3.198 \(\int (2 c x+3 d x^2) (a+c x^2+d x^3)^7 \, dx\)

Optimal. Leaf size=18 \[ \frac{1}{8} \left (a+c x^2+d x^3\right )^8 \]

[Out]

(a + c*x^2 + d*x^3)^8/8

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Rubi [A]  time = 0.0441323, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {1588} \[ \frac{1}{8} \left (a+c x^2+d x^3\right )^8 \]

Antiderivative was successfully verified.

[In]

Int[(2*c*x + 3*d*x^2)*(a + c*x^2 + d*x^3)^7,x]

[Out]

(a + c*x^2 + d*x^3)^8/8

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \left (2 c x+3 d x^2\right ) \left (a+c x^2+d x^3\right )^7 \, dx &=\frac{1}{8} \left (a+c x^2+d x^3\right )^8\\ \end{align*}

Mathematica [B]  time = 0.0532344, size = 115, normalized size = 6.39 \[ \frac{1}{8} x^2 (c+d x) \left (28 a^2 x^{10} (c+d x)^5+56 a^3 x^8 (c+d x)^4+70 a^4 x^6 (c+d x)^3+56 a^5 x^4 (c+d x)^2+28 a^6 x^2 (c+d x)+8 a^7+8 a x^{12} (c+d x)^6+x^{14} (c+d x)^7\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(2*c*x + 3*d*x^2)*(a + c*x^2 + d*x^3)^7,x]

[Out]

(x^2*(c + d*x)*(8*a^7 + 28*a^6*x^2*(c + d*x) + 56*a^5*x^4*(c + d*x)^2 + 70*a^4*x^6*(c + d*x)^3 + 56*a^3*x^8*(c
+ d*x)^4 + 28*a^2*x^10*(c + d*x)^5 + 8*a*x^12*(c + d*x)^6 + x^14*(c + d*x)^7))/8

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Maple [B]  time = 0.002, size = 2205, normalized size = 122.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*d*x^2+2*c*x)*(d*x^3+c*x^2+a)^7,x)

[Out]

1/8*d^8*x^24+c*d^7*x^23+7/2*c^2*d^6*x^22+1/21*(42*c^3*d^5+3*d*(a*d^6+15*c^3*d^4+d*(2*(3*a*d^2+c^3)*d^3+18*c^3*
d^3)))*x^21+1/20*(2*c*(a*d^6+15*c^3*d^4+d*(2*(3*a*d^2+c^3)*d^3+18*c^3*d^3))+3*d*(6*a*c*d^5+c*(2*(3*a*d^2+c^3)*
d^3+18*c^3*d^3)+d*(12*a*c*d^4+6*(3*a*d^2+c^3)*c*d^2+9*c^4*d^2)))*x^20+1/19*(2*c*(6*a*c*d^5+c*(2*(3*a*d^2+c^3)*
d^3+18*c^3*d^3)+d*(12*a*c*d^4+6*(3*a*d^2+c^3)*c*d^2+9*c^4*d^2))+3*d*(15*a*c^2*d^4+c*(12*a*c*d^4+6*(3*a*d^2+c^3
)*c*d^2+9*c^4*d^2)+d*(42*a*c^2*d^3+6*(3*a*d^2+c^3)*c^2*d)))*x^19+1/18*(2*c*(15*a*c^2*d^4+c*(12*a*c*d^4+6*(3*a*
d^2+c^3)*c*d^2+9*c^4*d^2)+d*(42*a*c^2*d^3+6*(3*a*d^2+c^3)*c^2*d))+3*d*(a*(2*(3*a*d^2+c^3)*d^3+18*c^3*d^3)+c*(4
2*a*c^2*d^3+6*(3*a*d^2+c^3)*c^2*d)+d*(6*a^2*d^4+54*a*c^3*d^2+(3*a*d^2+c^3)^2)))*x^18+1/17*(2*c*(a*(2*(3*a*d^2+
c^3)*d^3+18*c^3*d^3)+c*(42*a*c^2*d^3+6*(3*a*d^2+c^3)*c^2*d)+d*(6*a^2*d^4+54*a*c^3*d^2+(3*a*d^2+c^3)^2))+3*d*(a
*(12*a*c*d^4+6*(3*a*d^2+c^3)*c*d^2+9*c^4*d^2)+c*(6*a^2*d^4+54*a*c^3*d^2+(3*a*d^2+c^3)^2)+d*(24*a^2*c*d^3+18*c^
4*a*d+12*a*c*d*(3*a*d^2+c^3))))*x^17+1/16*(2*c*(a*(12*a*c*d^4+6*(3*a*d^2+c^3)*c*d^2+9*c^4*d^2)+c*(6*a^2*d^4+54
*a*c^3*d^2+(3*a*d^2+c^3)^2)+d*(24*a^2*c*d^3+18*c^4*a*d+12*a*c*d*(3*a*d^2+c^3)))+3*d*(a*(42*a*c^2*d^3+6*(3*a*d^
2+c^3)*c^2*d)+c*(24*a^2*c*d^3+18*c^4*a*d+12*a*c*d*(3*a*d^2+c^3))+d*(72*a^2*c^2*d^2+6*c^2*a*(3*a*d^2+c^3))))*x^
16+1/15*(2*c*(a*(42*a*c^2*d^3+6*(3*a*d^2+c^3)*c^2*d)+c*(24*a^2*c*d^3+18*c^4*a*d+12*a*c*d*(3*a*d^2+c^3))+d*(72*
a^2*c^2*d^2+6*c^2*a*(3*a*d^2+c^3)))+3*d*(a*(6*a^2*d^4+54*a*c^3*d^2+(3*a*d^2+c^3)^2)+c*(72*a^2*c^2*d^2+6*c^2*a*
(3*a*d^2+c^3))+d*(2*a^3*d^3+54*a^2*c^3*d+6*a^2*d*(3*a*d^2+c^3))))*x^15+1/14*(2*c*(a*(6*a^2*d^4+54*a*c^3*d^2+(3
*a*d^2+c^3)^2)+c*(72*a^2*c^2*d^2+6*c^2*a*(3*a*d^2+c^3))+d*(2*a^3*d^3+54*a^2*c^3*d+6*a^2*d*(3*a*d^2+c^3)))+3*d*
(a*(24*a^2*c*d^3+18*c^4*a*d+12*a*c*d*(3*a*d^2+c^3))+c*(2*a^3*d^3+54*a^2*c^3*d+6*a^2*d*(3*a*d^2+c^3))+d*(42*a^3
*c*d^2+6*a^2*c*(3*a*d^2+c^3)+9*c^4*a^2)))*x^14+1/13*(2*c*(a*(24*a^2*c*d^3+18*c^4*a*d+12*a*c*d*(3*a*d^2+c^3))+c
*(2*a^3*d^3+54*a^2*c^3*d+6*a^2*d*(3*a*d^2+c^3))+d*(42*a^3*c*d^2+6*a^2*c*(3*a*d^2+c^3)+9*c^4*a^2))+3*d*(a*(72*a
^2*c^2*d^2+6*c^2*a*(3*a*d^2+c^3))+c*(42*a^3*c*d^2+6*a^2*c*(3*a*d^2+c^3)+9*c^4*a^2)+60*a^3*c^2*d^2))*x^13+1/12*
(2*c*(a*(72*a^2*c^2*d^2+6*c^2*a*(3*a*d^2+c^3))+c*(42*a^3*c*d^2+6*a^2*c*(3*a*d^2+c^3)+9*c^4*a^2)+60*a^3*c^2*d^2
)+3*d*(a*(2*a^3*d^3+54*a^2*c^3*d+6*a^2*d*(3*a*d^2+c^3))+60*a^3*c^3*d+d*(2*a^3*(3*a*d^2+c^3)+18*a^3*c^3+9*a^4*d
^2)))*x^12+1/11*(2*c*(a*(2*a^3*d^3+54*a^2*c^3*d+6*a^2*d*(3*a*d^2+c^3))+60*a^3*c^3*d+d*(2*a^3*(3*a*d^2+c^3)+18*
a^3*c^3+9*a^4*d^2))+3*d*(a*(42*a^3*c*d^2+6*a^2*c*(3*a*d^2+c^3)+9*c^4*a^2)+c*(2*a^3*(3*a*d^2+c^3)+18*a^3*c^3+9*
a^4*d^2)+30*d^2*a^4*c))*x^11+1/10*(2*c*(a*(42*a^3*c*d^2+6*a^2*c*(3*a*d^2+c^3)+9*c^4*a^2)+c*(2*a^3*(3*a*d^2+c^3
)+18*a^3*c^3+9*a^4*d^2)+30*d^2*a^4*c)+315*d^2*a^4*c^2)*x^10+1/9*(210*c^3*a^4*d+3*d*(a*(2*a^3*(3*a*d^2+c^3)+18*
a^3*c^3+9*a^4*d^2)+15*c^3*a^4+6*d^2*a^5))*x^9+1/8*(2*c*(a*(2*a^3*(3*a*d^2+c^3)+18*a^3*c^3+9*a^4*d^2)+15*c^3*a^
4+6*d^2*a^5)+126*d^2*a^5*c)*x^8+21*c^2*a^5*d*x^7+1/6*(21*a^6*d^2+42*a^5*c^3)*x^6+7*c*d*a^6*x^5+7/2*c^2*a^6*x^4
+d*a^7*x^3+c*a^7*x^2

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Maxima [A]  time = 0.984774, size = 22, normalized size = 1.22 \begin{align*} \frac{1}{8} \,{\left (d x^{3} + c x^{2} + a\right )}^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*d*x^2+2*c*x)*(d*x^3+c*x^2+a)^7,x, algorithm="maxima")

[Out]

1/8*(d*x^3 + c*x^2 + a)^8

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Fricas [B]  time = 1.13598, size = 1085, normalized size = 60.28 \begin{align*} \frac{1}{8} x^{24} d^{8} + x^{23} d^{7} c + \frac{7}{2} x^{22} d^{6} c^{2} + 7 x^{21} d^{5} c^{3} + x^{21} d^{7} a + \frac{35}{4} x^{20} d^{4} c^{4} + 7 x^{20} d^{6} c a + 7 x^{19} d^{3} c^{5} + 21 x^{19} d^{5} c^{2} a + \frac{7}{2} x^{18} d^{2} c^{6} + 35 x^{18} d^{4} c^{3} a + \frac{7}{2} x^{18} d^{6} a^{2} + x^{17} d c^{7} + 35 x^{17} d^{3} c^{4} a + 21 x^{17} d^{5} c a^{2} + \frac{1}{8} x^{16} c^{8} + 21 x^{16} d^{2} c^{5} a + \frac{105}{2} x^{16} d^{4} c^{2} a^{2} + 7 x^{15} d c^{6} a + 70 x^{15} d^{3} c^{3} a^{2} + 7 x^{15} d^{5} a^{3} + x^{14} c^{7} a + \frac{105}{2} x^{14} d^{2} c^{4} a^{2} + 35 x^{14} d^{4} c a^{3} + 21 x^{13} d c^{5} a^{2} + 70 x^{13} d^{3} c^{2} a^{3} + \frac{7}{2} x^{12} c^{6} a^{2} + 70 x^{12} d^{2} c^{3} a^{3} + \frac{35}{4} x^{12} d^{4} a^{4} + 35 x^{11} d c^{4} a^{3} + 35 x^{11} d^{3} c a^{4} + 7 x^{10} c^{5} a^{3} + \frac{105}{2} x^{10} d^{2} c^{2} a^{4} + 35 x^{9} d c^{3} a^{4} + 7 x^{9} d^{3} a^{5} + \frac{35}{4} x^{8} c^{4} a^{4} + 21 x^{8} d^{2} c a^{5} + 21 x^{7} d c^{2} a^{5} + 7 x^{6} c^{3} a^{5} + \frac{7}{2} x^{6} d^{2} a^{6} + 7 x^{5} d c a^{6} + \frac{7}{2} x^{4} c^{2} a^{6} + x^{3} d a^{7} + x^{2} c a^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*d*x^2+2*c*x)*(d*x^3+c*x^2+a)^7,x, algorithm="fricas")

[Out]

1/8*x^24*d^8 + x^23*d^7*c + 7/2*x^22*d^6*c^2 + 7*x^21*d^5*c^3 + x^21*d^7*a + 35/4*x^20*d^4*c^4 + 7*x^20*d^6*c*
a + 7*x^19*d^3*c^5 + 21*x^19*d^5*c^2*a + 7/2*x^18*d^2*c^6 + 35*x^18*d^4*c^3*a + 7/2*x^18*d^6*a^2 + x^17*d*c^7
+ 35*x^17*d^3*c^4*a + 21*x^17*d^5*c*a^2 + 1/8*x^16*c^8 + 21*x^16*d^2*c^5*a + 105/2*x^16*d^4*c^2*a^2 + 7*x^15*d
*c^6*a + 70*x^15*d^3*c^3*a^2 + 7*x^15*d^5*a^3 + x^14*c^7*a + 105/2*x^14*d^2*c^4*a^2 + 35*x^14*d^4*c*a^3 + 21*x
^13*d*c^5*a^2 + 70*x^13*d^3*c^2*a^3 + 7/2*x^12*c^6*a^2 + 70*x^12*d^2*c^3*a^3 + 35/4*x^12*d^4*a^4 + 35*x^11*d*c
^4*a^3 + 35*x^11*d^3*c*a^4 + 7*x^10*c^5*a^3 + 105/2*x^10*d^2*c^2*a^4 + 35*x^9*d*c^3*a^4 + 7*x^9*d^3*a^5 + 35/4
*x^8*c^4*a^4 + 21*x^8*d^2*c*a^5 + 21*x^7*d*c^2*a^5 + 7*x^6*c^3*a^5 + 7/2*x^6*d^2*a^6 + 7*x^5*d*c*a^6 + 7/2*x^4
*c^2*a^6 + x^3*d*a^7 + x^2*c*a^7

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Sympy [B]  time = 0.152089, size = 484, normalized size = 26.89 \begin{align*} a^{7} c x^{2} + a^{7} d x^{3} + \frac{7 a^{6} c^{2} x^{4}}{2} + 7 a^{6} c d x^{5} + 21 a^{5} c^{2} d x^{7} + \frac{7 c^{2} d^{6} x^{22}}{2} + c d^{7} x^{23} + \frac{d^{8} x^{24}}{8} + x^{21} \left (a d^{7} + 7 c^{3} d^{5}\right ) + x^{20} \left (7 a c d^{6} + \frac{35 c^{4} d^{4}}{4}\right ) + x^{19} \left (21 a c^{2} d^{5} + 7 c^{5} d^{3}\right ) + x^{18} \left (\frac{7 a^{2} d^{6}}{2} + 35 a c^{3} d^{4} + \frac{7 c^{6} d^{2}}{2}\right ) + x^{17} \left (21 a^{2} c d^{5} + 35 a c^{4} d^{3} + c^{7} d\right ) + x^{16} \left (\frac{105 a^{2} c^{2} d^{4}}{2} + 21 a c^{5} d^{2} + \frac{c^{8}}{8}\right ) + x^{15} \left (7 a^{3} d^{5} + 70 a^{2} c^{3} d^{3} + 7 a c^{6} d\right ) + x^{14} \left (35 a^{3} c d^{4} + \frac{105 a^{2} c^{4} d^{2}}{2} + a c^{7}\right ) + x^{13} \left (70 a^{3} c^{2} d^{3} + 21 a^{2} c^{5} d\right ) + x^{12} \left (\frac{35 a^{4} d^{4}}{4} + 70 a^{3} c^{3} d^{2} + \frac{7 a^{2} c^{6}}{2}\right ) + x^{11} \left (35 a^{4} c d^{3} + 35 a^{3} c^{4} d\right ) + x^{10} \left (\frac{105 a^{4} c^{2} d^{2}}{2} + 7 a^{3} c^{5}\right ) + x^{9} \left (7 a^{5} d^{3} + 35 a^{4} c^{3} d\right ) + x^{8} \left (21 a^{5} c d^{2} + \frac{35 a^{4} c^{4}}{4}\right ) + x^{6} \left (\frac{7 a^{6} d^{2}}{2} + 7 a^{5} c^{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*d*x**2+2*c*x)*(d*x**3+c*x**2+a)**7,x)

[Out]

a**7*c*x**2 + a**7*d*x**3 + 7*a**6*c**2*x**4/2 + 7*a**6*c*d*x**5 + 21*a**5*c**2*d*x**7 + 7*c**2*d**6*x**22/2 +
c*d**7*x**23 + d**8*x**24/8 + x**21*(a*d**7 + 7*c**3*d**5) + x**20*(7*a*c*d**6 + 35*c**4*d**4/4) + x**19*(21*
a*c**2*d**5 + 7*c**5*d**3) + x**18*(7*a**2*d**6/2 + 35*a*c**3*d**4 + 7*c**6*d**2/2) + x**17*(21*a**2*c*d**5 +
35*a*c**4*d**3 + c**7*d) + x**16*(105*a**2*c**2*d**4/2 + 21*a*c**5*d**2 + c**8/8) + x**15*(7*a**3*d**5 + 70*a*
*2*c**3*d**3 + 7*a*c**6*d) + x**14*(35*a**3*c*d**4 + 105*a**2*c**4*d**2/2 + a*c**7) + x**13*(70*a**3*c**2*d**3
+ 21*a**2*c**5*d) + x**12*(35*a**4*d**4/4 + 70*a**3*c**3*d**2 + 7*a**2*c**6/2) + x**11*(35*a**4*c*d**3 + 35*a
**3*c**4*d) + x**10*(105*a**4*c**2*d**2/2 + 7*a**3*c**5) + x**9*(7*a**5*d**3 + 35*a**4*c**3*d) + x**8*(21*a**5
*c*d**2 + 35*a**4*c**4/4) + x**6*(7*a**6*d**2/2 + 7*a**5*c**3)

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Giac [B]  time = 1.21118, size = 659, normalized size = 36.61 \begin{align*} \frac{1}{8} \, d^{8} x^{24} + c d^{7} x^{23} + \frac{7}{2} \, c^{2} d^{6} x^{22} + 7 \, c^{3} d^{5} x^{21} + a d^{7} x^{21} + \frac{35}{4} \, c^{4} d^{4} x^{20} + 7 \, a c d^{6} x^{20} + 7 \, c^{5} d^{3} x^{19} + 21 \, a c^{2} d^{5} x^{19} + \frac{7}{2} \, c^{6} d^{2} x^{18} + 35 \, a c^{3} d^{4} x^{18} + \frac{7}{2} \, a^{2} d^{6} x^{18} + c^{7} d x^{17} + 35 \, a c^{4} d^{3} x^{17} + 21 \, a^{2} c d^{5} x^{17} + \frac{1}{8} \, c^{8} x^{16} + 21 \, a c^{5} d^{2} x^{16} + \frac{105}{2} \, a^{2} c^{2} d^{4} x^{16} + 7 \, a c^{6} d x^{15} + 70 \, a^{2} c^{3} d^{3} x^{15} + 7 \, a^{3} d^{5} x^{15} + a c^{7} x^{14} + \frac{105}{2} \, a^{2} c^{4} d^{2} x^{14} + 35 \, a^{3} c d^{4} x^{14} + 21 \, a^{2} c^{5} d x^{13} + 70 \, a^{3} c^{2} d^{3} x^{13} + \frac{7}{2} \, a^{2} c^{6} x^{12} + 70 \, a^{3} c^{3} d^{2} x^{12} + \frac{35}{4} \, a^{4} d^{4} x^{12} + 35 \, a^{3} c^{4} d x^{11} + 35 \, a^{4} c d^{3} x^{11} + 7 \, a^{3} c^{5} x^{10} + \frac{105}{2} \, a^{4} c^{2} d^{2} x^{10} + 35 \, a^{4} c^{3} d x^{9} + 7 \, a^{5} d^{3} x^{9} + \frac{35}{4} \, a^{4} c^{4} x^{8} + 21 \, a^{5} c d^{2} x^{8} + 21 \, a^{5} c^{2} d x^{7} + 7 \, a^{5} c^{3} x^{6} + \frac{7}{2} \, a^{6} d^{2} x^{6} + 7 \, a^{6} c d x^{5} + \frac{7}{2} \, a^{6} c^{2} x^{4} + a^{7} d x^{3} + a^{7} c x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*d*x^2+2*c*x)*(d*x^3+c*x^2+a)^7,x, algorithm="giac")

[Out]

1/8*d^8*x^24 + c*d^7*x^23 + 7/2*c^2*d^6*x^22 + 7*c^3*d^5*x^21 + a*d^7*x^21 + 35/4*c^4*d^4*x^20 + 7*a*c*d^6*x^2
0 + 7*c^5*d^3*x^19 + 21*a*c^2*d^5*x^19 + 7/2*c^6*d^2*x^18 + 35*a*c^3*d^4*x^18 + 7/2*a^2*d^6*x^18 + c^7*d*x^17
+ 35*a*c^4*d^3*x^17 + 21*a^2*c*d^5*x^17 + 1/8*c^8*x^16 + 21*a*c^5*d^2*x^16 + 105/2*a^2*c^2*d^4*x^16 + 7*a*c^6*
d*x^15 + 70*a^2*c^3*d^3*x^15 + 7*a^3*d^5*x^15 + a*c^7*x^14 + 105/2*a^2*c^4*d^2*x^14 + 35*a^3*c*d^4*x^14 + 21*a
^2*c^5*d*x^13 + 70*a^3*c^2*d^3*x^13 + 7/2*a^2*c^6*x^12 + 70*a^3*c^3*d^2*x^12 + 35/4*a^4*d^4*x^12 + 35*a^3*c^4*
d*x^11 + 35*a^4*c*d^3*x^11 + 7*a^3*c^5*x^10 + 105/2*a^4*c^2*d^2*x^10 + 35*a^4*c^3*d*x^9 + 7*a^5*d^3*x^9 + 35/4
*a^4*c^4*x^8 + 21*a^5*c*d^2*x^8 + 21*a^5*c^2*d*x^7 + 7*a^5*c^3*x^6 + 7/2*a^6*d^2*x^6 + 7*a^6*c*d*x^5 + 7/2*a^6
*c^2*x^4 + a^7*d*x^3 + a^7*c*x^2