3.584 \(\int \frac{(c x)^m (d+e x^n+f x^{2 n}+g x^{3 n})}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=305 \[ \frac{d (c x)^{m+1} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{c (m+1) \sqrt{a+b x^n}}+\frac{e x^{n+1} (c x)^m \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+n+1}{n};\frac{m+2 n+1}{n};-\frac{b x^n}{a}\right )}{(m+n+1) \sqrt{a+b x^n}}+\frac{f x^{2 n+1} (c x)^m \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+2 n+1}{n};\frac{m+3 n+1}{n};-\frac{b x^n}{a}\right )}{(m+2 n+1) \sqrt{a+b x^n}}+\frac{g x^{3 n+1} (c x)^m \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+3 n+1}{n};\frac{m+4 n+1}{n};-\frac{b x^n}{a}\right )}{(m+3 n+1) \sqrt{a+b x^n}} \]

[Out]

(d*(c*x)^(1 + m)*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[1/2, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(c*(1 + m
)*Sqrt[a + b*x^n]) + (e*x^(1 + n)*(c*x)^m*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[1/2, (1 + m + n)/n, (1 + m + 2
*n)/n, -((b*x^n)/a)])/((1 + m + n)*Sqrt[a + b*x^n]) + (f*x^(1 + 2*n)*(c*x)^m*Sqrt[1 + (b*x^n)/a]*Hypergeometri
c2F1[1/2, (1 + m + 2*n)/n, (1 + m + 3*n)/n, -((b*x^n)/a)])/((1 + m + 2*n)*Sqrt[a + b*x^n]) + (g*x^(1 + 3*n)*(c
*x)^m*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[1/2, (1 + m + 3*n)/n, (1 + m + 4*n)/n, -((b*x^n)/a)])/((1 + m + 3*
n)*Sqrt[a + b*x^n])

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Rubi [A]  time = 0.231849, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 4, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {1844, 365, 364, 20} \[ \frac{d (c x)^{m+1} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{c (m+1) \sqrt{a+b x^n}}+\frac{e x^{n+1} (c x)^m \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+n+1}{n};\frac{m+2 n+1}{n};-\frac{b x^n}{a}\right )}{(m+n+1) \sqrt{a+b x^n}}+\frac{f x^{2 n+1} (c x)^m \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+2 n+1}{n};\frac{m+3 n+1}{n};-\frac{b x^n}{a}\right )}{(m+2 n+1) \sqrt{a+b x^n}}+\frac{g x^{3 n+1} (c x)^m \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+3 n+1}{n};\frac{m+4 n+1}{n};-\frac{b x^n}{a}\right )}{(m+3 n+1) \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*(c*x)^(1 + m)*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[1/2, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(c*(1 + m
)*Sqrt[a + b*x^n]) + (e*x^(1 + n)*(c*x)^m*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[1/2, (1 + m + n)/n, (1 + m + 2
*n)/n, -((b*x^n)/a)])/((1 + m + n)*Sqrt[a + b*x^n]) + (f*x^(1 + 2*n)*(c*x)^m*Sqrt[1 + (b*x^n)/a]*Hypergeometri
c2F1[1/2, (1 + m + 2*n)/n, (1 + m + 3*n)/n, -((b*x^n)/a)])/((1 + m + 2*n)*Sqrt[a + b*x^n]) + (g*x^(1 + 3*n)*(c
*x)^m*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[1/2, (1 + m + 3*n)/n, (1 + m + 4*n)/n, -((b*x^n)/a)])/((1 + m + 3*
n)*Sqrt[a + b*x^n])

Rule 1844

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) &&  !IGtQ[m, 0]

Rule 365

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

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(b^IntPart[n]*(b*v)^FracPart[n])/(a^IntPart[n
]*(a*v)^FracPart[n]), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

Rubi steps

\begin{align*} \int \frac{(c x)^m \left (d+e x^n+f x^{2 n}+g x^{3 n}\right )}{\sqrt{a+b x^n}} \, dx &=\int \left (\frac{d (c x)^m}{\sqrt{a+b x^n}}+\frac{e x^n (c x)^m}{\sqrt{a+b x^n}}+\frac{f x^{2 n} (c x)^m}{\sqrt{a+b x^n}}+\frac{g x^{3 n} (c x)^m}{\sqrt{a+b x^n}}\right ) \, dx\\ &=d \int \frac{(c x)^m}{\sqrt{a+b x^n}} \, dx+e \int \frac{x^n (c x)^m}{\sqrt{a+b x^n}} \, dx+f \int \frac{x^{2 n} (c x)^m}{\sqrt{a+b x^n}} \, dx+g \int \frac{x^{3 n} (c x)^m}{\sqrt{a+b x^n}} \, dx\\ &=\left (e x^{-m} (c x)^m\right ) \int \frac{x^{m+n}}{\sqrt{a+b x^n}} \, dx+\left (f x^{-m} (c x)^m\right ) \int \frac{x^{m+2 n}}{\sqrt{a+b x^n}} \, dx+\left (g x^{-m} (c x)^m\right ) \int \frac{x^{m+3 n}}{\sqrt{a+b x^n}} \, dx+\frac{\left (d \sqrt{1+\frac{b x^n}{a}}\right ) \int \frac{(c x)^m}{\sqrt{1+\frac{b x^n}{a}}} \, dx}{\sqrt{a+b x^n}}\\ &=\frac{d (c x)^{1+m} \sqrt{1+\frac{b x^n}{a}} \, _2F_1\left (\frac{1}{2},\frac{1+m}{n};\frac{1+m+n}{n};-\frac{b x^n}{a}\right )}{c (1+m) \sqrt{a+b x^n}}+\frac{\left (e x^{-m} (c x)^m \sqrt{1+\frac{b x^n}{a}}\right ) \int \frac{x^{m+n}}{\sqrt{1+\frac{b x^n}{a}}} \, dx}{\sqrt{a+b x^n}}+\frac{\left (f x^{-m} (c x)^m \sqrt{1+\frac{b x^n}{a}}\right ) \int \frac{x^{m+2 n}}{\sqrt{1+\frac{b x^n}{a}}} \, dx}{\sqrt{a+b x^n}}+\frac{\left (g x^{-m} (c x)^m \sqrt{1+\frac{b x^n}{a}}\right ) \int \frac{x^{m+3 n}}{\sqrt{1+\frac{b x^n}{a}}} \, dx}{\sqrt{a+b x^n}}\\ &=\frac{d (c x)^{1+m} \sqrt{1+\frac{b x^n}{a}} \, _2F_1\left (\frac{1}{2},\frac{1+m}{n};\frac{1+m+n}{n};-\frac{b x^n}{a}\right )}{c (1+m) \sqrt{a+b x^n}}+\frac{e x^{1+n} (c x)^m \sqrt{1+\frac{b x^n}{a}} \, _2F_1\left (\frac{1}{2},\frac{1+m+n}{n};\frac{1+m+2 n}{n};-\frac{b x^n}{a}\right )}{(1+m+n) \sqrt{a+b x^n}}+\frac{f x^{1+2 n} (c x)^m \sqrt{1+\frac{b x^n}{a}} \, _2F_1\left (\frac{1}{2},\frac{1+m+2 n}{n};\frac{1+m+3 n}{n};-\frac{b x^n}{a}\right )}{(1+m+2 n) \sqrt{a+b x^n}}+\frac{g x^{1+3 n} (c x)^m \sqrt{1+\frac{b x^n}{a}} \, _2F_1\left (\frac{1}{2},\frac{1+m+3 n}{n};\frac{1+m+4 n}{n};-\frac{b x^n}{a}\right )}{(1+m+3 n) \sqrt{a+b x^n}}\\ \end{align*}

Mathematica [A]  time = 0.373558, size = 206, normalized size = 0.68 \[ \frac{x (c x)^m \sqrt{\frac{b x^n}{a}+1} \left (\frac{d \, _2F_1\left (\frac{1}{2},\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{m+1}+x^n \left (\frac{e \, _2F_1\left (\frac{1}{2},\frac{m+n+1}{n};\frac{m+2 n+1}{n};-\frac{b x^n}{a}\right )}{m+n+1}+x^n \left (\frac{f \, _2F_1\left (\frac{1}{2},\frac{m+2 n+1}{n};\frac{m+3 n+1}{n};-\frac{b x^n}{a}\right )}{m+2 n+1}+\frac{g x^n \, _2F_1\left (\frac{1}{2},\frac{m+3 n+1}{n};\frac{m+4 n+1}{n};-\frac{b x^n}{a}\right )}{m+3 n+1}\right )\right )\right )}{\sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(c*x)^m*Sqrt[1 + (b*x^n)/a]*((d*Hypergeometric2F1[1/2, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(1 + m) + x
^n*((e*Hypergeometric2F1[1/2, (1 + m + n)/n, (1 + m + 2*n)/n, -((b*x^n)/a)])/(1 + m + n) + x^n*((f*Hypergeomet
ric2F1[1/2, (1 + m + 2*n)/n, (1 + m + 3*n)/n, -((b*x^n)/a)])/(1 + m + 2*n) + (g*x^n*Hypergeometric2F1[1/2, (1
+ m + 3*n)/n, (1 + m + 4*n)/n, -((b*x^n)/a)])/(1 + m + 3*n)))))/Sqrt[a + b*x^n]

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Maple [F]  time = 0.464, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{m} \left ( d+e{x}^{n}+f{x}^{2\,n}+g{x}^{3\,n} \right ){\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^m*(d+e*x^n+f*x^(2*n)+g*x^(3*n))/(a+b*x^n)^(1/2),x)

[Out]

int((c*x)^m*(d+e*x^n+f*x^(2*n)+g*x^(3*n))/(a+b*x^n)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^m*(d+e*x^n+f*x^(2*n)+g*x^(3*n))/(a+b*x^n)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^m*(d+e*x^n+f*x^(2*n)+g*x^(3*n))/(a+b*x^n)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)**m*(d+e*x**n+f*x**(2*n)+g*x**(3*n))/(a+b*x**n)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (g x^{3 \, n} + f x^{2 \, n} + e x^{n} + d\right )} \left (c x\right )^{m}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^m*(d+e*x^n+f*x^(2*n)+g*x^(3*n))/(a+b*x^n)^(1/2),x, algorithm="giac")

[Out]

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