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]

1/8*(d*x^3+b*x+a)^8

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Rubi [A]  time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, 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*}

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Mathematica [B]  time = 0.06, size = 127, normalized size = 7.94 \[ \frac {1}{8} x \left (b+d x^2\right ) \left (8 a^7+28 a^6 x \left (b+d x^2\right )+56 a^5 x^2 \left (b+d x^2\right )^2+70 a^4 x^3 \left (b+d x^2\right )^3+56 a^3 x^4 \left (b+d x^2\right )^4+28 a^2 x^5 \left (b+d x^2\right )^5+8 a x^6 \left (b+d x^2\right )^6+x^7 \left (b+d x^2\right )^7\right ) \]

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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fricas [B]  time = 0.53, size = 486, normalized size = 30.38 \[ \frac {1}{8} x^{24} d^{8} + x^{22} d^{7} b + x^{21} d^{7} a + \frac {7}{2} x^{20} d^{6} b^{2} + 7 x^{19} d^{6} b a + 7 x^{18} d^{5} b^{3} + \frac {7}{2} x^{18} d^{6} a^{2} + 21 x^{17} d^{5} b^{2} a + \frac {35}{4} x^{16} d^{4} b^{4} + 21 x^{16} d^{5} b a^{2} + 35 x^{15} d^{4} b^{3} a + 7 x^{15} d^{5} a^{3} + 7 x^{14} d^{3} b^{5} + \frac {105}{2} x^{14} d^{4} b^{2} a^{2} + 35 x^{13} d^{3} b^{4} a + 35 x^{13} d^{4} b a^{3} + \frac {7}{2} x^{12} d^{2} b^{6} + 70 x^{12} d^{3} b^{3} a^{2} + \frac {35}{4} x^{12} d^{4} a^{4} + 21 x^{11} d^{2} b^{5} a + 70 x^{11} d^{3} b^{2} a^{3} + x^{10} d b^{7} + \frac {105}{2} x^{10} d^{2} b^{4} a^{2} + 35 x^{10} d^{3} b a^{4} + 7 x^{9} d b^{6} a + 70 x^{9} d^{2} b^{3} a^{3} + 7 x^{9} d^{3} a^{5} + \frac {1}{8} x^{8} b^{8} + 21 x^{8} d b^{5} a^{2} + \frac {105}{2} x^{8} d^{2} b^{2} a^{4} + x^{7} b^{7} a + 35 x^{7} d b^{4} a^{3} + 21 x^{7} d^{2} b a^{5} + \frac {7}{2} x^{6} b^{6} a^{2} + 35 x^{6} d b^{3} a^{4} + \frac {7}{2} x^{6} d^{2} a^{6} + 7 x^{5} b^{5} a^{3} + 21 x^{5} d b^{2} a^{5} + \frac {35}{4} x^{4} b^{4} a^{4} + 7 x^{4} d b a^{6} + 7 x^{3} b^{3} a^{5} + x^{3} d a^{7} + \frac {7}{2} x^{2} b^{2} a^{6} + x b a^{7} \]

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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giac [B]  time = 0.41, size = 120, normalized size = 7.50 \[ \frac {1}{8} \, {\left (d x^{3} + b x\right )}^{8} + {\left (d x^{3} + b x\right )}^{7} a + \frac {7}{2} \, {\left (d x^{3} + b x\right )}^{6} a^{2} + 7 \, {\left (d x^{3} + b x\right )}^{5} a^{3} + \frac {35}{4} \, {\left (d x^{3} + b x\right )}^{4} a^{4} + 7 \, {\left (d x^{3} + b x\right )}^{3} a^{5} + \frac {7}{2} \, {\left (d x^{3} + b x\right )}^{2} a^{6} + {\left (d x^{3} + b x\right )} a^{7} \]

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*x^3 + b*x)^8 + (d*x^3 + b*x)^7*a + 7/2*(d*x^3 + b*x)^6*a^2 + 7*(d*x^3 + b*x)^5*a^3 + 35/4*(d*x^3 + b*x)
^4*a^4 + 7*(d*x^3 + b*x)^3*a^5 + 7/2*(d*x^3 + b*x)^2*a^6 + (d*x^3 + b*x)*a^7

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maple [B]  time = 0.00, size = 2185, normalized size = 136.56 \[ \text {result too large to display} \]

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*b*a^2
*d^3+6*(3*a^2*d+b^3)*b*d^2+9*d^2*b^4)))*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*b^3*a*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*b*a^2*d^3+6*(3*a^2*d+b^3)*b*d^2+9*d^2*b^4))+3*d*(60*a^2*b^2*d^3+b*(42*b*a
^2*d^3+6*(3*a^2*d+b^3)*b*d^2+9*d^2*b^4)+d*(72*b^2*a^2*d^2+6*(3*a^2*d+b^3)*d*b^2)))*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*b^3*a*d^2+6*(3*a^2*d+b^3)*a*d^2))+3*d*(a*(42*b*a^
2*d^3+6*(3*a^2*d+b^3)*b*d^2+9*d^2*b^4)+b*(2*a^3*d^3+54*b^3*a*d^2+6*(3*a^2*d+b^3)*a*d^2)+d*(24*a^3*b*d^2+18*b^4
*a*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*b*a^2*d^3+6*(3*a^2*d+b^3)*b*d^2+9*d^2*b^4)+d
*(72*b^2*a^2*d^2+6*(3*a^2*d+b^3)*d*b^2))+3*d*(a*(2*a^3*d^3+54*b^3*a*d^2+6*(3*a^2*d+b^3)*a*d^2)+b*(72*b^2*a^2*d
^2+6*(3*a^2*d+b^3)*d*b^2)+d*(6*a^4*d^2+54*b^3*a^2*d+(3*a^2*d+b^3)^2)))*x^12+1/11*(b*(a*(42*b*a^2*d^3+6*(3*a^2*
d+b^3)*b*d^2+9*d^2*b^4)+b*(2*a^3*d^3+54*b^3*a*d^2+6*(3*a^2*d+b^3)*a*d^2)+d*(24*a^3*b*d^2+18*b^4*a*d+12*(3*a^2*
d+b^3)*d*a*b))+3*d*(a*(72*b^2*a^2*d^2+6*(3*a^2*d+b^3)*d*b^2)+b*(24*a^3*b*d^2+18*b^4*a*d+12*(3*a^2*d+b^3)*d*a*b
)+d*(42*a^3*d*b^2+6*b^2*a*(3*a^2*d+b^3))))*x^11+1/10*(b*(a*(2*a^3*d^3+54*b^3*a*d^2+6*(3*a^2*d+b^3)*a*d^2)+b*(7
2*b^2*a^2*d^2+6*(3*a^2*d+b^3)*d*b^2)+d*(6*a^4*d^2+54*b^3*a^2*d+(3*a^2*d+b^3)^2))+3*d*(a*(24*a^3*b*d^2+18*b^4*a
*d+12*(3*a^2*d+b^3)*d*a*b)+b*(6*a^4*d^2+54*b^3*a^2*d+(3*a^2*d+b^3)^2)+d*(12*a^4*d*b+6*b*a^2*(3*a^2*d+b^3)+9*b^
4*a^2)))*x^10+1/9*(b*(a*(72*b^2*a^2*d^2+6*(3*a^2*d+b^3)*d*b^2)+b*(24*a^3*b*d^2+18*b^4*a*d+12*(3*a^2*d+b^3)*d*a
*b)+d*(42*a^3*d*b^2+6*b^2*a*(3*a^2*d+b^3)))+3*d*(a*(6*a^4*d^2+54*b^3*a^2*d+(3*a^2*d+b^3)^2)+b*(42*a^3*d*b^2+6*
b^2*a*(3*a^2*d+b^3))+d*(2*a^3*(3*a^2*d+b^3)+18*b^3*a^3)))*x^9+1/8*(b*(a*(24*a^3*b*d^2+18*b^4*a*d+12*(3*a^2*d+b
^3)*d*a*b)+b*(6*a^4*d^2+54*b^3*a^2*d+(3*a^2*d+b^3)^2)+d*(12*a^4*d*b+6*b*a^2*(3*a^2*d+b^3)+9*b^4*a^2))+3*d*(a*(
42*a^3*d*b^2+6*b^2*a*(3*a^2*d+b^3))+b*(12*a^4*d*b+6*b*a^2*(3*a^2*d+b^3)+9*b^4*a^2)+15*d*b^2*a^4))*x^8+1/7*(b*(
a*(6*a^4*d^2+54*b^3*a^2*d+(3*a^2*d+b^3)^2)+b*(42*a^3*d*b^2+6*b^2*a*(3*a^2*d+b^3))+d*(2*a^3*(3*a^2*d+b^3)+18*b^
3*a^3))+3*d*(a*(12*a^4*d*b+6*b*a^2*(3*a^2*d+b^3)+9*b^4*a^2)+b*(2*a^3*(3*a^2*d+b^3)+18*b^3*a^3)+6*d*a^5*b))*x^7
+1/6*(b*(a*(42*a^3*d*b^2+6*b^2*a*(3*a^2*d+b^3))+b*(12*a^4*d*b+6*b*a^2*(3*a^2*d+b^3)+9*b^4*a^2)+15*d*b^2*a^4)+3
*d*(a*(2*a^3*(3*a^2*d+b^3)+18*b^3*a^3)+15*b^3*a^4+d*a^6))*x^6+1/5*(b*(a*(12*a^4*d*b+6*b*a^2*(3*a^2*d+b^3)+9*b^
4*a^2)+b*(2*a^3*(3*a^2*d+b^3)+18*b^3*a^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*b^3*a
^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 = 0.55, size = 14, normalized size = 0.88 \[ \frac {1}{8} \, {\left (d x^{3} + b x + a\right )}^{8} \]

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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mupad [B]  time = 2.63, size = 438, normalized size = 27.38 \[ x^{12}\,\left (\frac {35\,a^4\,d^4}{4}+70\,a^2\,b^3\,d^3+\frac {7\,b^6\,d^2}{2}\right )+x^4\,\left (7\,d\,a^6\,b+\frac {35\,a^4\,b^4}{4}\right )+x^{18}\,\left (\frac {7\,a^2\,d^6}{2}+7\,b^3\,d^5\right )+x^6\,\left (\frac {7\,a^6\,d^2}{2}+35\,a^4\,b^3\,d+\frac {7\,a^2\,b^6}{2}\right )+x^8\,\left (\frac {105\,a^4\,b^2\,d^2}{2}+21\,a^2\,b^5\,d+\frac {b^8}{8}\right )+\frac {d^8\,x^{24}}{8}+x^3\,\left (d\,a^7+7\,a^5\,b^3\right )+a\,d^7\,x^{21}+b\,d^7\,x^{22}+\frac {7\,a^6\,b^2\,x^2}{2}+\frac {7\,b^2\,d^6\,x^{20}}{2}+a^7\,b\,x+21\,a\,b^2\,d^5\,x^{17}+a\,b\,x^7\,\left (21\,a^4\,d^2+35\,a^2\,b^3\,d+b^6\right )+7\,a\,d\,x^9\,\left (a^4\,d^2+10\,a^2\,b^3\,d+b^6\right )+7\,a^3\,b^2\,x^5\,\left (3\,d\,a^2+b^3\right )+7\,a\,d^4\,x^{15}\,\left (d\,a^2+5\,b^3\right )+\frac {7\,b\,d^4\,x^{16}\,\left (12\,d\,a^2+5\,b^3\right )}{4}+\frac {b\,d\,x^{10}\,\left (70\,a^4\,d^2+105\,a^2\,b^3\,d+2\,b^6\right )}{2}+7\,a\,b\,d^6\,x^{19}+\frac {7\,b^2\,d^3\,x^{14}\,\left (15\,d\,a^2+2\,b^3\right )}{2}+7\,a\,b^2\,d^2\,x^{11}\,\left (10\,d\,a^2+3\,b^3\right )+35\,a\,b\,d^3\,x^{13}\,\left (d\,a^2+b^3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.17, size = 483, normalized size = 30.19 \[ a^{7} b x + \frac {7 a^{6} b^{2} x^{2}}{2} + 21 a b^{2} d^{5} x^{17} + 7 a b d^{6} x^{19} + a d^{7} x^{21} + \frac {7 b^{2} d^{6} x^{20}}{2} + b d^{7} x^{22} + \frac {d^{8} x^{24}}{8} + x^{18} \left (\frac {7 a^{2} d^{6}}{2} + 7 b^{3} d^{5}\right ) + x^{16} \left (21 a^{2} b d^{5} + \frac {35 b^{4} d^{4}}{4}\right ) + x^{15} \left (7 a^{3} d^{5} + 35 a b^{3} d^{4}\right ) + x^{14} \left (\frac {105 a^{2} b^{2} d^{4}}{2} + 7 b^{5} d^{3}\right ) + x^{13} \left (35 a^{3} b d^{4} + 35 a b^{4} d^{3}\right ) + x^{12} \left (\frac {35 a^{4} d^{4}}{4} + 70 a^{2} b^{3} d^{3} + \frac {7 b^{6} d^{2}}{2}\right ) + x^{11} \left (70 a^{3} b^{2} d^{3} + 21 a b^{5} d^{2}\right ) + x^{10} \left (35 a^{4} b d^{3} + \frac {105 a^{2} b^{4} d^{2}}{2} + b^{7} d\right ) + x^{9} \left (7 a^{5} d^{3} + 70 a^{3} b^{3} d^{2} + 7 a b^{6} d\right ) + x^{8} \left (\frac {105 a^{4} b^{2} d^{2}}{2} + 21 a^{2} b^{5} d + \frac {b^{8}}{8}\right ) + x^{7} \left (21 a^{5} b d^{2} + 35 a^{3} b^{4} d + a b^{7}\right ) + x^{6} \left (\frac {7 a^{6} d^{2}}{2} + 35 a^{4} b^{3} d + \frac {7 a^{2} b^{6}}{2}\right ) + x^{5} \left (21 a^{5} b^{2} d + 7 a^{3} b^{5}\right ) + x^{4} \left (7 a^{6} b d + \frac {35 a^{4} b^{4}}{4}\right ) + x^{3} \left (a^{7} d + 7 a^{5} b^{3}\right ) \]

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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