - 4*a*c])] - n*x^n*((b + Sqrt[b^2 - 4*a*c])*AppellF1[(1 + m
+ 2*n)/n, 1/2, 3/2, (1 + m + 3*n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x
^n)/(-b + Sqrt[b^2 - 4*a*c])] + (b - Sqrt[b^2 - 4*a*c])*AppellF1[(1 + m + 2*n)/n
, 3/2, 1/2, (1 + m + 3*n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b +
Sqrt[b^2 - 4*a*c])]))) + (8*a*b*c*m*(1 + m + 2*n)*x^(1 + n)*(d*x)^m*(b - Sqrt[b
^2 - 4*a*c] + 2*c*x^n)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)*AppellF1[(1 + m + n)/n,
1/2, 1/2, (1 + m + 2*n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b +
Sqrt[b^2 - 4*a*c])])/((-b^2 + 4*a*c)*(b - Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a
*c])*n*(1 + m + n)*(a + x^n*(b + c*x^n))^(3/2)*(4*a*(1 + m + 2*n)*AppellF1[(1 +
m + n)/n, 1/2, 1/2, (1 + m + 2*n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^
n)/(-b + Sqrt[b^2 - 4*a*c])] - n*x^n*((b + Sqrt[b^2 - 4*a*c])*AppellF1[(1 + m +
2*n)/n, 1/2, 3/2, (1 + m + 3*n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)
/(-b + Sqrt[b^2 - 4*a*c])] + (b - Sqrt[b^2 - 4*a*c])*AppellF1[(1 + m + 2*n)/n, 3
/2, 1/2, (1 + m + 3*n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sq
rt[b^2 - 4*a*c])])))
_______________________________________________________________________________________
Maple [F] time = 0.032, size = 0, normalized size = 0. \[ \int{ \left ( dx \right ) ^{m} \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{-{\frac{3}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x)^m/(a+b*x^n+c*x^(2*n))^(3/2),x)
[Out]
int((d*x)^m/(a+b*x^n+c*x^(2*n))^(3/2),x)
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d x\right )^{m}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)^m/(c*x^(2*n) + b*x^n + a)^(3/2),x, algorithm="maxima")
[Out]
integrate((d*x)^m/(c*x^(2*n) + b*x^n + a)^(3/2), x)
_______________________________________________________________________________________
Fricas [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((d*x)^m/(c*x^(2*n) + b*x^n + a)^(3/2),x, algorithm="fricas")
[Out]
Exception raised: TypeError
_______________________________________________________________________________________
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/(a+b*x**n+c*x**(2*n))**(3/2),x)
[Out]
Timed out
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d x\right )^{m}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)^m/(c*x^(2*n) + b*x^n + a)^(3/2),x, algorithm="giac")
[Out]
integrate((d*x)^m/(c*x^(2*n) + b*x^n + a)^(3/2), x)