3.196 \(\int (b+3 d x^2) (b x+d x^3)^7 \, dx\)

Optimal. Leaf size=15 \[ \frac{1}{8} \left (b x+d x^3\right )^8 \]

[Out]

(b*x + d*x^3)^8/8

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Rubi [A]  time = 0.0127609, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {1588} \[ \frac{1}{8} \left (b x+d x^3\right )^8 \]

Antiderivative was successfully verified.

[In]

Int[(b + 3*d*x^2)*(b*x + d*x^3)^7,x]

[Out]

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

Mathematica [B]  time = 0.0033504, size = 98, normalized size = 6.53 \[ \frac{7}{2} b^2 d^6 x^{20}+7 b^3 d^5 x^{18}+\frac{35}{4} b^4 d^4 x^{16}+7 b^5 d^3 x^{14}+\frac{7}{2} b^6 d^2 x^{12}+b^7 d x^{10}+\frac{b^8 x^8}{8}+b d^7 x^{22}+\frac{d^8 x^{24}}{8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^8*x^8)/8 + b^7*d*x^10 + (7*b^6*d^2*x^12)/2 + 7*b^5*d^3*x^14 + (35*b^4*d^4*x^16)/4 + 7*b^3*d^5*x^18 + (7*b^2
*d^6*x^20)/2 + b*d^7*x^22 + (d^8*x^24)/8

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Maple [B]  time = 0.002, size = 89, normalized size = 5.9 \begin{align*}{\frac{{d}^{8}{x}^{24}}{8}}+b{d}^{7}{x}^{22}+{\frac{7\,{b}^{2}{d}^{6}{x}^{20}}{2}}+7\,{b}^{3}{d}^{5}{x}^{18}+{\frac{35\,{b}^{4}{d}^{4}{x}^{16}}{4}}+7\,{b}^{5}{d}^{3}{x}^{14}+{\frac{7\,{b}^{6}{d}^{2}{x}^{12}}{2}}+d{b}^{7}{x}^{10}+{\frac{{b}^{8}{x}^{8}}{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*d^8*x^24+b*d^7*x^22+7/2*b^2*d^6*x^20+7*b^3*d^5*x^18+35/4*b^4*d^4*x^16+7*b^5*d^3*x^14+7/2*b^6*d^2*x^12+d*b^
7*x^10+1/8*b^8*x^8

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Maxima [A]  time = 1.01034, size = 18, normalized size = 1.2 \begin{align*} \frac{1}{8} \,{\left (d x^{3} + b x\right )}^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(d*x^3 + b*x)^8

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Fricas [B]  time = 1.12616, size = 197, normalized size = 13.13 \begin{align*} \frac{1}{8} x^{24} d^{8} + x^{22} d^{7} b + \frac{7}{2} x^{20} d^{6} b^{2} + 7 x^{18} d^{5} b^{3} + \frac{35}{4} x^{16} d^{4} b^{4} + 7 x^{14} d^{3} b^{5} + \frac{7}{2} x^{12} d^{2} b^{6} + x^{10} d b^{7} + \frac{1}{8} x^{8} b^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x^24*d^8 + x^22*d^7*b + 7/2*x^20*d^6*b^2 + 7*x^18*d^5*b^3 + 35/4*x^16*d^4*b^4 + 7*x^14*d^3*b^5 + 7/2*x^12*
d^2*b^6 + x^10*d*b^7 + 1/8*x^8*b^8

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Sympy [B]  time = 0.089477, size = 97, normalized size = 6.47 \begin{align*} \frac{b^{8} x^{8}}{8} + b^{7} d x^{10} + \frac{7 b^{6} d^{2} x^{12}}{2} + 7 b^{5} d^{3} x^{14} + \frac{35 b^{4} d^{4} x^{16}}{4} + 7 b^{3} d^{5} x^{18} + \frac{7 b^{2} d^{6} x^{20}}{2} + b d^{7} x^{22} + \frac{d^{8} x^{24}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*d*x**2+b)*(d*x**3+b*x)**7,x)

[Out]

b**8*x**8/8 + b**7*d*x**10 + 7*b**6*d**2*x**12/2 + 7*b**5*d**3*x**14 + 35*b**4*d**4*x**16/4 + 7*b**3*d**5*x**1
8 + 7*b**2*d**6*x**20/2 + b*d**7*x**22 + d**8*x**24/8

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Giac [B]  time = 1.19029, size = 119, normalized size = 7.93 \begin{align*} \frac{1}{8} \, d^{8} x^{24} + b d^{7} x^{22} + \frac{7}{2} \, b^{2} d^{6} x^{20} + 7 \, b^{3} d^{5} x^{18} + \frac{35}{4} \, b^{4} d^{4} x^{16} + 7 \, b^{5} d^{3} x^{14} + \frac{7}{2} \, b^{6} d^{2} x^{12} + b^{7} d x^{10} + \frac{1}{8} \, b^{8} x^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*d^8*x^24 + b*d^7*x^22 + 7/2*b^2*d^6*x^20 + 7*b^3*d^5*x^18 + 35/4*b^4*d^4*x^16 + 7*b^5*d^3*x^14 + 7/2*b^6*d
^2*x^12 + b^7*d*x^10 + 1/8*b^8*x^8