3.603 \(\int (d x)^m \sqrt{a+b x^n+c x^{2 n}} \, dx\)

Optimal. Leaf size=160 \[ \frac{(d x)^{m+1} \sqrt{a+b x^n+c x^{2 n}} F_1\left (\frac{m+1}{n};-\frac{1}{2},-\frac{1}{2};\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{d (m+1) \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}} \]

[Out]

((d*x)^(1 + m)*Sqrt[a + b*x^n + c*x^(2*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])
])/(d*(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])])

_______________________________________________________________________________________

Rubi [A]  time = 0.471221, antiderivative size = 160, 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 \[ \frac{(d x)^{m+1} \sqrt{a+b x^n+c x^{2 n}} F_1\left (\frac{m+1}{n};-\frac{1}{2},-\frac{1}{2};\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{d (m+1) \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1}} \]

Antiderivative was successfully verified.

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

[Out]

((d*x)^(1 + m)*Sqrt[a + b*x^n + c*x^(2*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])
])/(d*(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])])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 35.7589, size = 138, normalized size = 0.86 \[ \frac{\left (d x\right )^{m + 1} \sqrt{a + b x^{n} + c x^{2 n}} \operatorname{appellf_{1}}{\left (\frac{m + 1}{n},- \frac{1}{2},- \frac{1}{2},\frac{m + n + 1}{n},- \frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{d \left (m + 1\right ) \sqrt{\frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d*x)**(m + 1)*sqrt(a + b*x**n + c*x**(2*n))*appellf1((m + 1)/n, -1/2, -1/2, (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)))/(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 = 6.09769, size = 930, normalized size = 5.81 \[ \frac{x \sqrt{b x^n+c x^{2 n}+a} (d x)^m}{m+n+1}+\frac{4 a^3 n x \left (2 c x^n+b-\sqrt{b^2-4 a c}\right ) \left (2 c x^n+b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{m+1}{n};\frac{1}{2},\frac{1}{2};\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right ) (d x)^m}{\left (b-\sqrt{b^2-4 a c}\right ) \left (b+\sqrt{b^2-4 a c}\right ) (m+1) \left (\left (c x^n+b\right ) x^n+a\right )^{3/2} \left (4 a (m+n+1) F_1\left (\frac{m+1}{n};\frac{1}{2},\frac{1}{2};\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )-n x^n \left (\left (b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{m+n+1}{n};\frac{1}{2},\frac{3}{2};\frac{m+2 n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{m+n+1}{n};\frac{3}{2},\frac{1}{2};\frac{m+2 n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right )\right )}+\frac{2 a^2 b n (m+2 n+1) x^{n+1} \left (2 c x^n+b-\sqrt{b^2-4 a c}\right ) \left (2 c x^n+b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{m+n+1}{n};\frac{1}{2},\frac{1}{2};\frac{m+2 n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right ) (d x)^m}{\left (b-\sqrt{b^2-4 a c}\right ) \left (b+\sqrt{b^2-4 a c}\right ) (m+n+1)^2 \left (\left (c x^n+b\right ) x^n+a\right )^{3/2} \left (4 a (m+2 n+1) F_1\left (\frac{m+n+1}{n};\frac{1}{2},\frac{1}{2};\frac{m+2 n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )-n x^n \left (\left (b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{m+2 n+1}{n};\frac{1}{2},\frac{3}{2};\frac{m+3 n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{m+2 n+1}{n};\frac{3}{2},\frac{1}{2};\frac{m+3 n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x*(d*x)^m*Sqrt[a + b*x^n + c*x^(2*n)])/(1 + m + n) + (4*a^3*n*x*(d*x)^m*(b - Sq
rt[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 + Sq
rt[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])]))) + (2*a^2*b*n*(
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 - Sqrt[b^2 -
4*a*c])*(b + Sqrt[b^2 - 4*a*c])*(1 + m + n)^2*(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 + Sq
rt[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])])))

_______________________________________________________________________________________

Maple [F]  time = 0.187, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m}\sqrt{a+b{x}^{n}+c{x}^{2\,n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(c*x^(2*n) + b*x^n + a)*(d*x)^m, 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(sqrt(c*x^(2*n) + b*x^n + a)*(d*x)^m,x, algorithm="fricas")

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d*x)**m*sqrt(a + b*x**n + c*x**(2*n)), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(c*x^(2*n) + b*x^n + a)*(d*x)^m, x)