3.876 \(\int (\frac {a+b+c x^2}{d})^m \, dx\)

Optimal. Leaf size=49 \[ \frac {d x \left (\frac {a+b}{d}+\frac {c x^2}{d}\right )^{m+1} \, _2F_1\left (1,m+\frac {3}{2};\frac {3}{2};-\frac {c x^2}{a+b}\right )}{a+b} \]

[Out]

d*x*((a+b)/d+x^2*c/d)^(1+m)*hypergeom([1, 3/2+m],[3/2],-c*x^2/(a+b))/(a+b)

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Rubi [A]  time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.16, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1972, 246, 245} \[ x \left (\frac {c x^2}{a+b}+1\right )^{-m} \left (\frac {a+b}{d}+\frac {c x^2}{d}\right )^m \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};-\frac {c x^2}{a+b}\right ) \]

Antiderivative was successfully verified.

[In]

Int[((a + b + c*x^2)/d)^m,x]

[Out]

(x*((a + b)/d + (c*x^2)/d)^m*Hypergeometric2F1[1/2, -m, 3/2, -((c*x^2)/(a + b))])/(1 + (c*x^2)/(a + b))^m

Rule 245

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

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^Fr
acPart[p], Int[(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 1972

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && BinomialQ[u, x] &&  !BinomialMatchQ[
u, x]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 53, normalized size = 1.08 \[ x \left (\frac {c x^2}{a+b}+1\right )^{-m} \left (\frac {a+b+c x^2}{d}\right )^m \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};-\frac {c x^2}{a+b}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b + c*x^2)/d)^m,x]

[Out]

(x*((a + b + c*x^2)/d)^m*Hypergeometric2F1[1/2, -m, 3/2, -((c*x^2)/(a + b))])/(1 + (c*x^2)/(a + b))^m

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fricas [F]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (\frac {c x^{2} + a + b}{d}\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(((c*x^2 + a + b)/d)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((c*x^2 + a + b)/d)^m, x)

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[ \int \left (\frac {c \,x^{2}+a +b}{d}\right )^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((c*x^2 + a + b)/d)^m, x)

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mupad [B]  time = 4.67, size = 54, normalized size = 1.10 \[ \frac {x\,{\left (\frac {a+b}{d}+\frac {c\,x^2}{d}\right )}^m\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},-m;\ \frac {3}{2};\ -\frac {c\,x^2}{a+b}\right )}{{\left (\frac {c\,x^2}{a+b}+1\right )}^m} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b + c*x^2)/d)^m,x)

[Out]

(x*((a + b)/d + (c*x^2)/d)^m*hypergeom([1/2, -m], 3/2, -(c*x^2)/(a + b)))/((c*x^2)/(a + b) + 1)^m

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(((a + b + c*x**2)/d)**m, x)

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