3.2848 \(\int (c+d x)^4 (a+b (c+d x)^2)^p \, dx\)

Optimal. Leaf size=55 \[ \frac {(c+d x)^5 \left (a+b (c+d x)^2\right )^{p+1} \, _2F_1\left (1,p+\frac {7}{2};\frac {7}{2};-\frac {b (c+d x)^2}{a}\right )}{5 a d} \]

[Out]

1/5*(d*x+c)^5*(a+b*(d*x+c)^2)^(1+p)*hypergeom([1, 7/2+p],[7/2],-b*(d*x+c)^2/a)/a/d

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Rubi [A]  time = 0.05, antiderivative size = 68, normalized size of antiderivative = 1.24, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {372, 365, 364} \[ \frac {(c+d x)^5 \left (a+b (c+d x)^2\right )^p \left (\frac {b (c+d x)^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {5}{2},-p;\frac {7}{2};-\frac {b (c+d x)^2}{a}\right )}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^4*(a + b*(c + d*x)^2)^p,x]

[Out]

((c + d*x)^5*(a + b*(c + d*x)^2)^p*Hypergeometric2F1[5/2, -p, 7/2, -((b*(c + d*x)^2)/a)])/(5*d*(1 + (b*(c + d*
x)^2)/a)^p)

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 372

Int[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m), Subst[Int[x^m
*(a + b*x^n)^p, x], x, v], x] /; FreeQ[{a, b, m, n, p}, x] && LinearPairQ[u, v, x]

Rubi steps

\begin {align*} \int (c+d x)^4 \left (a+b (c+d x)^2\right )^p \, dx &=\frac {\operatorname {Subst}\left (\int x^4 \left (a+b x^2\right )^p \, dx,x,c+d x\right )}{d}\\ &=\frac {\left (\left (a+b (c+d x)^2\right )^p \left (1+\frac {b (c+d x)^2}{a}\right )^{-p}\right ) \operatorname {Subst}\left (\int x^4 \left (1+\frac {b x^2}{a}\right )^p \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x)^5 \left (a+b (c+d x)^2\right )^p \left (1+\frac {b (c+d x)^2}{a}\right )^{-p} \, _2F_1\left (\frac {5}{2},-p;\frac {7}{2};-\frac {b (c+d x)^2}{a}\right )}{5 d}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 68, normalized size = 1.24 \[ \frac {(c+d x)^5 \left (a+b (c+d x)^2\right )^p \left (\frac {b (c+d x)^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {5}{2},-p;\frac {7}{2};-\frac {b (c+d x)^2}{a}\right )}{5 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^4*(a + b*(c + d*x)^2)^p,x]

[Out]

((c + d*x)^5*(a + b*(c + d*x)^2)^p*Hypergeometric2F1[5/2, -p, 7/2, -((b*(c + d*x)^2)/a)])/(5*d*(1 + (b*(c + d*
x)^2)/a)^p)

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fricas [F]  time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}\right )} {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4*(a+b*(d*x+c)^2)^p,x, algorithm="fricas")

[Out]

integral((d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2*x^2 + 4*c^3*d*x + c^4)*(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a)^p, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{4} {\left ({\left (d x + c\right )}^{2} b + a\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4*(a+b*(d*x+c)^2)^p,x, algorithm="giac")

[Out]

integrate((d*x + c)^4*((d*x + c)^2*b + a)^p, x)

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maple [F]  time = 0.45, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right )^{4} \left (a +\left (d x +c \right )^{2} b \right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^4*(a+b*(d*x+c)^2)^p,x)

[Out]

int((d*x+c)^4*(a+b*(d*x+c)^2)^p,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{4} {\left ({\left (d x + c\right )}^{2} b + a\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4*(a+b*(d*x+c)^2)^p,x, algorithm="maxima")

[Out]

integrate((d*x + c)^4*((d*x + c)^2*b + a)^p, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (a+b\,{\left (c+d\,x\right )}^2\right )}^p\,{\left (c+d\,x\right )}^4 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*(c + d*x)^2)^p*(c + d*x)^4,x)

[Out]

int((a + b*(c + d*x)^2)^p*(c + d*x)^4, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**4*(a+b*(d*x+c)**2)**p,x)

[Out]

Timed out

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