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

Optimal. Leaf size=17 \[ \frac{1}{8} \left (c x^2+d x^3\right )^8 \]

[Out]

(c*x^2 + d*x^3)^8/8

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Rubi [A]  time = 0.0251481, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {1588} \[ \frac{1}{8} \left (c x^2+d x^3\right )^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [B]  time = 0.0030102, size = 98, normalized size = 5.76 \[ \frac{7}{2} c^2 d^6 x^{22}+7 c^3 d^5 x^{21}+\frac{35}{4} c^4 d^4 x^{20}+7 c^5 d^3 x^{19}+\frac{7}{2} c^6 d^2 x^{18}+c^7 d x^{17}+\frac{c^8 x^{16}}{8}+c d^7 x^{23}+\frac{d^8 x^{24}}{8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^8*x^16)/8 + c^7*d*x^17 + (7*c^6*d^2*x^18)/2 + 7*c^5*d^3*x^19 + (35*c^4*d^4*x^20)/4 + 7*c^3*d^5*x^21 + (7*c^
2*d^6*x^22)/2 + c*d^7*x^23 + (d^8*x^24)/8

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Maple [B]  time = 0.002, size = 89, normalized size = 5.2 \begin{align*}{\frac{{d}^{8}{x}^{24}}{8}}+c{d}^{7}{x}^{23}+{\frac{7\,{c}^{2}{d}^{6}{x}^{22}}{2}}+7\,{c}^{3}{d}^{5}{x}^{21}+{\frac{35\,{c}^{4}{d}^{4}{x}^{20}}{4}}+7\,{c}^{5}{d}^{3}{x}^{19}+{\frac{7\,{c}^{6}{d}^{2}{x}^{18}}{2}}+{c}^{7}d{x}^{17}+{\frac{{c}^{8}{x}^{16}}{8}} \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)^7,x)

[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+35/4*c^4*d^4*x^20+7*c^5*d^3*x^19+7/2*c^6*d^2*x^18+c^7*
d*x^17+1/8*c^8*x^16

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Maxima [A]  time = 0.977805, size = 20, normalized size = 1.18 \begin{align*} \frac{1}{8} \,{\left (d x^{3} + c x^{2}\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)^7,x, algorithm="maxima")

[Out]

1/8*(d*x^3 + c*x^2)^8

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Fricas [B]  time = 1.13133, size = 198, normalized size = 11.65 \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} + \frac{35}{4} x^{20} d^{4} c^{4} + 7 x^{19} d^{3} c^{5} + \frac{7}{2} x^{18} d^{2} c^{6} + x^{17} d c^{7} + \frac{1}{8} x^{16} c^{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)^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 + 35/4*x^20*d^4*c^4 + 7*x^19*d^3*c^5 + 7/2*x^18*
d^2*c^6 + x^17*d*c^7 + 1/8*x^16*c^8

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Sympy [B]  time = 0.090522, size = 97, normalized size = 5.71 \begin{align*} \frac{c^{8} x^{16}}{8} + c^{7} d x^{17} + \frac{7 c^{6} d^{2} x^{18}}{2} + 7 c^{5} d^{3} x^{19} + \frac{35 c^{4} d^{4} x^{20}}{4} + 7 c^{3} d^{5} x^{21} + \frac{7 c^{2} d^{6} x^{22}}{2} + c d^{7} x^{23} + \frac{d^{8} x^{24}}{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)**7,x)

[Out]

c**8*x**16/8 + c**7*d*x**17 + 7*c**6*d**2*x**18/2 + 7*c**5*d**3*x**19 + 35*c**4*d**4*x**20/4 + 7*c**3*d**5*x**
21 + 7*c**2*d**6*x**22/2 + c*d**7*x**23 + d**8*x**24/8

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Giac [B]  time = 1.31019, size = 119, normalized size = 7. \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} + \frac{35}{4} \, c^{4} d^{4} x^{20} + 7 \, c^{5} d^{3} x^{19} + \frac{7}{2} \, c^{6} d^{2} x^{18} + c^{7} d x^{17} + \frac{1}{8} \, c^{8} x^{16} \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)^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 + 35/4*c^4*d^4*x^20 + 7*c^5*d^3*x^19 + 7/2*c^6*d
^2*x^18 + c^7*d*x^17 + 1/8*c^8*x^16