\(\int (b+3 d x^2) (a+b x+d x^3)^7 \, dx\) [195]

Optimal result

Integrand size = 21, antiderivative size = 16 \[ \int \left (b+3 d x^2\right ) \left (a+b x+d x^3\right )^7 \, dx=\frac {1}{8} \left (a+b x+d x^3\right )^8 \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {1602} \[ \int \left (b+3 d x^2\right ) \left (a+b x+d x^3\right )^7 \, dx=\frac {1}{8} \left (a+b x+d x^3\right )^8 \]

[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 1602

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(127\) vs. \(2(16)=32\).

Time = 0.04 (sec) , antiderivative size = 127, normalized size of antiderivative = 7.94 \[ \int \left (b+3 d x^2\right ) \left (a+b x+d x^3\right )^7 \, dx=\frac {1}{8} x \left (b+d x^2\right ) \left (8 a^7+28 a^6 x \left (b+d x^2\right )+56 a^5 x^2 \left (b+d x^2\right )^2+70 a^4 x^3 \left (b+d x^2\right )^3+56 a^3 x^4 \left (b+d x^2\right )^4+28 a^2 x^5 \left (b+d x^2\right )^5+8 a x^6 \left (b+d x^2\right )^6+x^7 \left (b+d x^2\right )^7\right ) \]

[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

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94

method	result	size
default	\(\frac {\left (x^{3} d +b x +a \right )^{8}}{8}\)	\(15\)
norman	\(\frac {7 x^{20} b^{2} d^{6}}{2}+a \,d^{7} x^{21}+x^{22} b \,d^{7}+\frac {d^{8} x^{24}}{8}+7 a b \,d^{6} x^{19}+\left (7 a^{3} d^{5}+35 b^{3} a \,d^{4}\right ) x^{15}+\left (21 a^{2} b \,d^{5}+\frac {35}{4} b^{4} d^{4}\right ) x^{16}+21 a \,b^{2} d^{5} x^{17}+\left (\frac {7}{2} a^{2} d^{6}+7 b^{3} d^{5}\right ) x^{18}+\left (35 a^{4} b \,d^{3}+\frac {105}{2} a^{2} b^{4} d^{2}+b^{7} d \right ) x^{10}+\left (70 a^{3} b^{2} d^{3}+21 a \,b^{5} d^{2}\right ) x^{11}+\left (\frac {35}{4} a^{4} d^{4}+70 a^{2} b^{3} d^{3}+\frac {7}{2} b^{6} d^{2}\right ) x^{12}+\left (35 a^{3} b \,d^{4}+35 a \,b^{4} d^{3}\right ) x^{13}+\left (\frac {105}{2} a^{2} b^{2} d^{4}+7 b^{5} d^{3}\right ) x^{14}+\left (\frac {7}{2} a^{6} d^{2}+35 a^{4} b^{3} d +\frac {7}{2} a^{2} b^{6}\right ) x^{6}+\left (21 a^{5} d^{2} b +35 a^{3} b^{4} d +a \,b^{7}\right ) x^{7}+\left (\frac {105}{2} a^{4} b^{2} d^{2}+21 a^{2} b^{5} d +\frac {1}{8} b^{8}\right ) x^{8}+\left (7 a^{5} d^{3}+70 a^{3} b^{3} d^{2}+7 a \,b^{6} d \right ) x^{9}+\left (d \,a^{7}+7 a^{5} b^{3}\right ) x^{3}+\left (7 d \,a^{6} b +\frac {35}{4} a^{4} b^{4}\right ) x^{4}+\left (21 d \,b^{2} a^{5}+7 a^{3} b^{5}\right ) x^{5}+a^{7} b x +\frac {7 x^{2} b^{2} a^{6}}{2}\)	\(454\)
gosper	\(\frac {7}{2} x^{12} b^{6} d^{2}+\frac {35}{4} x^{4} a^{4} b^{4}+\frac {7}{2} x^{6} a^{6} d^{2}+\frac {7}{2} x^{6} a^{2} b^{6}+\frac {7}{2} x^{2} b^{2} a^{6}+70 a^{3} b^{3} d^{2} x^{9}+7 a \,b^{6} d \,x^{9}+70 a^{3} b^{2} d^{3} x^{11}+a^{7} b x +a \,d^{7} x^{21}+21 a \,b^{2} d^{5} x^{17}+a^{7} d \,x^{3}+7 a^{5} b^{3} x^{3}+7 a^{3} b^{5} x^{5}+a \,b^{7} x^{7}+7 a^{3} d^{5} x^{15}+\frac {35}{4} x^{16} b^{4} d^{4}+\frac {7}{2} x^{20} b^{2} d^{6}+x^{22} b \,d^{7}+7 a^{5} d^{3} x^{9}+7 x^{14} b^{5} d^{3}+\frac {1}{8} x^{8} b^{8}+35 a^{3} b \,d^{4} x^{13}+70 x^{12} a^{2} b^{3} d^{3}+21 a^{5} b^{2} d \,x^{5}+21 a^{5} b \,d^{2} x^{7}+35 a^{3} b^{4} d \,x^{7}+35 a \,b^{3} d^{4} x^{15}+\frac {1}{8} d^{8} x^{24}+\frac {105}{2} x^{8} a^{4} b^{2} d^{2}+21 x^{8} a^{2} b^{5} d +35 x^{10} a^{4} b \,d^{3}+\frac {105}{2} x^{10} a^{2} b^{4} d^{2}+\frac {7}{2} x^{18} a^{2} d^{6}+7 x^{18} b^{3} d^{5}+x^{10} b^{7} d +21 x^{16} a^{2} b \,d^{5}+\frac {35}{4} x^{12} a^{4} d^{4}+21 a \,b^{5} d^{2} x^{11}+7 x^{4} d \,a^{6} b +35 x^{6} a^{4} b^{3} d +\frac {105}{2} x^{14} a^{2} b^{2} d^{4}+35 a \,b^{4} d^{3} x^{13}+7 a b \,d^{6} x^{19}\)	\(487\)
parallelrisch	\(\frac {7}{2} x^{12} b^{6} d^{2}+\frac {35}{4} x^{4} a^{4} b^{4}+\frac {7}{2} x^{6} a^{6} d^{2}+\frac {7}{2} x^{6} a^{2} b^{6}+\frac {7}{2} x^{2} b^{2} a^{6}+70 a^{3} b^{3} d^{2} x^{9}+7 a \,b^{6} d \,x^{9}+70 a^{3} b^{2} d^{3} x^{11}+a^{7} b x +a \,d^{7} x^{21}+21 a \,b^{2} d^{5} x^{17}+a^{7} d \,x^{3}+7 a^{5} b^{3} x^{3}+7 a^{3} b^{5} x^{5}+a \,b^{7} x^{7}+7 a^{3} d^{5} x^{15}+\frac {35}{4} x^{16} b^{4} d^{4}+\frac {7}{2} x^{20} b^{2} d^{6}+x^{22} b \,d^{7}+7 a^{5} d^{3} x^{9}+7 x^{14} b^{5} d^{3}+\frac {1}{8} x^{8} b^{8}+35 a^{3} b \,d^{4} x^{13}+70 x^{12} a^{2} b^{3} d^{3}+21 a^{5} b^{2} d \,x^{5}+21 a^{5} b \,d^{2} x^{7}+35 a^{3} b^{4} d \,x^{7}+35 a \,b^{3} d^{4} x^{15}+\frac {1}{8} d^{8} x^{24}+\frac {105}{2} x^{8} a^{4} b^{2} d^{2}+21 x^{8} a^{2} b^{5} d +35 x^{10} a^{4} b \,d^{3}+\frac {105}{2} x^{10} a^{2} b^{4} d^{2}+\frac {7}{2} x^{18} a^{2} d^{6}+7 x^{18} b^{3} d^{5}+x^{10} b^{7} d +21 x^{16} a^{2} b \,d^{5}+\frac {35}{4} x^{12} a^{4} d^{4}+21 a \,b^{5} d^{2} x^{11}+7 x^{4} d \,a^{6} b +35 x^{6} a^{4} b^{3} d +\frac {105}{2} x^{14} a^{2} b^{2} d^{4}+35 a \,b^{4} d^{3} x^{13}+7 a b \,d^{6} x^{19}\)	\(487\)
risch	\(\frac {7}{2} x^{12} b^{6} d^{2}+\frac {35}{4} x^{4} a^{4} b^{4}+\frac {7}{2} x^{6} a^{6} d^{2}+\frac {7}{2} x^{6} a^{2} b^{6}+\frac {7}{2} x^{2} b^{2} a^{6}+70 a^{3} b^{3} d^{2} x^{9}+7 a \,b^{6} d \,x^{9}+70 a^{3} b^{2} d^{3} x^{11}+a^{7} b x +a \,d^{7} x^{21}+21 a \,b^{2} d^{5} x^{17}+a^{7} d \,x^{3}+7 a^{5} b^{3} x^{3}+7 a^{3} b^{5} x^{5}+a \,b^{7} x^{7}+7 a^{3} d^{5} x^{15}+\frac {35}{4} x^{16} b^{4} d^{4}+\frac {7}{2} x^{20} b^{2} d^{6}+x^{22} b \,d^{7}+7 a^{5} d^{3} x^{9}+7 x^{14} b^{5} d^{3}+\frac {1}{8} x^{8} b^{8}+35 a^{3} b \,d^{4} x^{13}+70 x^{12} a^{2} b^{3} d^{3}+21 a^{5} b^{2} d \,x^{5}+21 a^{5} b \,d^{2} x^{7}+35 a^{3} b^{4} d \,x^{7}+35 a \,b^{3} d^{4} x^{15}+\frac {1}{8} d^{8} x^{24}+\frac {105}{2} x^{8} a^{4} b^{2} d^{2}+21 x^{8} a^{2} b^{5} d +35 x^{10} a^{4} b \,d^{3}+\frac {105}{2} x^{10} a^{2} b^{4} d^{2}+\frac {7}{2} x^{18} a^{2} d^{6}+7 x^{18} b^{3} d^{5}+x^{10} b^{7} d +21 x^{16} a^{2} b \,d^{5}+\frac {35}{4} x^{12} a^{4} d^{4}+21 a \,b^{5} d^{2} x^{11}+7 x^{4} d \,a^{6} b +35 x^{6} a^{4} b^{3} d +\frac {105}{2} x^{14} a^{2} b^{2} d^{4}+\frac {1}{8} a^{8}+35 a \,b^{4} d^{3} x^{13}+7 a b \,d^{6} x^{19}\)	\(492\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (14) = 28\).

Time = 0.24 (sec) , antiderivative size = 456, normalized size of antiderivative = 28.50 \[ \int \left (b+3 d x^2\right ) \left (a+b x+d x^3\right )^7 \, dx=\frac {1}{8} \, d^{8} x^{24} + b d^{7} x^{22} + a d^{7} x^{21} + \frac {7}{2} \, b^{2} d^{6} x^{20} + 7 \, a b d^{6} x^{19} + 21 \, a b^{2} d^{5} x^{17} + \frac {7}{2} \, {\left (2 \, b^{3} d^{5} + a^{2} d^{6}\right )} x^{18} + \frac {7}{4} \, {\left (5 \, b^{4} d^{4} + 12 \, a^{2} b d^{5}\right )} x^{16} + 7 \, {\left (5 \, a b^{3} d^{4} + a^{3} d^{5}\right )} x^{15} + \frac {7}{2} \, {\left (2 \, b^{5} d^{3} + 15 \, a^{2} b^{2} d^{4}\right )} x^{14} + 35 \, {\left (a b^{4} d^{3} + a^{3} b d^{4}\right )} x^{13} + \frac {7}{4} \, {\left (2 \, b^{6} d^{2} + 40 \, a^{2} b^{3} d^{3} + 5 \, a^{4} d^{4}\right )} x^{12} + 7 \, {\left (3 \, a b^{5} d^{2} + 10 \, a^{3} b^{2} d^{3}\right )} x^{11} + \frac {1}{2} \, {\left (2 \, b^{7} d + 105 \, a^{2} b^{4} d^{2} + 70 \, a^{4} b d^{3}\right )} x^{10} + \frac {7}{2} \, a^{6} b^{2} x^{2} + 7 \, {\left (a b^{6} d + 10 \, a^{3} b^{3} d^{2} + a^{5} d^{3}\right )} x^{9} + a^{7} b x + \frac {1}{8} \, {\left (b^{8} + 168 \, a^{2} b^{5} d + 420 \, a^{4} b^{2} d^{2}\right )} x^{8} + {\left (a b^{7} + 35 \, a^{3} b^{4} d + 21 \, a^{5} b d^{2}\right )} x^{7} + \frac {7}{2} \, {\left (a^{2} b^{6} + 10 \, a^{4} b^{3} d + a^{6} d^{2}\right )} x^{6} + 7 \, {\left (a^{3} b^{5} + 3 \, a^{5} b^{2} d\right )} x^{5} + \frac {7}{4} \, {\left (5 \, a^{4} b^{4} + 4 \, a^{6} b d\right )} x^{4} + {\left (7 \, a^{5} b^{3} + a^{7} d\right )} x^{3} \]

[In]

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

[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 + 21*a*b^2*d^5*x^17 + 7/2*(2*b^3*d^
5 + a^2*d^6)*x^18 + 7/4*(5*b^4*d^4 + 12*a^2*b*d^5)*x^16 + 7*(5*a*b^3*d^4 + a^3*d^5)*x^15 + 7/2*(2*b^5*d^3 + 15
*a^2*b^2*d^4)*x^14 + 35*(a*b^4*d^3 + a^3*b*d^4)*x^13 + 7/4*(2*b^6*d^2 + 40*a^2*b^3*d^3 + 5*a^4*d^4)*x^12 + 7*(
3*a*b^5*d^2 + 10*a^3*b^2*d^3)*x^11 + 1/2*(2*b^7*d + 105*a^2*b^4*d^2 + 70*a^4*b*d^3)*x^10 + 7/2*a^6*b^2*x^2 + 7
*(a*b^6*d + 10*a^3*b^3*d^2 + a^5*d^3)*x^9 + a^7*b*x + 1/8*(b^8 + 168*a^2*b^5*d + 420*a^4*b^2*d^2)*x^8 + (a*b^7
+ 35*a^3*b^4*d + 21*a^5*b*d^2)*x^7 + 7/2*(a^2*b^6 + 10*a^4*b^3*d + a^6*d^2)*x^6 + 7*(a^3*b^5 + 3*a^5*b^2*d)*x
^5 + 7/4*(5*a^4*b^4 + 4*a^6*b*d)*x^4 + (7*a^5*b^3 + a^7*d)*x^3

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 483 vs. \(2 (12) = 24\).

Time = 0.08 (sec) , antiderivative size = 483, normalized size of antiderivative = 30.19 \[ \int \left (b+3 d x^2\right ) \left (a+b x+d x^3\right )^7 \, dx=a^{7} b x + \frac {7 a^{6} b^{2} x^{2}}{2} + 21 a b^{2} d^{5} x^{17} + 7 a b d^{6} x^{19} + a d^{7} x^{21} + \frac {7 b^{2} d^{6} x^{20}}{2} + b d^{7} x^{22} + \frac {d^{8} x^{24}}{8} + x^{18} \cdot \left (\frac {7 a^{2} d^{6}}{2} + 7 b^{3} d^{5}\right ) + x^{16} \cdot \left (21 a^{2} b d^{5} + \frac {35 b^{4} d^{4}}{4}\right ) + x^{15} \cdot \left (7 a^{3} d^{5} + 35 a b^{3} d^{4}\right ) + x^{14} \cdot \left (\frac {105 a^{2} b^{2} d^{4}}{2} + 7 b^{5} d^{3}\right ) + x^{13} \cdot \left (35 a^{3} b d^{4} + 35 a b^{4} d^{3}\right ) + x^{12} \cdot \left (\frac {35 a^{4} d^{4}}{4} + 70 a^{2} b^{3} d^{3} + \frac {7 b^{6} d^{2}}{2}\right ) + x^{11} \cdot \left (70 a^{3} b^{2} d^{3} + 21 a b^{5} d^{2}\right ) + x^{10} \cdot \left (35 a^{4} b d^{3} + \frac {105 a^{2} b^{4} d^{2}}{2} + b^{7} d\right ) + x^{9} \cdot \left (7 a^{5} d^{3} + 70 a^{3} b^{3} d^{2} + 7 a b^{6} d\right ) + x^{8} \cdot \left (\frac {105 a^{4} b^{2} d^{2}}{2} + 21 a^{2} b^{5} d + \frac {b^{8}}{8}\right ) + x^{7} \cdot \left (21 a^{5} b d^{2} + 35 a^{3} b^{4} d + a b^{7}\right ) + x^{6} \cdot \left (\frac {7 a^{6} d^{2}}{2} + 35 a^{4} b^{3} d + \frac {7 a^{2} b^{6}}{2}\right ) + x^{5} \cdot \left (21 a^{5} b^{2} d + 7 a^{3} b^{5}\right ) + x^{4} \cdot \left (7 a^{6} b d + \frac {35 a^{4} b^{4}}{4}\right ) + x^{3} \left (a^{7} d + 7 a^{5} b^{3}\right ) \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \left (b+3 d x^2\right ) \left (a+b x+d x^3\right )^7 \, dx=\frac {1}{8} \, {\left (d x^{3} + b x + a\right )}^{8} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (14) = 28\).

Time = 0.27 (sec) , antiderivative size = 120, normalized size of antiderivative = 7.50 \[ \int \left (b+3 d x^2\right ) \left (a+b x+d x^3\right )^7 \, dx=\frac {1}{8} \, {\left (d x^{3} + b x\right )}^{8} + {\left (d x^{3} + b x\right )}^{7} a + \frac {7}{2} \, {\left (d x^{3} + b x\right )}^{6} a^{2} + 7 \, {\left (d x^{3} + b x\right )}^{5} a^{3} + \frac {35}{4} \, {\left (d x^{3} + b x\right )}^{4} a^{4} + 7 \, {\left (d x^{3} + b x\right )}^{3} a^{5} + \frac {7}{2} \, {\left (d x^{3} + b x\right )}^{2} a^{6} + {\left (d x^{3} + b x\right )} a^{7} \]

[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

Mupad [B] (verification not implemented)

Time = 9.69 (sec) , antiderivative size = 438, normalized size of antiderivative = 27.38 \[ \int \left (b+3 d x^2\right ) \left (a+b x+d x^3\right )^7 \, dx=x^{12}\,\left (\frac {35\,a^4\,d^4}{4}+70\,a^2\,b^3\,d^3+\frac {7\,b^6\,d^2}{2}\right )+x^4\,\left (7\,d\,a^6\,b+\frac {35\,a^4\,b^4}{4}\right )+x^{18}\,\left (\frac {7\,a^2\,d^6}{2}+7\,b^3\,d^5\right )+x^6\,\left (\frac {7\,a^6\,d^2}{2}+35\,a^4\,b^3\,d+\frac {7\,a^2\,b^6}{2}\right )+x^8\,\left (\frac {105\,a^4\,b^2\,d^2}{2}+21\,a^2\,b^5\,d+\frac {b^8}{8}\right )+\frac {d^8\,x^{24}}{8}+x^3\,\left (d\,a^7+7\,a^5\,b^3\right )+a\,d^7\,x^{21}+b\,d^7\,x^{22}+\frac {7\,a^6\,b^2\,x^2}{2}+\frac {7\,b^2\,d^6\,x^{20}}{2}+a^7\,b\,x+21\,a\,b^2\,d^5\,x^{17}+a\,b\,x^7\,\left (21\,a^4\,d^2+35\,a^2\,b^3\,d+b^6\right )+7\,a\,d\,x^9\,\left (a^4\,d^2+10\,a^2\,b^3\,d+b^6\right )+7\,a^3\,b^2\,x^5\,\left (3\,d\,a^2+b^3\right )+7\,a\,d^4\,x^{15}\,\left (d\,a^2+5\,b^3\right )+\frac {7\,b\,d^4\,x^{16}\,\left (12\,d\,a^2+5\,b^3\right )}{4}+\frac {b\,d\,x^{10}\,\left (70\,a^4\,d^2+105\,a^2\,b^3\,d+2\,b^6\right )}{2}+7\,a\,b\,d^6\,x^{19}+\frac {7\,b^2\,d^3\,x^{14}\,\left (15\,d\,a^2+2\,b^3\right )}{2}+7\,a\,b^2\,d^2\,x^{11}\,\left (10\,d\,a^2+3\,b^3\right )+35\,a\,b\,d^3\,x^{13}\,\left (d\,a^2+b^3\right ) \]

[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)