3.2843 \(\int (c+d x)^3 (a+b (c+d x)^2)^3 \, dx\)

Optimal. Leaf size=48 \[ \frac{\left (a+b (c+d x)^2\right )^5}{10 b^2 d}-\frac{a \left (a+b (c+d x)^2\right )^4}{8 b^2 d} \]

[Out]

-(a*(a + b*(c + d*x)^2)^4)/(8*b^2*d) + (a + b*(c + d*x)^2)^5/(10*b^2*d)

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Rubi [A]  time = 0.217787, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {372, 266, 43} \[ \frac{\left (a+b (c+d x)^2\right )^5}{10 b^2 d}-\frac{a \left (a+b (c+d x)^2\right )^4}{8 b^2 d} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3*(a + b*(c + d*x)^2)^3,x]

[Out]

-(a*(a + b*(c + d*x)^2)^4)/(8*b^2*d) + (a + b*(c + d*x)^2)^5/(10*b^2*d)

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]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (c+d x)^3 \left (a+b (c+d x)^2\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int x^3 \left (a+b x^2\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int x (a+b x)^3 \, dx,x,(c+d x)^2\right )}{2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^3}{b}+\frac{(a+b x)^4}{b}\right ) \, dx,x,(c+d x)^2\right )}{2 d}\\ &=-\frac{a \left (a+b (c+d x)^2\right )^4}{8 b^2 d}+\frac{\left (a+b (c+d x)^2\right )^5}{10 b^2 d}\\ \end{align*}

Mathematica [B]  time = 0.0341876, size = 249, normalized size = 5.19 \[ \frac{1}{2} b d^5 x^6 \left (a^2+21 a b c^2+42 b^2 c^4\right )+\frac{3}{5} b c d^4 x^5 \left (5 a^2+35 a b c^2+42 b^2 c^4\right )+\frac{1}{4} d^3 x^4 \left (30 a^2 b c^2+a^3+105 a b^2 c^4+84 b^3 c^6\right )+c d^2 x^3 \left (10 a^2 b c^2+a^3+21 a b^2 c^4+12 b^3 c^6\right )+\frac{3}{8} b^2 d^7 x^8 \left (a+12 b c^2\right )+3 b^2 c d^6 x^7 \left (a+4 b c^2\right )+\frac{3}{2} c^2 d x^2 \left (a+b c^2\right )^2 \left (a+3 b c^2\right )+c^3 x \left (a+b c^2\right )^3+b^3 c d^8 x^9+\frac{1}{10} b^3 d^9 x^{10} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^3*(a + b*(c + d*x)^2)^3,x]

[Out]

c^3*(a + b*c^2)^3*x + (3*c^2*(a + b*c^2)^2*(a + 3*b*c^2)*d*x^2)/2 + c*(a^3 + 10*a^2*b*c^2 + 21*a*b^2*c^4 + 12*
b^3*c^6)*d^2*x^3 + ((a^3 + 30*a^2*b*c^2 + 105*a*b^2*c^4 + 84*b^3*c^6)*d^3*x^4)/4 + (3*b*c*(5*a^2 + 35*a*b*c^2
+ 42*b^2*c^4)*d^4*x^5)/5 + (b*(a^2 + 21*a*b*c^2 + 42*b^2*c^4)*d^5*x^6)/2 + 3*b^2*c*(a + 4*b*c^2)*d^6*x^7 + (3*
b^2*(a + 12*b*c^2)*d^7*x^8)/8 + b^3*c*d^8*x^9 + (b^3*d^9*x^10)/10

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Maple [B]  time = 0.001, size = 960, normalized size = 20. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3*(a+b*(d*x+c)^2)^3,x)

[Out]

1/10*d^9*b^3*x^10+c*d^8*b^3*x^9+1/8*(21*c^2*d^7*b^3+d^3*((b*c^2+a)*b^2*d^4+8*b^3*c^2*d^4+b*d^2*(2*(b*c^2+a)*b*
d^2+4*b^2*c^2*d^2)))*x^8+1/7*(19*c^3*b^3*d^6+3*c*d^2*((b*c^2+a)*b^2*d^4+8*b^3*c^2*d^4+b*d^2*(2*(b*c^2+a)*b*d^2
+4*b^2*c^2*d^2))+d^3*(8*(b*c^2+a)*b^2*c*d^3+2*b*c*d*(2*(b*c^2+a)*b*d^2+4*b^2*c^2*d^2)))*x^7+1/6*(6*c^4*b^3*d^5
+3*c^2*d*((b*c^2+a)*b^2*d^4+8*b^3*c^2*d^4+b*d^2*(2*(b*c^2+a)*b*d^2+4*b^2*c^2*d^2))+3*c*d^2*(8*(b*c^2+a)*b^2*c*
d^3+2*b*c*d*(2*(b*c^2+a)*b*d^2+4*b^2*c^2*d^2))+d^3*((b*c^2+a)*(2*(b*c^2+a)*b*d^2+4*b^2*c^2*d^2)+8*b^2*c^2*d^2*
(b*c^2+a)+b*d^2*(b*c^2+a)^2))*x^6+1/5*(c^3*((b*c^2+a)*b^2*d^4+8*b^3*c^2*d^4+b*d^2*(2*(b*c^2+a)*b*d^2+4*b^2*c^2
*d^2))+3*c^2*d*(8*(b*c^2+a)*b^2*c*d^3+2*b*c*d*(2*(b*c^2+a)*b*d^2+4*b^2*c^2*d^2))+3*c*d^2*((b*c^2+a)*(2*(b*c^2+
a)*b*d^2+4*b^2*c^2*d^2)+8*b^2*c^2*d^2*(b*c^2+a)+b*d^2*(b*c^2+a)^2)+6*d^4*(b*c^2+a)^2*b*c)*x^5+1/4*(c^3*(8*(b*c
^2+a)*b^2*c*d^3+2*b*c*d*(2*(b*c^2+a)*b*d^2+4*b^2*c^2*d^2))+3*c^2*d*((b*c^2+a)*(2*(b*c^2+a)*b*d^2+4*b^2*c^2*d^2
)+8*b^2*c^2*d^2*(b*c^2+a)+b*d^2*(b*c^2+a)^2)+18*c^2*d^3*(b*c^2+a)^2*b+d^3*(b*c^2+a)^3)*x^4+1/3*(c^3*((b*c^2+a)
*(2*(b*c^2+a)*b*d^2+4*b^2*c^2*d^2)+8*b^2*c^2*d^2*(b*c^2+a)+b*d^2*(b*c^2+a)^2)+18*c^3*d^2*(b*c^2+a)^2*b+3*c*d^2
*(b*c^2+a)^3)*x^3+1/2*(6*c^4*(b*c^2+a)^2*b*d+3*c^2*d*(b*c^2+a)^3)*x^2+c^3*(b*c^2+a)^3*x

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Maxima [B]  time = 1.05115, size = 383, normalized size = 7.98 \begin{align*} \frac{1}{10} \, b^{3} d^{9} x^{10} + b^{3} c d^{8} x^{9} + \frac{3}{8} \,{\left (12 \, b^{3} c^{2} + a b^{2}\right )} d^{7} x^{8} + 3 \,{\left (4 \, b^{3} c^{3} + a b^{2} c\right )} d^{6} x^{7} + \frac{1}{2} \,{\left (42 \, b^{3} c^{4} + 21 \, a b^{2} c^{2} + a^{2} b\right )} d^{5} x^{6} + \frac{3}{5} \,{\left (42 \, b^{3} c^{5} + 35 \, a b^{2} c^{3} + 5 \, a^{2} b c\right )} d^{4} x^{5} + \frac{1}{4} \,{\left (84 \, b^{3} c^{6} + 105 \, a b^{2} c^{4} + 30 \, a^{2} b c^{2} + a^{3}\right )} d^{3} x^{4} +{\left (12 \, b^{3} c^{7} + 21 \, a b^{2} c^{5} + 10 \, a^{2} b c^{3} + a^{3} c\right )} d^{2} x^{3} + \frac{3}{2} \,{\left (3 \, b^{3} c^{8} + 7 \, a b^{2} c^{6} + 5 \, a^{2} b c^{4} + a^{3} c^{2}\right )} d x^{2} +{\left (b^{3} c^{9} + 3 \, a b^{2} c^{7} + 3 \, a^{2} b c^{5} + a^{3} c^{3}\right )} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*(a+b*(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

1/10*b^3*d^9*x^10 + b^3*c*d^8*x^9 + 3/8*(12*b^3*c^2 + a*b^2)*d^7*x^8 + 3*(4*b^3*c^3 + a*b^2*c)*d^6*x^7 + 1/2*(
42*b^3*c^4 + 21*a*b^2*c^2 + a^2*b)*d^5*x^6 + 3/5*(42*b^3*c^5 + 35*a*b^2*c^3 + 5*a^2*b*c)*d^4*x^5 + 1/4*(84*b^3
*c^6 + 105*a*b^2*c^4 + 30*a^2*b*c^2 + a^3)*d^3*x^4 + (12*b^3*c^7 + 21*a*b^2*c^5 + 10*a^2*b*c^3 + a^3*c)*d^2*x^
3 + 3/2*(3*b^3*c^8 + 7*a*b^2*c^6 + 5*a^2*b*c^4 + a^3*c^2)*d*x^2 + (b^3*c^9 + 3*a*b^2*c^7 + 3*a^2*b*c^5 + a^3*c
^3)*x

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Fricas [B]  time = 1.33458, size = 764, normalized size = 15.92 \begin{align*} \frac{1}{10} x^{10} d^{9} b^{3} + x^{9} d^{8} c b^{3} + \frac{9}{2} x^{8} d^{7} c^{2} b^{3} + 12 x^{7} d^{6} c^{3} b^{3} + 21 x^{6} d^{5} c^{4} b^{3} + \frac{3}{8} x^{8} d^{7} b^{2} a + \frac{126}{5} x^{5} d^{4} c^{5} b^{3} + 3 x^{7} d^{6} c b^{2} a + 21 x^{4} d^{3} c^{6} b^{3} + \frac{21}{2} x^{6} d^{5} c^{2} b^{2} a + 12 x^{3} d^{2} c^{7} b^{3} + 21 x^{5} d^{4} c^{3} b^{2} a + \frac{9}{2} x^{2} d c^{8} b^{3} + \frac{105}{4} x^{4} d^{3} c^{4} b^{2} a + \frac{1}{2} x^{6} d^{5} b a^{2} + x c^{9} b^{3} + 21 x^{3} d^{2} c^{5} b^{2} a + 3 x^{5} d^{4} c b a^{2} + \frac{21}{2} x^{2} d c^{6} b^{2} a + \frac{15}{2} x^{4} d^{3} c^{2} b a^{2} + 3 x c^{7} b^{2} a + 10 x^{3} d^{2} c^{3} b a^{2} + \frac{15}{2} x^{2} d c^{4} b a^{2} + \frac{1}{4} x^{4} d^{3} a^{3} + 3 x c^{5} b a^{2} + x^{3} d^{2} c a^{3} + \frac{3}{2} x^{2} d c^{2} a^{3} + x c^{3} a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*(a+b*(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

1/10*x^10*d^9*b^3 + x^9*d^8*c*b^3 + 9/2*x^8*d^7*c^2*b^3 + 12*x^7*d^6*c^3*b^3 + 21*x^6*d^5*c^4*b^3 + 3/8*x^8*d^
7*b^2*a + 126/5*x^5*d^4*c^5*b^3 + 3*x^7*d^6*c*b^2*a + 21*x^4*d^3*c^6*b^3 + 21/2*x^6*d^5*c^2*b^2*a + 12*x^3*d^2
*c^7*b^3 + 21*x^5*d^4*c^3*b^2*a + 9/2*x^2*d*c^8*b^3 + 105/4*x^4*d^3*c^4*b^2*a + 1/2*x^6*d^5*b*a^2 + x*c^9*b^3
+ 21*x^3*d^2*c^5*b^2*a + 3*x^5*d^4*c*b*a^2 + 21/2*x^2*d*c^6*b^2*a + 15/2*x^4*d^3*c^2*b*a^2 + 3*x*c^7*b^2*a + 1
0*x^3*d^2*c^3*b*a^2 + 15/2*x^2*d*c^4*b*a^2 + 1/4*x^4*d^3*a^3 + 3*x*c^5*b*a^2 + x^3*d^2*c*a^3 + 3/2*x^2*d*c^2*a
^3 + x*c^3*a^3

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Sympy [B]  time = 0.131286, size = 357, normalized size = 7.44 \begin{align*} b^{3} c d^{8} x^{9} + \frac{b^{3} d^{9} x^{10}}{10} + x^{8} \left (\frac{3 a b^{2} d^{7}}{8} + \frac{9 b^{3} c^{2} d^{7}}{2}\right ) + x^{7} \left (3 a b^{2} c d^{6} + 12 b^{3} c^{3} d^{6}\right ) + x^{6} \left (\frac{a^{2} b d^{5}}{2} + \frac{21 a b^{2} c^{2} d^{5}}{2} + 21 b^{3} c^{4} d^{5}\right ) + x^{5} \left (3 a^{2} b c d^{4} + 21 a b^{2} c^{3} d^{4} + \frac{126 b^{3} c^{5} d^{4}}{5}\right ) + x^{4} \left (\frac{a^{3} d^{3}}{4} + \frac{15 a^{2} b c^{2} d^{3}}{2} + \frac{105 a b^{2} c^{4} d^{3}}{4} + 21 b^{3} c^{6} d^{3}\right ) + x^{3} \left (a^{3} c d^{2} + 10 a^{2} b c^{3} d^{2} + 21 a b^{2} c^{5} d^{2} + 12 b^{3} c^{7} d^{2}\right ) + x^{2} \left (\frac{3 a^{3} c^{2} d}{2} + \frac{15 a^{2} b c^{4} d}{2} + \frac{21 a b^{2} c^{6} d}{2} + \frac{9 b^{3} c^{8} d}{2}\right ) + x \left (a^{3} c^{3} + 3 a^{2} b c^{5} + 3 a b^{2} c^{7} + b^{3} c^{9}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**3*(a+b*(d*x+c)**2)**3,x)

[Out]

b**3*c*d**8*x**9 + b**3*d**9*x**10/10 + x**8*(3*a*b**2*d**7/8 + 9*b**3*c**2*d**7/2) + x**7*(3*a*b**2*c*d**6 +
12*b**3*c**3*d**6) + x**6*(a**2*b*d**5/2 + 21*a*b**2*c**2*d**5/2 + 21*b**3*c**4*d**5) + x**5*(3*a**2*b*c*d**4
+ 21*a*b**2*c**3*d**4 + 126*b**3*c**5*d**4/5) + x**4*(a**3*d**3/4 + 15*a**2*b*c**2*d**3/2 + 105*a*b**2*c**4*d*
*3/4 + 21*b**3*c**6*d**3) + x**3*(a**3*c*d**2 + 10*a**2*b*c**3*d**2 + 21*a*b**2*c**5*d**2 + 12*b**3*c**7*d**2)
+ x**2*(3*a**3*c**2*d/2 + 15*a**2*b*c**4*d/2 + 21*a*b**2*c**6*d/2 + 9*b**3*c**8*d/2) + x*(a**3*c**3 + 3*a**2*
b*c**5 + 3*a*b**2*c**7 + b**3*c**9)

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Giac [B]  time = 1.09657, size = 479, normalized size = 9.98 \begin{align*} \frac{1}{10} \, b^{3} d^{9} x^{10} + b^{3} c d^{8} x^{9} + \frac{9}{2} \, b^{3} c^{2} d^{7} x^{8} + 12 \, b^{3} c^{3} d^{6} x^{7} + 21 \, b^{3} c^{4} d^{5} x^{6} + \frac{3}{8} \, a b^{2} d^{7} x^{8} + \frac{126}{5} \, b^{3} c^{5} d^{4} x^{5} + 3 \, a b^{2} c d^{6} x^{7} + 21 \, b^{3} c^{6} d^{3} x^{4} + \frac{21}{2} \, a b^{2} c^{2} d^{5} x^{6} + 12 \, b^{3} c^{7} d^{2} x^{3} + 21 \, a b^{2} c^{3} d^{4} x^{5} + \frac{9}{2} \, b^{3} c^{8} d x^{2} + \frac{105}{4} \, a b^{2} c^{4} d^{3} x^{4} + \frac{1}{2} \, a^{2} b d^{5} x^{6} + b^{3} c^{9} x + 21 \, a b^{2} c^{5} d^{2} x^{3} + 3 \, a^{2} b c d^{4} x^{5} + \frac{21}{2} \, a b^{2} c^{6} d x^{2} + \frac{15}{2} \, a^{2} b c^{2} d^{3} x^{4} + 3 \, a b^{2} c^{7} x + 10 \, a^{2} b c^{3} d^{2} x^{3} + \frac{15}{2} \, a^{2} b c^{4} d x^{2} + \frac{1}{4} \, a^{3} d^{3} x^{4} + 3 \, a^{2} b c^{5} x + a^{3} c d^{2} x^{3} + \frac{3}{2} \, a^{3} c^{2} d x^{2} + a^{3} c^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*(a+b*(d*x+c)^2)^3,x, algorithm="giac")

[Out]

1/10*b^3*d^9*x^10 + b^3*c*d^8*x^9 + 9/2*b^3*c^2*d^7*x^8 + 12*b^3*c^3*d^6*x^7 + 21*b^3*c^4*d^5*x^6 + 3/8*a*b^2*
d^7*x^8 + 126/5*b^3*c^5*d^4*x^5 + 3*a*b^2*c*d^6*x^7 + 21*b^3*c^6*d^3*x^4 + 21/2*a*b^2*c^2*d^5*x^6 + 12*b^3*c^7
*d^2*x^3 + 21*a*b^2*c^3*d^4*x^5 + 9/2*b^3*c^8*d*x^2 + 105/4*a*b^2*c^4*d^3*x^4 + 1/2*a^2*b*d^5*x^6 + b^3*c^9*x
+ 21*a*b^2*c^5*d^2*x^3 + 3*a^2*b*c*d^4*x^5 + 21/2*a*b^2*c^6*d*x^2 + 15/2*a^2*b*c^2*d^3*x^4 + 3*a*b^2*c^7*x + 1
0*a^2*b*c^3*d^2*x^3 + 15/2*a^2*b*c^4*d*x^2 + 1/4*a^3*d^3*x^4 + 3*a^2*b*c^5*x + a^3*c*d^2*x^3 + 3/2*a^3*c^2*d*x
^2 + a^3*c^3*x