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

Optimal. Leaf size=16 \[ \frac{1}{8} \left (a+b x+d x^3\right )^8 \]

[Out]

(a + b*x + d*x^3)^8/8

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Rubi [A]  time = 0.0243779, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {1588} \[ \frac{1}{8} \left (a+b x+d x^3\right )^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [B]  time = 0.0574379, size = 127, normalized size = 7.94 \[ \frac{1}{8} x \left (b+d x^2\right ) \left (28 a^6 x \left (b+d x^2\right )+56 a^5 x^2 \left (b+d x^2\right )^2+70 a^4 x^3 \left (b+d x^2\right )^3+56 a^3 x^4 \left (b+d x^2\right )^4+28 a^2 x^5 \left (b+d x^2\right )^5+8 a^7+8 a x^6 \left (b+d x^2\right )^6+x^7 \left (b+d x^2\right )^7\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(b + d*x^2)*(8*a^7 + 28*a^6*x*(b + d*x^2) + 56*a^5*x^2*(b + d*x^2)^2 + 70*a^4*x^3*(b + d*x^2)^3 + 56*a^3*x^
4*(b + d*x^2)^4 + 28*a^2*x^5*(b + d*x^2)^5 + 8*a*x^6*(b + d*x^2)^6 + x^7*(b + d*x^2)^7))/8

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Maple [B]  time = 0.002, size = 2185, normalized size = 136.6 \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+b)*(d*x^3+b*x+a)^7,x)

[Out]

1/8*d^8*x^24+b*d^7*x^22+d^7*a*x^21+7/2*b^2*d^6*x^20+7*b*a*d^6*x^19+1/18*(21*b^3*d^5+3*d*(6*a^2*d^5+15*b^3*d^4+
d*(2*(3*a^2*d+b^3)*d^3+18*b^3*d^3+9*a^2*d^4)))*x^18+21*b^2*a*d^5*x^17+1/16*(b*(6*a^2*d^5+15*b^3*d^4+d*(2*(3*a^
2*d+b^3)*d^3+18*b^3*d^3+9*a^2*d^4))+3*d*(30*a^2*d^4*b+b*(2*(3*a^2*d+b^3)*d^3+18*b^3*d^3+9*a^2*d^4)+d*(42*a^2*b
*d^3+6*(3*a^2*d+b^3)*b*d^2+9*b^4*d^2)))*x^16+1/15*(105*b^3*a*d^4+3*d*(a*(2*(3*a^2*d+b^3)*d^3+18*b^3*d^3+9*a^2*
d^4)+60*b^3*a*d^3+d*(2*a^3*d^3+54*a*b^3*d^2+6*(3*a^2*d+b^3)*a*d^2)))*x^15+1/14*(b*(30*a^2*d^4*b+b*(2*(3*a^2*d+
b^3)*d^3+18*b^3*d^3+9*a^2*d^4)+d*(42*a^2*b*d^3+6*(3*a^2*d+b^3)*b*d^2+9*b^4*d^2))+3*d*(60*a^2*b^2*d^3+b*(42*a^2
*b*d^3+6*(3*a^2*d+b^3)*b*d^2+9*b^4*d^2)+d*(72*a^2*b^2*d^2+6*(3*a^2*d+b^3)*b^2*d)))*x^14+1/13*(b*(a*(2*(3*a^2*d
+b^3)*d^3+18*b^3*d^3+9*a^2*d^4)+60*b^3*a*d^3+d*(2*a^3*d^3+54*a*b^3*d^2+6*(3*a^2*d+b^3)*a*d^2))+3*d*(a*(42*a^2*
b*d^3+6*(3*a^2*d+b^3)*b*d^2+9*b^4*d^2)+b*(2*a^3*d^3+54*a*b^3*d^2+6*(3*a^2*d+b^3)*a*d^2)+d*(24*a^3*b*d^2+18*a*b
^4*d+12*(3*a^2*d+b^3)*d*a*b)))*x^13+1/12*(b*(60*a^2*b^2*d^3+b*(42*a^2*b*d^3+6*(3*a^2*d+b^3)*b*d^2+9*b^4*d^2)+d
*(72*a^2*b^2*d^2+6*(3*a^2*d+b^3)*b^2*d))+3*d*(a*(2*a^3*d^3+54*a*b^3*d^2+6*(3*a^2*d+b^3)*a*d^2)+b*(72*a^2*b^2*d
^2+6*(3*a^2*d+b^3)*b^2*d)+d*(6*a^4*d^2+54*a^2*b^3*d+(3*a^2*d+b^3)^2)))*x^12+1/11*(b*(a*(42*a^2*b*d^3+6*(3*a^2*
d+b^3)*b*d^2+9*b^4*d^2)+b*(2*a^3*d^3+54*a*b^3*d^2+6*(3*a^2*d+b^3)*a*d^2)+d*(24*a^3*b*d^2+18*a*b^4*d+12*(3*a^2*
d+b^3)*d*a*b))+3*d*(a*(72*a^2*b^2*d^2+6*(3*a^2*d+b^3)*b^2*d)+b*(24*a^3*b*d^2+18*a*b^4*d+12*(3*a^2*d+b^3)*d*a*b
)+d*(42*a^3*b^2*d+6*a*b^2*(3*a^2*d+b^3))))*x^11+1/10*(b*(a*(2*a^3*d^3+54*a*b^3*d^2+6*(3*a^2*d+b^3)*a*d^2)+b*(7
2*a^2*b^2*d^2+6*(3*a^2*d+b^3)*b^2*d)+d*(6*a^4*d^2+54*a^2*b^3*d+(3*a^2*d+b^3)^2))+3*d*(a*(24*a^3*b*d^2+18*a*b^4
*d+12*(3*a^2*d+b^3)*d*a*b)+b*(6*a^4*d^2+54*a^2*b^3*d+(3*a^2*d+b^3)^2)+d*(12*a^4*d*b+6*a^2*b*(3*a^2*d+b^3)+9*a^
2*b^4)))*x^10+1/9*(b*(a*(72*a^2*b^2*d^2+6*(3*a^2*d+b^3)*b^2*d)+b*(24*a^3*b*d^2+18*a*b^4*d+12*(3*a^2*d+b^3)*d*a
*b)+d*(42*a^3*b^2*d+6*a*b^2*(3*a^2*d+b^3)))+3*d*(a*(6*a^4*d^2+54*a^2*b^3*d+(3*a^2*d+b^3)^2)+b*(42*a^3*b^2*d+6*
a*b^2*(3*a^2*d+b^3))+d*(2*a^3*(3*a^2*d+b^3)+18*a^3*b^3)))*x^9+1/8*(b*(a*(24*a^3*b*d^2+18*a*b^4*d+12*(3*a^2*d+b
^3)*d*a*b)+b*(6*a^4*d^2+54*a^2*b^3*d+(3*a^2*d+b^3)^2)+d*(12*a^4*d*b+6*a^2*b*(3*a^2*d+b^3)+9*a^2*b^4))+3*d*(a*(
42*a^3*b^2*d+6*a*b^2*(3*a^2*d+b^3))+b*(12*a^4*d*b+6*a^2*b*(3*a^2*d+b^3)+9*a^2*b^4)+15*d*a^4*b^2))*x^8+1/7*(b*(
a*(6*a^4*d^2+54*a^2*b^3*d+(3*a^2*d+b^3)^2)+b*(42*a^3*b^2*d+6*a*b^2*(3*a^2*d+b^3))+d*(2*a^3*(3*a^2*d+b^3)+18*a^
3*b^3))+3*d*(a*(12*a^4*d*b+6*a^2*b*(3*a^2*d+b^3)+9*a^2*b^4)+b*(2*a^3*(3*a^2*d+b^3)+18*a^3*b^3)+6*d*a^5*b))*x^7
+1/6*(b*(a*(42*a^3*b^2*d+6*a*b^2*(3*a^2*d+b^3))+b*(12*a^4*d*b+6*a^2*b*(3*a^2*d+b^3)+9*a^2*b^4)+15*d*a^4*b^2)+3
*d*(a*(2*a^3*(3*a^2*d+b^3)+18*a^3*b^3)+15*b^3*a^4+d*a^6))*x^6+1/5*(b*(a*(12*a^4*d*b+6*a^2*b*(3*a^2*d+b^3)+9*a^
2*b^4)+b*(2*a^3*(3*a^2*d+b^3)+18*a^3*b^3)+6*d*a^5*b)+63*d*b^2*a^5)*x^5+1/4*(b*(a*(2*a^3*(3*a^2*d+b^3)+18*a^3*b
^3)+15*b^3*a^4+d*a^6)+21*d*a^6*b)*x^4+1/3*(3*a^7*d+21*a^5*b^3)*x^3+7/2*b^2*a^6*x^2+b*a^7*x

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Maxima [A]  time = 0.986591, size = 19, normalized size = 1.19 \begin{align*} \frac{1}{8} \,{\left (d x^{3} + b x + a\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+a)^7,x, algorithm="maxima")

[Out]

1/8*(d*x^3 + b*x + a)^8

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Fricas [B]  time = 1.17532, size = 1071, normalized size = 66.94 \begin{align*} \frac{1}{8} x^{24} d^{8} + x^{22} d^{7} b + x^{21} d^{7} a + \frac{7}{2} x^{20} d^{6} b^{2} + 7 x^{19} d^{6} b a + 7 x^{18} d^{5} b^{3} + \frac{7}{2} x^{18} d^{6} a^{2} + 21 x^{17} d^{5} b^{2} a + \frac{35}{4} x^{16} d^{4} b^{4} + 21 x^{16} d^{5} b a^{2} + 35 x^{15} d^{4} b^{3} a + 7 x^{15} d^{5} a^{3} + 7 x^{14} d^{3} b^{5} + \frac{105}{2} x^{14} d^{4} b^{2} a^{2} + 35 x^{13} d^{3} b^{4} a + 35 x^{13} d^{4} b a^{3} + \frac{7}{2} x^{12} d^{2} b^{6} + 70 x^{12} d^{3} b^{3} a^{2} + \frac{35}{4} x^{12} d^{4} a^{4} + 21 x^{11} d^{2} b^{5} a + 70 x^{11} d^{3} b^{2} a^{3} + x^{10} d b^{7} + \frac{105}{2} x^{10} d^{2} b^{4} a^{2} + 35 x^{10} d^{3} b a^{4} + 7 x^{9} d b^{6} a + 70 x^{9} d^{2} b^{3} a^{3} + 7 x^{9} d^{3} a^{5} + \frac{1}{8} x^{8} b^{8} + 21 x^{8} d b^{5} a^{2} + \frac{105}{2} x^{8} d^{2} b^{2} a^{4} + x^{7} b^{7} a + 35 x^{7} d b^{4} a^{3} + 21 x^{7} d^{2} b a^{5} + \frac{7}{2} x^{6} b^{6} a^{2} + 35 x^{6} d b^{3} a^{4} + \frac{7}{2} x^{6} d^{2} a^{6} + 7 x^{5} b^{5} a^{3} + 21 x^{5} d b^{2} a^{5} + \frac{35}{4} x^{4} b^{4} a^{4} + 7 x^{4} d b a^{6} + 7 x^{3} b^{3} a^{5} + x^{3} d a^{7} + \frac{7}{2} x^{2} b^{2} a^{6} + x b a^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.151129, size = 483, normalized size = 30.19 \begin{align*} a^{7} b x + \frac{7 a^{6} b^{2} x^{2}}{2} + 21 a b^{2} d^{5} x^{17} + 7 a b d^{6} x^{19} + a d^{7} x^{21} + \frac{7 b^{2} d^{6} x^{20}}{2} + b d^{7} x^{22} + \frac{d^{8} x^{24}}{8} + x^{18} \left (\frac{7 a^{2} d^{6}}{2} + 7 b^{3} d^{5}\right ) + x^{16} \left (21 a^{2} b d^{5} + \frac{35 b^{4} d^{4}}{4}\right ) + x^{15} \left (7 a^{3} d^{5} + 35 a b^{3} d^{4}\right ) + x^{14} \left (\frac{105 a^{2} b^{2} d^{4}}{2} + 7 b^{5} d^{3}\right ) + x^{13} \left (35 a^{3} b d^{4} + 35 a b^{4} d^{3}\right ) + x^{12} \left (\frac{35 a^{4} d^{4}}{4} + 70 a^{2} b^{3} d^{3} + \frac{7 b^{6} d^{2}}{2}\right ) + x^{11} \left (70 a^{3} b^{2} d^{3} + 21 a b^{5} d^{2}\right ) + x^{10} \left (35 a^{4} b d^{3} + \frac{105 a^{2} b^{4} d^{2}}{2} + b^{7} d\right ) + x^{9} \left (7 a^{5} d^{3} + 70 a^{3} b^{3} d^{2} + 7 a b^{6} d\right ) + x^{8} \left (\frac{105 a^{4} b^{2} d^{2}}{2} + 21 a^{2} b^{5} d + \frac{b^{8}}{8}\right ) + x^{7} \left (21 a^{5} b d^{2} + 35 a^{3} b^{4} d + a b^{7}\right ) + x^{6} \left (\frac{7 a^{6} d^{2}}{2} + 35 a^{4} b^{3} d + \frac{7 a^{2} b^{6}}{2}\right ) + x^{5} \left (21 a^{5} b^{2} d + 7 a^{3} b^{5}\right ) + x^{4} \left (7 a^{6} b d + \frac{35 a^{4} b^{4}}{4}\right ) + x^{3} \left (a^{7} d + 7 a^{5} b^{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.21341, size = 656, normalized size = 41. \begin{align*} \frac{1}{8} \, d^{8} x^{24} + b d^{7} x^{22} + a d^{7} x^{21} + \frac{7}{2} \, b^{2} d^{6} x^{20} + 7 \, a b d^{6} x^{19} + 7 \, b^{3} d^{5} x^{18} + \frac{7}{2} \, a^{2} d^{6} x^{18} + 21 \, a b^{2} d^{5} x^{17} + \frac{35}{4} \, b^{4} d^{4} x^{16} + 21 \, a^{2} b d^{5} x^{16} + 35 \, a b^{3} d^{4} x^{15} + 7 \, a^{3} d^{5} x^{15} + 7 \, b^{5} d^{3} x^{14} + \frac{105}{2} \, a^{2} b^{2} d^{4} x^{14} + 35 \, a b^{4} d^{3} x^{13} + 35 \, a^{3} b d^{4} x^{13} + \frac{7}{2} \, b^{6} d^{2} x^{12} + 70 \, a^{2} b^{3} d^{3} x^{12} + \frac{35}{4} \, a^{4} d^{4} x^{12} + 21 \, a b^{5} d^{2} x^{11} + 70 \, a^{3} b^{2} d^{3} x^{11} + b^{7} d x^{10} + \frac{105}{2} \, a^{2} b^{4} d^{2} x^{10} + 35 \, a^{4} b d^{3} x^{10} + 7 \, a b^{6} d x^{9} + 70 \, a^{3} b^{3} d^{2} x^{9} + 7 \, a^{5} d^{3} x^{9} + \frac{1}{8} \, b^{8} x^{8} + 21 \, a^{2} b^{5} d x^{8} + \frac{105}{2} \, a^{4} b^{2} d^{2} x^{8} + a b^{7} x^{7} + 35 \, a^{3} b^{4} d x^{7} + 21 \, a^{5} b d^{2} x^{7} + \frac{7}{2} \, a^{2} b^{6} x^{6} + 35 \, a^{4} b^{3} d x^{6} + \frac{7}{2} \, a^{6} d^{2} x^{6} + 7 \, a^{3} b^{5} x^{5} + 21 \, a^{5} b^{2} d x^{5} + \frac{35}{4} \, a^{4} b^{4} x^{4} + 7 \, a^{6} b d x^{4} + 7 \, a^{5} b^{3} x^{3} + a^{7} d x^{3} + \frac{7}{2} \, a^{6} b^{2} x^{2} + a^{7} b x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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