3.654 \(\int x^2 (a+b \sqrt{c+d x})^p \, dx\)

Optimal. Leaf size=242 \[ \frac{2 \left (-6 a^2 b^2 c+5 a^4+b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{p+2}}{b^6 d^3 (p+2)}-\frac{2 a \left (a^2-b^2 c\right )^2 \left (a+b \sqrt{c+d x}\right )^{p+1}}{b^6 d^3 (p+1)}-\frac{4 a \left (5 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+3}}{b^6 d^3 (p+3)}+\frac{4 \left (5 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+4}}{b^6 d^3 (p+4)}-\frac{10 a \left (a+b \sqrt{c+d x}\right )^{p+5}}{b^6 d^3 (p+5)}+\frac{2 \left (a+b \sqrt{c+d x}\right )^{p+6}}{b^6 d^3 (p+6)} \]

[Out]

(-2*a*(a^2 - b^2*c)^2*(a + b*Sqrt[c + d*x])^(1 + p))/(b^6*d^3*(1 + p)) + (2*(5*a^4 - 6*a^2*b^2*c + b^4*c^2)*(a
+ b*Sqrt[c + d*x])^(2 + p))/(b^6*d^3*(2 + p)) - (4*a*(5*a^2 - 3*b^2*c)*(a + b*Sqrt[c + d*x])^(3 + p))/(b^6*d^
3*(3 + p)) + (4*(5*a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(4 + p))/(b^6*d^3*(4 + p)) - (10*a*(a + b*Sqrt[c + d*x])
^(5 + p))/(b^6*d^3*(5 + p)) + (2*(a + b*Sqrt[c + d*x])^(6 + p))/(b^6*d^3*(6 + p))

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Rubi [A]  time = 0.18244, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {371, 1398, 772} \[ \frac{2 \left (-6 a^2 b^2 c+5 a^4+b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{p+2}}{b^6 d^3 (p+2)}-\frac{2 a \left (a^2-b^2 c\right )^2 \left (a+b \sqrt{c+d x}\right )^{p+1}}{b^6 d^3 (p+1)}-\frac{4 a \left (5 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+3}}{b^6 d^3 (p+3)}+\frac{4 \left (5 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+4}}{b^6 d^3 (p+4)}-\frac{10 a \left (a+b \sqrt{c+d x}\right )^{p+5}}{b^6 d^3 (p+5)}+\frac{2 \left (a+b \sqrt{c+d x}\right )^{p+6}}{b^6 d^3 (p+6)} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*Sqrt[c + d*x])^p,x]

[Out]

(-2*a*(a^2 - b^2*c)^2*(a + b*Sqrt[c + d*x])^(1 + p))/(b^6*d^3*(1 + p)) + (2*(5*a^4 - 6*a^2*b^2*c + b^4*c^2)*(a
+ b*Sqrt[c + d*x])^(2 + p))/(b^6*d^3*(2 + p)) - (4*a*(5*a^2 - 3*b^2*c)*(a + b*Sqrt[c + d*x])^(3 + p))/(b^6*d^
3*(3 + p)) + (4*(5*a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(4 + p))/(b^6*d^3*(4 + p)) - (10*a*(a + b*Sqrt[c + d*x])
^(5 + p))/(b^6*d^3*(5 + p)) + (2*(a + b*Sqrt[c + d*x])^(6 + p))/(b^6*d^3*(6 + p))

Rule 371

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coefficient[v, x, 0], d = Coefficient[v,
x, 1]}, Dist[1/d^(m + 1), Subst[Int[SimplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]
] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]

Rule 1398

Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, D
ist[g, Subst[Int[x^(g - 1)*(d + e*x^(g*n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, p
, q}, x] && EqQ[n2, 2*n] && FractionQ[n]

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int x^2 \left (a+b \sqrt{c+d x}\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b \sqrt{x}\right )^p (-c+x)^2 \, dx,x,c+d x\right )}{d^3}\\ &=\frac{2 \operatorname{Subst}\left (\int x (a+b x)^p \left (-c+x^2\right )^2 \, dx,x,\sqrt{c+d x}\right )}{d^3}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (-\frac{a \left (a^2-b^2 c\right )^2 (a+b x)^p}{b^5}+\frac{\left (5 a^4-6 a^2 b^2 c+b^4 c^2\right ) (a+b x)^{1+p}}{b^5}-\frac{2 \left (5 a^3-3 a b^2 c\right ) (a+b x)^{2+p}}{b^5}-\frac{2 \left (-5 a^2+b^2 c\right ) (a+b x)^{3+p}}{b^5}-\frac{5 a (a+b x)^{4+p}}{b^5}+\frac{(a+b x)^{5+p}}{b^5}\right ) \, dx,x,\sqrt{c+d x}\right )}{d^3}\\ &=-\frac{2 a \left (a^2-b^2 c\right )^2 \left (a+b \sqrt{c+d x}\right )^{1+p}}{b^6 d^3 (1+p)}+\frac{2 \left (5 a^4-6 a^2 b^2 c+b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{2+p}}{b^6 d^3 (2+p)}-\frac{4 a \left (5 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{3+p}}{b^6 d^3 (3+p)}+\frac{4 \left (5 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{4+p}}{b^6 d^3 (4+p)}-\frac{10 a \left (a+b \sqrt{c+d x}\right )^{5+p}}{b^6 d^3 (5+p)}+\frac{2 \left (a+b \sqrt{c+d x}\right )^{6+p}}{b^6 d^3 (6+p)}\\ \end{align*}

Mathematica [A]  time = 0.388854, size = 284, normalized size = 1.17 \[ -\frac{2 \left (a+b \sqrt{c+d x}\right )^{p+1} \left (12 a^3 b^2 \left (4 c \left (p^2+p-5\right )+5 d \left (p^2+3 p+2\right ) x\right )-4 a^2 b^3 (p+1) \sqrt{c+d x} \left (2 c \left (p^2-4 p-30\right )+5 d \left (p^2+5 p+6\right ) x\right )-120 a^4 b (p+1) \sqrt{c+d x}+120 a^5+a b^4 \left (-8 c^2 \left (2 p^3+12 p^2+10 p-15\right )+4 c d \left (p^4+4 p^3-10 p^2-43 p-30\right ) x+5 d^2 \left (p^4+10 p^3+35 p^2+50 p+24\right ) x^2\right )-b^5 \left (p^3+9 p^2+23 p+15\right ) \sqrt{c+d x} \left (8 c^2-4 c d (p+2) x+d^2 \left (p^2+6 p+8\right ) x^2\right )\right )}{b^6 d^3 (p+1) (p+2) (p+3) (p+4) (p+5) (p+6)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a + b*Sqrt[c + d*x])^(1 + p)*(120*a^5 - 120*a^4*b*(1 + p)*Sqrt[c + d*x] + 12*a^3*b^2*(4*c*(-5 + p + p^2)
+ 5*d*(2 + 3*p + p^2)*x) - 4*a^2*b^3*(1 + p)*Sqrt[c + d*x]*(2*c*(-30 - 4*p + p^2) + 5*d*(6 + 5*p + p^2)*x) - b
^5*(15 + 23*p + 9*p^2 + p^3)*Sqrt[c + d*x]*(8*c^2 - 4*c*d*(2 + p)*x + d^2*(8 + 6*p + p^2)*x^2) + a*b^4*(-8*c^2
*(-15 + 10*p + 12*p^2 + 2*p^3) + 4*c*d*(-30 - 43*p - 10*p^2 + 4*p^3 + p^4)*x + 5*d^2*(24 + 50*p + 35*p^2 + 10*
p^3 + p^4)*x^2)))/(b^6*d^3*(1 + p)*(2 + p)*(3 + p)*(4 + p)*(5 + p)*(6 + p))

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Maple [F]  time = 0.004, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b\sqrt{dx+c} \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*(d*x+c)^(1/2))^p,x)

[Out]

int(x^2*(a+b*(d*x+c)^(1/2))^p,x)

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Maxima [A]  time = 1.145, size = 543, normalized size = 2.24 \begin{align*} \frac{2 \,{\left (\frac{{\left ({\left (d x + c\right )} b^{2}{\left (p + 1\right )} + \sqrt{d x + c} a b p - a^{2}\right )}{\left (\sqrt{d x + c} b + a\right )}^{p} c^{2}}{{\left (p^{2} + 3 \, p + 2\right )} b^{2}} - \frac{2 \,{\left ({\left (p^{3} + 6 \, p^{2} + 11 \, p + 6\right )}{\left (d x + c\right )}^{2} b^{4} +{\left (p^{3} + 3 \, p^{2} + 2 \, p\right )}{\left (d x + c\right )}^{\frac{3}{2}} a b^{3} - 3 \,{\left (p^{2} + p\right )}{\left (d x + c\right )} a^{2} b^{2} + 6 \, \sqrt{d x + c} a^{3} b p - 6 \, a^{4}\right )}{\left (\sqrt{d x + c} b + a\right )}^{p} c}{{\left (p^{4} + 10 \, p^{3} + 35 \, p^{2} + 50 \, p + 24\right )} b^{4}} + \frac{{\left ({\left (p^{5} + 15 \, p^{4} + 85 \, p^{3} + 225 \, p^{2} + 274 \, p + 120\right )}{\left (d x + c\right )}^{3} b^{6} +{\left (p^{5} + 10 \, p^{4} + 35 \, p^{3} + 50 \, p^{2} + 24 \, p\right )}{\left (d x + c\right )}^{\frac{5}{2}} a b^{5} - 5 \,{\left (p^{4} + 6 \, p^{3} + 11 \, p^{2} + 6 \, p\right )}{\left (d x + c\right )}^{2} a^{2} b^{4} + 20 \,{\left (p^{3} + 3 \, p^{2} + 2 \, p\right )}{\left (d x + c\right )}^{\frac{3}{2}} a^{3} b^{3} - 60 \,{\left (p^{2} + p\right )}{\left (d x + c\right )} a^{4} b^{2} + 120 \, \sqrt{d x + c} a^{5} b p - 120 \, a^{6}\right )}{\left (\sqrt{d x + c} b + a\right )}^{p}}{{\left (p^{6} + 21 \, p^{5} + 175 \, p^{4} + 735 \, p^{3} + 1624 \, p^{2} + 1764 \, p + 720\right )} b^{6}}\right )}}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(((d*x + c)*b^2*(p + 1) + sqrt(d*x + c)*a*b*p - a^2)*(sqrt(d*x + c)*b + a)^p*c^2/((p^2 + 3*p + 2)*b^2) - 2*(
(p^3 + 6*p^2 + 11*p + 6)*(d*x + c)^2*b^4 + (p^3 + 3*p^2 + 2*p)*(d*x + c)^(3/2)*a*b^3 - 3*(p^2 + p)*(d*x + c)*a
^2*b^2 + 6*sqrt(d*x + c)*a^3*b*p - 6*a^4)*(sqrt(d*x + c)*b + a)^p*c/((p^4 + 10*p^3 + 35*p^2 + 50*p + 24)*b^4)
+ ((p^5 + 15*p^4 + 85*p^3 + 225*p^2 + 274*p + 120)*(d*x + c)^3*b^6 + (p^5 + 10*p^4 + 35*p^3 + 50*p^2 + 24*p)*(
d*x + c)^(5/2)*a*b^5 - 5*(p^4 + 6*p^3 + 11*p^2 + 6*p)*(d*x + c)^2*a^2*b^4 + 20*(p^3 + 3*p^2 + 2*p)*(d*x + c)^(
3/2)*a^3*b^3 - 60*(p^2 + p)*(d*x + c)*a^4*b^2 + 120*sqrt(d*x + c)*a^5*b*p - 120*a^6)*(sqrt(d*x + c)*b + a)^p/(
(p^6 + 21*p^5 + 175*p^4 + 735*p^3 + 1624*p^2 + 1764*p + 720)*b^6))/d^3

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Fricas [B]  time = 2.45443, size = 1492, normalized size = 6.17 \begin{align*} \frac{2 \,{\left (120 \, b^{6} c^{3} - 360 \, a^{2} b^{4} c^{2} + 360 \, a^{4} b^{2} c - 120 \, a^{6} + 8 \,{\left (b^{6} c^{3} + 3 \, a^{2} b^{4} c^{2}\right )} p^{3} +{\left (b^{6} d^{3} p^{5} + 15 \, b^{6} d^{3} p^{4} + 85 \, b^{6} d^{3} p^{3} + 225 \, b^{6} d^{3} p^{2} + 274 \, b^{6} d^{3} p + 120 \, b^{6} d^{3}\right )} x^{3} + 24 \,{\left (3 \, b^{6} c^{3} + 3 \, a^{2} b^{4} c^{2} - 2 \, a^{4} b^{2} c\right )} p^{2} +{\left (b^{6} c d^{2} p^{5} +{\left (11 \, b^{6} c - 5 \, a^{2} b^{4}\right )} d^{2} p^{4} +{\left (41 \, b^{6} c - 30 \, a^{2} b^{4}\right )} d^{2} p^{3} +{\left (61 \, b^{6} c - 55 \, a^{2} b^{4}\right )} d^{2} p^{2} + 30 \,{\left (b^{6} c - a^{2} b^{4}\right )} d^{2} p\right )} x^{2} + 8 \,{\left (23 \, b^{6} c^{3} - 24 \, a^{2} b^{4} c^{2} + 9 \, a^{4} b^{2} c\right )} p - 4 \,{\left ({\left (b^{6} c^{2} + a^{2} b^{4} c\right )} d p^{4} + 3 \,{\left (3 \, b^{6} c^{2} - a^{2} b^{4} c\right )} d p^{3} +{\left (23 \, b^{6} c^{2} - 34 \, a^{2} b^{4} c + 15 \, a^{4} b^{2}\right )} d p^{2} + 15 \,{\left (b^{6} c^{2} - 2 \, a^{2} b^{4} c + a^{4} b^{2}\right )} d p\right )} x +{\left (8 \,{\left (3 \, a b^{5} c^{2} + a^{3} b^{3} c\right )} p^{3} + 24 \,{\left (7 \, a b^{5} c^{2} - 3 \, a^{3} b^{3} c\right )} p^{2} +{\left (a b^{5} d^{2} p^{5} + 10 \, a b^{5} d^{2} p^{4} + 35 \, a b^{5} d^{2} p^{3} + 50 \, a b^{5} d^{2} p^{2} + 24 \, a b^{5} d^{2} p\right )} x^{2} + 8 \,{\left (33 \, a b^{5} c^{2} - 40 \, a^{3} b^{3} c + 15 \, a^{5} b\right )} p - 4 \,{\left (2 \, a b^{5} c d p^{4} + 5 \,{\left (3 \, a b^{5} c - a^{3} b^{3}\right )} d p^{3} +{\left (31 \, a b^{5} c - 15 \, a^{3} b^{3}\right )} d p^{2} + 2 \,{\left (9 \, a b^{5} c - 5 \, a^{3} b^{3}\right )} d p\right )} x\right )} \sqrt{d x + c}\right )}{\left (\sqrt{d x + c} b + a\right )}^{p}}{b^{6} d^{3} p^{6} + 21 \, b^{6} d^{3} p^{5} + 175 \, b^{6} d^{3} p^{4} + 735 \, b^{6} d^{3} p^{3} + 1624 \, b^{6} d^{3} p^{2} + 1764 \, b^{6} d^{3} p + 720 \, b^{6} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(120*b^6*c^3 - 360*a^2*b^4*c^2 + 360*a^4*b^2*c - 120*a^6 + 8*(b^6*c^3 + 3*a^2*b^4*c^2)*p^3 + (b^6*d^3*p^5 +
15*b^6*d^3*p^4 + 85*b^6*d^3*p^3 + 225*b^6*d^3*p^2 + 274*b^6*d^3*p + 120*b^6*d^3)*x^3 + 24*(3*b^6*c^3 + 3*a^2*b
^4*c^2 - 2*a^4*b^2*c)*p^2 + (b^6*c*d^2*p^5 + (11*b^6*c - 5*a^2*b^4)*d^2*p^4 + (41*b^6*c - 30*a^2*b^4)*d^2*p^3
+ (61*b^6*c - 55*a^2*b^4)*d^2*p^2 + 30*(b^6*c - a^2*b^4)*d^2*p)*x^2 + 8*(23*b^6*c^3 - 24*a^2*b^4*c^2 + 9*a^4*b
^2*c)*p - 4*((b^6*c^2 + a^2*b^4*c)*d*p^4 + 3*(3*b^6*c^2 - a^2*b^4*c)*d*p^3 + (23*b^6*c^2 - 34*a^2*b^4*c + 15*a
^4*b^2)*d*p^2 + 15*(b^6*c^2 - 2*a^2*b^4*c + a^4*b^2)*d*p)*x + (8*(3*a*b^5*c^2 + a^3*b^3*c)*p^3 + 24*(7*a*b^5*c
^2 - 3*a^3*b^3*c)*p^2 + (a*b^5*d^2*p^5 + 10*a*b^5*d^2*p^4 + 35*a*b^5*d^2*p^3 + 50*a*b^5*d^2*p^2 + 24*a*b^5*d^2
*p)*x^2 + 8*(33*a*b^5*c^2 - 40*a^3*b^3*c + 15*a^5*b)*p - 4*(2*a*b^5*c*d*p^4 + 5*(3*a*b^5*c - a^3*b^3)*d*p^3 +
(31*a*b^5*c - 15*a^3*b^3)*d*p^2 + 2*(9*a*b^5*c - 5*a^3*b^3)*d*p)*x)*sqrt(d*x + c))*(sqrt(d*x + c)*b + a)^p/(b^
6*d^3*p^6 + 21*b^6*d^3*p^5 + 175*b^6*d^3*p^4 + 735*b^6*d^3*p^3 + 1624*b^6*d^3*p^2 + 1764*b^6*d^3*p + 720*b^6*d
^3)

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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(x**2*(a+b*(d*x+c)**(1/2))**p,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError