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)*hypergeom([1/2, (1+m)/n],[(1+m+n)/n],-b*x^n/a)*(1+b*x^n/a)^(1/2)/c/(1+m)/(a+b*x^n)^(1/2)+e*x^(1+
n)*(c*x)^m*hypergeom([1/2, (1+m+n)/n],[(1+m+2*n)/n],-b*x^n/a)*(1+b*x^n/a)^(1/2)/(1+m+n)/(a+b*x^n)^(1/2)+f*x^(1
+2*n)*(c*x)^m*hypergeom([1/2, (1+m+2*n)/n],[(1+m+3*n)/n],-b*x^n/a)*(1+b*x^n/a)^(1/2)/(1+m+2*n)/(a+b*x^n)^(1/2)
+g*x^(1+3*n)*(c*x)^m*hypergeom([1/2, (1+m+3*n)/n],[(1+m+4*n)/n],-b*x^n/a)*(1+b*x^n/a)^(1/2)/(1+m+3*n)/(a+b*x^n
)^(1/2)

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Rubi [A]  time = 0.23, antiderivative size = 305, normalized size of antiderivative = 1.00, 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 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]

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 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 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]

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*}

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Mathematica [A]  time = 0.56, 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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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x^{3 \, n} + f x^{2 \, n} + e x^{n} + d\right )} \left (c x\right )^{m}}{\sqrt {b x^{n} + a}}\,{d x} \]

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)

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maple [F]  time = 0.70, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \,x^{n}+f \,x^{2 n}+g \,x^{3 n}+d \right ) \left (c x \right )^{m}}{\sqrt {b \,x^{n}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x^{3 \, n} + f x^{2 \, n} + e x^{n} + d\right )} \left (c x\right )^{m}}{\sqrt {b x^{n} + a}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,x\right )}^m\,\left (d+e\,x^n+f\,x^{2\,n}+g\,x^{3\,n}\right )}{\sqrt {a+b\,x^n}} \,d x \]

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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sympy [C]  time = 55.36, size = 274, normalized size = 0.90 \[ \frac {c^{m} d x x^{m} \Gamma \left (\frac {m}{n} + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{n} + \frac {1}{n} \\ \frac {m}{n} + 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{\sqrt {a} n \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )} + \frac {c^{m} e x x^{m} x^{n} \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{n} + 1 + \frac {1}{n} \\ \frac {m}{n} + 2 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{\sqrt {a} n \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )} + \frac {c^{m} f x x^{m} x^{2 n} \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{n} + 2 + \frac {1}{n} \\ \frac {m}{n} + 3 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{\sqrt {a} n \Gamma \left (\frac {m}{n} + 3 + \frac {1}{n}\right )} + \frac {c^{m} g x x^{m} x^{3 n} \Gamma \left (\frac {m}{n} + 3 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{n} + 3 + \frac {1}{n} \\ \frac {m}{n} + 4 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{\sqrt {a} n \Gamma \left (\frac {m}{n} + 4 + \frac {1}{n}\right )} \]

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]

c**m*d*x*x**m*gamma(m/n + 1/n)*hyper((1/2, m/n + 1/n), (m/n + 1 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(sqrt(a)*n*
gamma(m/n + 1 + 1/n)) + c**m*e*x*x**m*x**n*gamma(m/n + 1 + 1/n)*hyper((1/2, m/n + 1 + 1/n), (m/n + 2 + 1/n,),
b*x**n*exp_polar(I*pi)/a)/(sqrt(a)*n*gamma(m/n + 2 + 1/n)) + c**m*f*x*x**m*x**(2*n)*gamma(m/n + 2 + 1/n)*hyper
((1/2, m/n + 2 + 1/n), (m/n + 3 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(sqrt(a)*n*gamma(m/n + 3 + 1/n)) + c**m*g*x
*x**m*x**(3*n)*gamma(m/n + 3 + 1/n)*hyper((1/2, m/n + 3 + 1/n), (m/n + 4 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(s
qrt(a)*n*gamma(m/n + 4 + 1/n))

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