3.6.29 \(\int \frac {(d x)^m}{\sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \, dx\) [529]

Optimal. Leaf size=76 \[ \frac {(d x)^{1+m} \left (a+b x^n\right ) \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{a d (1+m) \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \]

[Out]

(d*x)^(1+m)*(a+b*x^n)*hypergeom([1, (1+m)/n],[(1+m+n)/n],-b*x^n/a)/a/d/(1+m)/(a^2+2*a*b*x^n+b^2*x^(2*n))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1369, 371} \begin {gather*} \frac {(d x)^{m+1} \left (a+b x^n\right ) \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{a d (m+1) \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d*x)^(1 + m)*(a + b*x^n)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(a*d*(1 + m)*Sqrt[a^2
+ 2*a*b*x^n + b^2*x^(2*n)])

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 1369

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^
(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; Fr
eeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rubi steps

\begin {align*} \int \frac {(d x)^m}{\sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \, dx &=\frac {\left (a b+b^2 x^n\right ) \int \frac {(d x)^m}{a b+b^2 x^n} \, dx}{\sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=\frac {(d x)^{1+m} \left (a+b x^n\right ) \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{a d (1+m) \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.03, size = 62, normalized size = 0.82 \begin {gather*} \frac {x (d x)^m \left (a+b x^n\right ) \, _2F_1\left (1,\frac {1+m}{n};1+\frac {1+m}{n};-\frac {b x^n}{a}\right )}{a (1+m) \sqrt {\left (a+b x^n\right )^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(d*x)^m*(a + b*x^n)*Hypergeometric2F1[1, (1 + m)/n, 1 + (1 + m)/n, -((b*x^n)/a)])/(a*(1 + m)*Sqrt[(a + b*x^
n)^2])

________________________________________________________________________________________

Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (d x \right )^{m}}{\sqrt {a^{2}+2 a b \,x^{n}+b^{2} x^{2 n}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m/(a^2+2*a*b*x^n+b^2*x^(2*n))^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m/(a^2+2*a*b*x^n+b^2*x^(2*n))^(1/2),x, algorithm="fricas")

[Out]

integral((d*x)^m/sqrt(b^2*x^(2*n) + 2*a*b*x^n + a^2), x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d x\right )^{m}}{\sqrt {\left (a + b x^{n}\right )^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m/(a**2+2*a*b*x**n+b**2*x**(2*n))**(1/2),x)

[Out]

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m/(a^2+2*a*b*x^n+b^2*x^(2*n))^(1/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d\,x\right )}^m}{\sqrt {a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________