3.653 \(\int x^3 (a+b \sqrt{c+d x})^p \, dx\)
Optimal. Leaf size=350 \[ \frac{2 \left (-30 a^2 b^2 c+35 a^4+3 b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{p+4}}{b^8 d^4 (p+4)}-\frac{2 a \left (a^2-b^2 c\right )^3 \left (a+b \sqrt{c+d x}\right )^{p+1}}{b^8 d^4 (p+1)}+\frac{2 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+2}}{b^8 d^4 (p+2)}-\frac{6 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+3}}{b^8 d^4 (p+3)}-\frac{10 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+5}}{b^8 d^4 (p+5)}+\frac{6 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+6}}{b^8 d^4 (p+6)}-\frac{14 a \left (a+b \sqrt{c+d x}\right )^{p+7}}{b^8 d^4 (p+7)}+\frac{2 \left (a+b \sqrt{c+d x}\right )^{p+8}}{b^8 d^4 (p+8)} \]
[Out]
(-2*a*(a^2 - b^2*c)^3*(a + b*Sqrt[c + d*x])^(1 + p))/(b^8*d^4*(1 + p)) + (2*(a^2 - b^2*c)^2*(7*a^2 - b^2*c)*(a
+ b*Sqrt[c + d*x])^(2 + p))/(b^8*d^4*(2 + p)) - (6*a*(7*a^2 - 3*b^2*c)*(a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(3
+ p))/(b^8*d^4*(3 + p)) + (2*(35*a^4 - 30*a^2*b^2*c + 3*b^4*c^2)*(a + b*Sqrt[c + d*x])^(4 + p))/(b^8*d^4*(4 +
p)) - (10*a*(7*a^2 - 3*b^2*c)*(a + b*Sqrt[c + d*x])^(5 + p))/(b^8*d^4*(5 + p)) + (6*(7*a^2 - b^2*c)*(a + b*Sq
rt[c + d*x])^(6 + p))/(b^8*d^4*(6 + p)) - (14*a*(a + b*Sqrt[c + d*x])^(7 + p))/(b^8*d^4*(7 + p)) + (2*(a + b*S
qrt[c + d*x])^(8 + p))/(b^8*d^4*(8 + p))
________________________________________________________________________________________
Rubi [A] time = 0.278758, antiderivative size = 350, 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} \[ \frac{2 \left (-30 a^2 b^2 c+35 a^4+3 b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{p+4}}{b^8 d^4 (p+4)}-\frac{2 a \left (a^2-b^2 c\right )^3 \left (a+b \sqrt{c+d x}\right )^{p+1}}{b^8 d^4 (p+1)}+\frac{2 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+2}}{b^8 d^4 (p+2)}-\frac{6 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+3}}{b^8 d^4 (p+3)}-\frac{10 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+5}}{b^8 d^4 (p+5)}+\frac{6 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{p+6}}{b^8 d^4 (p+6)}-\frac{14 a \left (a+b \sqrt{c+d x}\right )^{p+7}}{b^8 d^4 (p+7)}+\frac{2 \left (a+b \sqrt{c+d x}\right )^{p+8}}{b^8 d^4 (p+8)} \]
Antiderivative was successfully verified.
[In]
Int[x^3*(a + b*Sqrt[c + d*x])^p,x]
[Out]
(-2*a*(a^2 - b^2*c)^3*(a + b*Sqrt[c + d*x])^(1 + p))/(b^8*d^4*(1 + p)) + (2*(a^2 - b^2*c)^2*(7*a^2 - b^2*c)*(a
+ b*Sqrt[c + d*x])^(2 + p))/(b^8*d^4*(2 + p)) - (6*a*(7*a^2 - 3*b^2*c)*(a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(3
+ p))/(b^8*d^4*(3 + p)) + (2*(35*a^4 - 30*a^2*b^2*c + 3*b^4*c^2)*(a + b*Sqrt[c + d*x])^(4 + p))/(b^8*d^4*(4 +
p)) - (10*a*(7*a^2 - 3*b^2*c)*(a + b*Sqrt[c + d*x])^(5 + p))/(b^8*d^4*(5 + p)) + (6*(7*a^2 - b^2*c)*(a + b*Sq
rt[c + d*x])^(6 + p))/(b^8*d^4*(6 + p)) - (14*a*(a + b*Sqrt[c + d*x])^(7 + p))/(b^8*d^4*(7 + p)) + (2*(a + b*S
qrt[c + d*x])^(8 + p))/(b^8*d^4*(8 + 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^3 \left (a+b \sqrt{c+d x}\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b \sqrt{x}\right )^p (-c+x)^3 \, dx,x,c+d x\right )}{d^4}\\ &=\frac{2 \operatorname{Subst}\left (\int x (a+b x)^p \left (-c+x^2\right )^3 \, dx,x,\sqrt{c+d x}\right )}{d^4}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (-\frac{a \left (a^2-b^2 c\right )^3 (a+b x)^p}{b^7}-\frac{\left (-7 a^2+b^2 c\right ) \left (-a^2+b^2 c\right )^2 (a+b x)^{1+p}}{b^7}-\frac{3 \left (7 a^5-10 a^3 b^2 c+3 a b^4 c^2\right ) (a+b x)^{2+p}}{b^7}+\frac{\left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) (a+b x)^{3+p}}{b^7}-\frac{5 a \left (7 a^2-3 b^2 c\right ) (a+b x)^{4+p}}{b^7}-\frac{3 \left (-7 a^2+b^2 c\right ) (a+b x)^{5+p}}{b^7}-\frac{7 a (a+b x)^{6+p}}{b^7}+\frac{(a+b x)^{7+p}}{b^7}\right ) \, dx,x,\sqrt{c+d x}\right )}{d^4}\\ &=-\frac{2 a \left (a^2-b^2 c\right )^3 \left (a+b \sqrt{c+d x}\right )^{1+p}}{b^8 d^4 (1+p)}+\frac{2 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{2+p}}{b^8 d^4 (2+p)}-\frac{6 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{3+p}}{b^8 d^4 (3+p)}+\frac{2 \left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) \left (a+b \sqrt{c+d x}\right )^{4+p}}{b^8 d^4 (4+p)}-\frac{10 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{5+p}}{b^8 d^4 (5+p)}+\frac{6 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )^{6+p}}{b^8 d^4 (6+p)}-\frac{14 a \left (a+b \sqrt{c+d x}\right )^{7+p}}{b^8 d^4 (7+p)}+\frac{2 \left (a+b \sqrt{c+d x}\right )^{8+p}}{b^8 d^4 (8+p)}\\ \end{align*}
Mathematica [A] time = 0.870672, size = 555, normalized size = 1.59 \[ -\frac{2 \left (a+b \sqrt{c+d x}\right )^{p+1} \left (6 a^3 b^4 \left (8 c^2 \left (p^4-14 p^3-139 p^2-124 p+315\right )+40 c d \left (p^4+4 p^3-16 p^2-61 p-42\right ) x+35 d^2 \left (p^4+10 p^3+35 p^2+50 p+24\right ) x^2\right )-6 a^2 b^5 (p+1) \sqrt{c+d x} \left (-24 c^2 \left (p^3+5 p^2-24 p-105\right )+4 c d \left (p^4-p^3-94 p^2-386 p-420\right ) x+7 d^2 \left (p^4+14 p^3+71 p^2+154 p+120\right ) x^2\right )+360 a^5 b^2 \left (6 c \left (p^2+p-7\right )+7 d \left (p^2+3 p+2\right ) x\right )-120 a^4 b^3 (p+1) \sqrt{c+d x} \left (2 c \left (2 p^2-5 p-63\right )+7 d \left (p^2+5 p+6\right ) x\right )-5040 a^6 b (p+1) \sqrt{c+d x}+5040 a^7+a b^6 \left (-24 c^2 d \left (2 p^5+24 p^4+74 p^3-21 p^2-283 p-210\right ) x+48 c^3 \left (3 p^4+38 p^3+138 p^2+103 p-105\right )+6 c d^2 \left (p^6+11 p^5+10 p^4-265 p^3-1151 p^2-1726 p-840\right ) x^2+7 d^3 \left (p^6+21 p^5+175 p^4+735 p^3+1624 p^2+1764 p+720\right ) x^3\right )-b^7 \left (p^4+16 p^3+86 p^2+176 p+105\right ) \sqrt{c+d x} \left (24 c^2 d (p+2) x-48 c^3-6 c d^2 \left (p^2+6 p+8\right ) x^2+d^3 \left (p^3+12 p^2+44 p+48\right ) x^3\right )\right )}{b^8 d^4 (p+1) (p+2) (p+3) (p+4) (p+5) (p+6) (p+7) (p+8)} \]
Antiderivative was successfully verified.
[In]
Integrate[x^3*(a + b*Sqrt[c + d*x])^p,x]
[Out]
(-2*(a + b*Sqrt[c + d*x])^(1 + p)*(5040*a^7 - 5040*a^6*b*(1 + p)*Sqrt[c + d*x] + 360*a^5*b^2*(6*c*(-7 + p + p^
2) + 7*d*(2 + 3*p + p^2)*x) - 120*a^4*b^3*(1 + p)*Sqrt[c + d*x]*(2*c*(-63 - 5*p + 2*p^2) + 7*d*(6 + 5*p + p^2)
*x) + 6*a^3*b^4*(8*c^2*(315 - 124*p - 139*p^2 - 14*p^3 + p^4) + 40*c*d*(-42 - 61*p - 16*p^2 + 4*p^3 + p^4)*x +
35*d^2*(24 + 50*p + 35*p^2 + 10*p^3 + p^4)*x^2) - 6*a^2*b^5*(1 + p)*Sqrt[c + d*x]*(-24*c^2*(-105 - 24*p + 5*p
^2 + p^3) + 4*c*d*(-420 - 386*p - 94*p^2 - p^3 + p^4)*x + 7*d^2*(120 + 154*p + 71*p^2 + 14*p^3 + p^4)*x^2) - b
^7*(105 + 176*p + 86*p^2 + 16*p^3 + p^4)*Sqrt[c + d*x]*(-48*c^3 + 24*c^2*d*(2 + p)*x - 6*c*d^2*(8 + 6*p + p^2)
*x^2 + d^3*(48 + 44*p + 12*p^2 + p^3)*x^3) + a*b^6*(48*c^3*(-105 + 103*p + 138*p^2 + 38*p^3 + 3*p^4) - 24*c^2*
d*(-210 - 283*p - 21*p^2 + 74*p^3 + 24*p^4 + 2*p^5)*x + 6*c*d^2*(-840 - 1726*p - 1151*p^2 - 265*p^3 + 10*p^4 +
11*p^5 + p^6)*x^2 + 7*d^3*(720 + 1764*p + 1624*p^2 + 735*p^3 + 175*p^4 + 21*p^5 + p^6)*x^3)))/(b^8*d^4*(1 + p
)*(2 + p)*(3 + p)*(4 + p)*(5 + p)*(6 + p)*(7 + p)*(8 + p))
________________________________________________________________________________________
Maple [F] time = 0.004, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( a+b\sqrt{dx+c} \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^3*(a+b*(d*x+c)^(1/2))^p,x)
[Out]
int(x^3*(a+b*(d*x+c)^(1/2))^p,x)
________________________________________________________________________________________
Maxima [B] time = 1.11213, size = 983, normalized size = 2.81 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*(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^3/((p^2 + 3*p + 2)*b^2) - 3*
((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^2/((p^4 + 10*p^3 + 35*p^2 + 50*p + 24)*b^
4) + 3*((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*c/((p^6 + 21*p^5 + 175*p^4 + 735*p^3 + 1624*p^2 + 1764*p + 720)*b^6) - ((p^7 + 28*p^6 + 322*p^5 + 1960*p^4
+ 6769*p^3 + 13132*p^2 + 13068*p + 5040)*(d*x + c)^4*b^8 + (p^7 + 21*p^6 + 175*p^5 + 735*p^4 + 1624*p^3 + 176
4*p^2 + 720*p)*(d*x + c)^(7/2)*a*b^7 - 7*(p^6 + 15*p^5 + 85*p^4 + 225*p^3 + 274*p^2 + 120*p)*(d*x + c)^3*a^2*b
^6 + 42*(p^5 + 10*p^4 + 35*p^3 + 50*p^2 + 24*p)*(d*x + c)^(5/2)*a^3*b^5 - 210*(p^4 + 6*p^3 + 11*p^2 + 6*p)*(d*
x + c)^2*a^4*b^4 + 840*(p^3 + 3*p^2 + 2*p)*(d*x + c)^(3/2)*a^5*b^3 - 2520*(p^2 + p)*(d*x + c)*a^6*b^2 + 5040*s
qrt(d*x + c)*a^7*b*p - 5040*a^8)*(sqrt(d*x + c)*b + a)^p/((p^8 + 36*p^7 + 546*p^6 + 4536*p^5 + 22449*p^4 + 672
84*p^3 + 118124*p^2 + 109584*p + 40320)*b^8))/d^4
________________________________________________________________________________________
Fricas [B] time = 2.92685, size = 3067, normalized size = 8.76 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*(a+b*(d*x+c)^(1/2))^p,x, algorithm="fricas")
[Out]
-2*(5040*b^8*c^4 - 20160*a^2*b^6*c^3 + 30240*a^4*b^4*c^2 - 20160*a^6*b^2*c + 5040*a^8 + 48*(b^8*c^4 + 6*a^2*b^
6*c^3 + a^4*b^4*c^2)*p^4 - (b^8*d^4*p^7 + 28*b^8*d^4*p^6 + 322*b^8*d^4*p^5 + 1960*b^8*d^4*p^4 + 6769*b^8*d^4*p
^3 + 13132*b^8*d^4*p^2 + 13068*b^8*d^4*p + 5040*b^8*d^4)*x^4 + 384*(2*b^8*c^4 + 7*a^2*b^6*c^3 - 3*a^4*b^4*c^2)
*p^3 - (b^8*c*d^3*p^7 + (22*b^8*c - 7*a^2*b^6)*d^3*p^6 + 5*(38*b^8*c - 21*a^2*b^6)*d^3*p^5 + 5*(164*b^8*c - 11
9*a^2*b^6)*d^3*p^4 + (1849*b^8*c - 1575*a^2*b^6)*d^3*p^3 + 2*(1019*b^8*c - 959*a^2*b^6)*d^3*p^2 + 840*(b^8*c -
a^2*b^6)*d^3*p)*x^3 + 48*(86*b^8*c^4 + 81*a^2*b^6*c^3 - 124*a^4*b^4*c^2 + 45*a^6*b^2*c)*p^2 + 6*(18*b^8*c^2*d
^2*p^5 + (b^8*c^2 + a^2*b^6*c)*d^2*p^6 + (118*b^8*c^2 - 95*a^2*b^6*c + 35*a^4*b^4)*d^2*p^4 + 6*(58*b^8*c^2 - 8
0*a^2*b^6*c + 35*a^4*b^4)*d^2*p^3 + (457*b^8*c^2 - 806*a^2*b^6*c + 385*a^4*b^4)*d^2*p^2 + 210*(b^8*c^2 - 2*a^2
*b^6*c + a^4*b^4)*d^2*p)*x^2 + 192*(44*b^8*c^4 - 71*a^2*b^6*c^3 + 54*a^4*b^4*c^2 - 15*a^6*b^2*c)*p - 24*((b^8*
c^3 + 3*a^2*b^6*c^2)*d*p^5 + 2*(8*b^8*c^3 + 9*a^2*b^6*c^2 - 5*a^4*b^4*c)*d*p^4 + (86*b^8*c^3 - 57*a^2*b^6*c^2
+ 15*a^4*b^4*c)*d*p^3 + (176*b^8*c^3 - 387*a^2*b^6*c^2 + 340*a^4*b^4*c - 105*a^6*b^2)*d*p^2 + 105*(b^8*c^3 - 3
*a^2*b^6*c^2 + 3*a^4*b^4*c - a^6*b^2)*d*p)*x + (192*(a*b^7*c^3 + a^3*b^5*c^2)*p^4 + 96*(27*a*b^7*c^3 + 2*a^3*b
^5*c^2 - 5*a^5*b^3*c)*p^3 - (a*b^7*d^3*p^7 + 21*a*b^7*d^3*p^6 + 175*a*b^7*d^3*p^5 + 735*a*b^7*d^3*p^4 + 1624*a
*b^7*d^3*p^3 + 1764*a*b^7*d^3*p^2 + 720*a*b^7*d^3*p)*x^3 + 192*(56*a*b^7*c^3 - 49*a^3*b^5*c^2 + 15*a^5*b^3*c)*
p^2 + 6*(2*a*b^7*c*d^2*p^6 + (33*a*b^7*c - 7*a^3*b^5)*d^2*p^5 + 10*(20*a*b^7*c - 7*a^3*b^5)*d^2*p^4 + 5*(111*a
*b^7*c - 49*a^3*b^5)*d^2*p^3 + 2*(349*a*b^7*c - 175*a^3*b^5)*d^2*p^2 + 24*(13*a*b^7*c - 7*a^3*b^5)*d^2*p)*x^2
+ 48*(279*a*b^7*c^3 - 511*a^3*b^5*c^2 + 385*a^5*b^3*c - 105*a^7*b)*p - 24*((3*a*b^7*c^2 + a^3*b^5*c)*d*p^5 + 2
*(21*a*b^7*c^2 - 5*a^3*b^5*c)*d*p^4 + (192*a*b^7*c^2 - 135*a^3*b^5*c + 35*a^5*b^3)*d*p^3 + (327*a*b^7*c^2 - 32
0*a^3*b^5*c + 105*a^5*b^3)*d*p^2 + 2*(87*a*b^7*c^2 - 98*a^3*b^5*c + 35*a^5*b^3)*d*p)*x)*sqrt(d*x + c))*(sqrt(d
*x + c)*b + a)^p/(b^8*d^4*p^8 + 36*b^8*d^4*p^7 + 546*b^8*d^4*p^6 + 4536*b^8*d^4*p^5 + 22449*b^8*d^4*p^4 + 6728
4*b^8*d^4*p^3 + 118124*b^8*d^4*p^2 + 109584*b^8*d^4*p + 40320*b^8*d^4)
________________________________________________________________________________________
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**3*(a+b*(d*x+c)**(1/2))**p,x)
[Out]
Timed out
________________________________________________________________________________________
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^3*(a+b*(d*x+c)^(1/2))^p,x, algorithm="giac")
[Out]
Exception raised: NotImplementedError