3.472 \(\int (d+e x)^6 (a+c x^2)^3 \, dx\)

Optimal. Leaf size=190 \[ \frac{3 c^2 (d+e x)^{11} \left (a e^2+5 c d^2\right )}{11 e^7}-\frac{2 c^2 d (d+e x)^{10} \left (3 a e^2+5 c d^2\right )}{5 e^7}+\frac{c (d+e x)^9 \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{3 e^7}-\frac{3 c d (d+e x)^8 \left (a e^2+c d^2\right )^2}{4 e^7}+\frac{(d+e x)^7 \left (a e^2+c d^2\right )^3}{7 e^7}+\frac{c^3 (d+e x)^{13}}{13 e^7}-\frac{c^3 d (d+e x)^{12}}{2 e^7} \]

[Out]

((c*d^2 + a*e^2)^3*(d + e*x)^7)/(7*e^7) - (3*c*d*(c*d^2 + a*e^2)^2*(d + e*x)^8)/(4*e^7) + (c*(c*d^2 + a*e^2)*(
5*c*d^2 + a*e^2)*(d + e*x)^9)/(3*e^7) - (2*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^10)/(5*e^7) + (3*c^2*(5*c*d^2 +
a*e^2)*(d + e*x)^11)/(11*e^7) - (c^3*d*(d + e*x)^12)/(2*e^7) + (c^3*(d + e*x)^13)/(13*e^7)

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Rubi [A]  time = 0.329189, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {697} \[ \frac{3 c^2 (d+e x)^{11} \left (a e^2+5 c d^2\right )}{11 e^7}-\frac{2 c^2 d (d+e x)^{10} \left (3 a e^2+5 c d^2\right )}{5 e^7}+\frac{c (d+e x)^9 \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{3 e^7}-\frac{3 c d (d+e x)^8 \left (a e^2+c d^2\right )^2}{4 e^7}+\frac{(d+e x)^7 \left (a e^2+c d^2\right )^3}{7 e^7}+\frac{c^3 (d+e x)^{13}}{13 e^7}-\frac{c^3 d (d+e x)^{12}}{2 e^7} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^6*(a + c*x^2)^3,x]

[Out]

((c*d^2 + a*e^2)^3*(d + e*x)^7)/(7*e^7) - (3*c*d*(c*d^2 + a*e^2)^2*(d + e*x)^8)/(4*e^7) + (c*(c*d^2 + a*e^2)*(
5*c*d^2 + a*e^2)*(d + e*x)^9)/(3*e^7) - (2*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^10)/(5*e^7) + (3*c^2*(5*c*d^2 +
a*e^2)*(d + e*x)^11)/(11*e^7) - (c^3*d*(d + e*x)^12)/(2*e^7) + (c^3*(d + e*x)^13)/(13*e^7)

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int (d+e x)^6 \left (a+c x^2\right )^3 \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^3 (d+e x)^6}{e^6}-\frac{6 c d \left (c d^2+a e^2\right )^2 (d+e x)^7}{e^6}+\frac{3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^8}{e^6}-\frac{4 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^9}{e^6}+\frac{3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{10}}{e^6}-\frac{6 c^3 d (d+e x)^{11}}{e^6}+\frac{c^3 (d+e x)^{12}}{e^6}\right ) \, dx\\ &=\frac{\left (c d^2+a e^2\right )^3 (d+e x)^7}{7 e^7}-\frac{3 c d \left (c d^2+a e^2\right )^2 (d+e x)^8}{4 e^7}+\frac{c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^9}{3 e^7}-\frac{2 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{10}}{5 e^7}+\frac{3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{11}}{11 e^7}-\frac{c^3 d (d+e x)^{12}}{2 e^7}+\frac{c^3 (d+e x)^{13}}{13 e^7}\\ \end{align*}

Mathematica [A]  time = 0.0481254, size = 338, normalized size = 1.78 \[ \frac{1}{3} c e^2 x^9 \left (a^2 e^4+15 a c d^2 e^2+5 c^2 d^4\right )+\frac{3}{4} c d e x^8 \left (3 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right )+\frac{1}{7} x^7 \left (45 a^2 c d^2 e^4+a^3 e^6+45 a c^2 d^4 e^2+c^3 d^6\right )+a d e x^6 \left (a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right )+\frac{3}{5} a d^2 x^5 \left (5 a^2 e^4+15 a c d^2 e^2+c^2 d^4\right )+\frac{1}{2} a^2 d^3 e x^4 \left (10 a e^2+9 c d^2\right )+a^2 d^4 x^3 \left (5 a e^2+c d^2\right )+3 a^3 d^5 e x^2+a^3 d^6 x+\frac{3}{11} c^2 e^4 x^{11} \left (a e^2+5 c d^2\right )+\frac{1}{5} c^2 d e^3 x^{10} \left (9 a e^2+10 c d^2\right )+\frac{1}{2} c^3 d e^5 x^{12}+\frac{1}{13} c^3 e^6 x^{13} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^6*(a + c*x^2)^3,x]

[Out]

a^3*d^6*x + 3*a^3*d^5*e*x^2 + a^2*d^4*(c*d^2 + 5*a*e^2)*x^3 + (a^2*d^3*e*(9*c*d^2 + 10*a*e^2)*x^4)/2 + (3*a*d^
2*(c^2*d^4 + 15*a*c*d^2*e^2 + 5*a^2*e^4)*x^5)/5 + a*d*e*(3*c^2*d^4 + 10*a*c*d^2*e^2 + a^2*e^4)*x^6 + ((c^3*d^6
+ 45*a*c^2*d^4*e^2 + 45*a^2*c*d^2*e^4 + a^3*e^6)*x^7)/7 + (3*c*d*e*(c^2*d^4 + 10*a*c*d^2*e^2 + 3*a^2*e^4)*x^8
)/4 + (c*e^2*(5*c^2*d^4 + 15*a*c*d^2*e^2 + a^2*e^4)*x^9)/3 + (c^2*d*e^3*(10*c*d^2 + 9*a*e^2)*x^10)/5 + (3*c^2*
e^4*(5*c*d^2 + a*e^2)*x^11)/11 + (c^3*d*e^5*x^12)/2 + (c^3*e^6*x^13)/13

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Maple [A]  time = 0.043, size = 345, normalized size = 1.8 \begin{align*}{\frac{{e}^{6}{c}^{3}{x}^{13}}{13}}+{\frac{d{e}^{5}{c}^{3}{x}^{12}}{2}}+{\frac{ \left ( 3\,{e}^{6}a{c}^{2}+15\,{d}^{2}{e}^{4}{c}^{3} \right ){x}^{11}}{11}}+{\frac{ \left ( 18\,d{e}^{5}a{c}^{2}+20\,{d}^{3}{e}^{3}{c}^{3} \right ){x}^{10}}{10}}+{\frac{ \left ( 3\,{e}^{6}{a}^{2}c+45\,{d}^{2}{e}^{4}a{c}^{2}+15\,{d}^{4}{e}^{2}{c}^{3} \right ){x}^{9}}{9}}+{\frac{ \left ( 18\,d{e}^{5}{a}^{2}c+60\,{d}^{3}{e}^{3}a{c}^{2}+6\,{d}^{5}e{c}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ({e}^{6}{a}^{3}+45\,{d}^{2}{e}^{4}{a}^{2}c+45\,{d}^{4}{e}^{2}a{c}^{2}+{d}^{6}{c}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,d{e}^{5}{a}^{3}+60\,{d}^{3}{e}^{3}{a}^{2}c+18\,{d}^{5}ea{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 15\,{d}^{2}{e}^{4}{a}^{3}+45\,{d}^{4}{e}^{2}{a}^{2}c+3\,{d}^{6}a{c}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 20\,{d}^{3}{e}^{3}{a}^{3}+18\,{d}^{5}e{a}^{2}c \right ){x}^{4}}{4}}+{\frac{ \left ( 15\,{d}^{4}{e}^{2}{a}^{3}+3\,{d}^{6}{a}^{2}c \right ){x}^{3}}{3}}+3\,{d}^{5}e{a}^{3}{x}^{2}+{d}^{6}{a}^{3}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^6*(c*x^2+a)^3,x)

[Out]

1/13*e^6*c^3*x^13+1/2*d*e^5*c^3*x^12+1/11*(3*a*c^2*e^6+15*c^3*d^2*e^4)*x^11+1/10*(18*a*c^2*d*e^5+20*c^3*d^3*e^
3)*x^10+1/9*(3*a^2*c*e^6+45*a*c^2*d^2*e^4+15*c^3*d^4*e^2)*x^9+1/8*(18*a^2*c*d*e^5+60*a*c^2*d^3*e^3+6*c^3*d^5*e
)*x^8+1/7*(a^3*e^6+45*a^2*c*d^2*e^4+45*a*c^2*d^4*e^2+c^3*d^6)*x^7+1/6*(6*a^3*d*e^5+60*a^2*c*d^3*e^3+18*a*c^2*d
^5*e)*x^6+1/5*(15*a^3*d^2*e^4+45*a^2*c*d^4*e^2+3*a*c^2*d^6)*x^5+1/4*(20*a^3*d^3*e^3+18*a^2*c*d^5*e)*x^4+1/3*(1
5*a^3*d^4*e^2+3*a^2*c*d^6)*x^3+3*d^5*e*a^3*x^2+d^6*a^3*x

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Maxima [A]  time = 1.17997, size = 454, normalized size = 2.39 \begin{align*} \frac{1}{13} \, c^{3} e^{6} x^{13} + \frac{1}{2} \, c^{3} d e^{5} x^{12} + \frac{3}{11} \,{\left (5 \, c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )} x^{11} + 3 \, a^{3} d^{5} e x^{2} + \frac{1}{5} \,{\left (10 \, c^{3} d^{3} e^{3} + 9 \, a c^{2} d e^{5}\right )} x^{10} + a^{3} d^{6} x + \frac{1}{3} \,{\left (5 \, c^{3} d^{4} e^{2} + 15 \, a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{9} + \frac{3}{4} \,{\left (c^{3} d^{5} e + 10 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x^{8} + \frac{1}{7} \,{\left (c^{3} d^{6} + 45 \, a c^{2} d^{4} e^{2} + 45 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} x^{7} +{\left (3 \, a c^{2} d^{5} e + 10 \, a^{2} c d^{3} e^{3} + a^{3} d e^{5}\right )} x^{6} + \frac{3}{5} \,{\left (a c^{2} d^{6} + 15 \, a^{2} c d^{4} e^{2} + 5 \, a^{3} d^{2} e^{4}\right )} x^{5} + \frac{1}{2} \,{\left (9 \, a^{2} c d^{5} e + 10 \, a^{3} d^{3} e^{3}\right )} x^{4} +{\left (a^{2} c d^{6} + 5 \, a^{3} d^{4} e^{2}\right )} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6*(c*x^2+a)^3,x, algorithm="maxima")

[Out]

1/13*c^3*e^6*x^13 + 1/2*c^3*d*e^5*x^12 + 3/11*(5*c^3*d^2*e^4 + a*c^2*e^6)*x^11 + 3*a^3*d^5*e*x^2 + 1/5*(10*c^3
*d^3*e^3 + 9*a*c^2*d*e^5)*x^10 + a^3*d^6*x + 1/3*(5*c^3*d^4*e^2 + 15*a*c^2*d^2*e^4 + a^2*c*e^6)*x^9 + 3/4*(c^3
*d^5*e + 10*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*x^8 + 1/7*(c^3*d^6 + 45*a*c^2*d^4*e^2 + 45*a^2*c*d^2*e^4 + a^3*e^6)
*x^7 + (3*a*c^2*d^5*e + 10*a^2*c*d^3*e^3 + a^3*d*e^5)*x^6 + 3/5*(a*c^2*d^6 + 15*a^2*c*d^4*e^2 + 5*a^3*d^2*e^4)
*x^5 + 1/2*(9*a^2*c*d^5*e + 10*a^3*d^3*e^3)*x^4 + (a^2*c*d^6 + 5*a^3*d^4*e^2)*x^3

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Fricas [B]  time = 1.77748, size = 779, normalized size = 4.1 \begin{align*} \frac{1}{13} x^{13} e^{6} c^{3} + \frac{1}{2} x^{12} e^{5} d c^{3} + \frac{15}{11} x^{11} e^{4} d^{2} c^{3} + \frac{3}{11} x^{11} e^{6} c^{2} a + 2 x^{10} e^{3} d^{3} c^{3} + \frac{9}{5} x^{10} e^{5} d c^{2} a + \frac{5}{3} x^{9} e^{2} d^{4} c^{3} + 5 x^{9} e^{4} d^{2} c^{2} a + \frac{1}{3} x^{9} e^{6} c a^{2} + \frac{3}{4} x^{8} e d^{5} c^{3} + \frac{15}{2} x^{8} e^{3} d^{3} c^{2} a + \frac{9}{4} x^{8} e^{5} d c a^{2} + \frac{1}{7} x^{7} d^{6} c^{3} + \frac{45}{7} x^{7} e^{2} d^{4} c^{2} a + \frac{45}{7} x^{7} e^{4} d^{2} c a^{2} + \frac{1}{7} x^{7} e^{6} a^{3} + 3 x^{6} e d^{5} c^{2} a + 10 x^{6} e^{3} d^{3} c a^{2} + x^{6} e^{5} d a^{3} + \frac{3}{5} x^{5} d^{6} c^{2} a + 9 x^{5} e^{2} d^{4} c a^{2} + 3 x^{5} e^{4} d^{2} a^{3} + \frac{9}{2} x^{4} e d^{5} c a^{2} + 5 x^{4} e^{3} d^{3} a^{3} + x^{3} d^{6} c a^{2} + 5 x^{3} e^{2} d^{4} a^{3} + 3 x^{2} e d^{5} a^{3} + x d^{6} a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6*(c*x^2+a)^3,x, algorithm="fricas")

[Out]

1/13*x^13*e^6*c^3 + 1/2*x^12*e^5*d*c^3 + 15/11*x^11*e^4*d^2*c^3 + 3/11*x^11*e^6*c^2*a + 2*x^10*e^3*d^3*c^3 + 9
/5*x^10*e^5*d*c^2*a + 5/3*x^9*e^2*d^4*c^3 + 5*x^9*e^4*d^2*c^2*a + 1/3*x^9*e^6*c*a^2 + 3/4*x^8*e*d^5*c^3 + 15/2
*x^8*e^3*d^3*c^2*a + 9/4*x^8*e^5*d*c*a^2 + 1/7*x^7*d^6*c^3 + 45/7*x^7*e^2*d^4*c^2*a + 45/7*x^7*e^4*d^2*c*a^2 +
1/7*x^7*e^6*a^3 + 3*x^6*e*d^5*c^2*a + 10*x^6*e^3*d^3*c*a^2 + x^6*e^5*d*a^3 + 3/5*x^5*d^6*c^2*a + 9*x^5*e^2*d^
4*c*a^2 + 3*x^5*e^4*d^2*a^3 + 9/2*x^4*e*d^5*c*a^2 + 5*x^4*e^3*d^3*a^3 + x^3*d^6*c*a^2 + 5*x^3*e^2*d^4*a^3 + 3*
x^2*e*d^5*a^3 + x*d^6*a^3

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Sympy [B]  time = 0.113902, size = 371, normalized size = 1.95 \begin{align*} a^{3} d^{6} x + 3 a^{3} d^{5} e x^{2} + \frac{c^{3} d e^{5} x^{12}}{2} + \frac{c^{3} e^{6} x^{13}}{13} + x^{11} \left (\frac{3 a c^{2} e^{6}}{11} + \frac{15 c^{3} d^{2} e^{4}}{11}\right ) + x^{10} \left (\frac{9 a c^{2} d e^{5}}{5} + 2 c^{3} d^{3} e^{3}\right ) + x^{9} \left (\frac{a^{2} c e^{6}}{3} + 5 a c^{2} d^{2} e^{4} + \frac{5 c^{3} d^{4} e^{2}}{3}\right ) + x^{8} \left (\frac{9 a^{2} c d e^{5}}{4} + \frac{15 a c^{2} d^{3} e^{3}}{2} + \frac{3 c^{3} d^{5} e}{4}\right ) + x^{7} \left (\frac{a^{3} e^{6}}{7} + \frac{45 a^{2} c d^{2} e^{4}}{7} + \frac{45 a c^{2} d^{4} e^{2}}{7} + \frac{c^{3} d^{6}}{7}\right ) + x^{6} \left (a^{3} d e^{5} + 10 a^{2} c d^{3} e^{3} + 3 a c^{2} d^{5} e\right ) + x^{5} \left (3 a^{3} d^{2} e^{4} + 9 a^{2} c d^{4} e^{2} + \frac{3 a c^{2} d^{6}}{5}\right ) + x^{4} \left (5 a^{3} d^{3} e^{3} + \frac{9 a^{2} c d^{5} e}{2}\right ) + x^{3} \left (5 a^{3} d^{4} e^{2} + a^{2} c d^{6}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**6*(c*x**2+a)**3,x)

[Out]

a**3*d**6*x + 3*a**3*d**5*e*x**2 + c**3*d*e**5*x**12/2 + c**3*e**6*x**13/13 + x**11*(3*a*c**2*e**6/11 + 15*c**
3*d**2*e**4/11) + x**10*(9*a*c**2*d*e**5/5 + 2*c**3*d**3*e**3) + x**9*(a**2*c*e**6/3 + 5*a*c**2*d**2*e**4 + 5*
c**3*d**4*e**2/3) + x**8*(9*a**2*c*d*e**5/4 + 15*a*c**2*d**3*e**3/2 + 3*c**3*d**5*e/4) + x**7*(a**3*e**6/7 + 4
5*a**2*c*d**2*e**4/7 + 45*a*c**2*d**4*e**2/7 + c**3*d**6/7) + x**6*(a**3*d*e**5 + 10*a**2*c*d**3*e**3 + 3*a*c*
*2*d**5*e) + x**5*(3*a**3*d**2*e**4 + 9*a**2*c*d**4*e**2 + 3*a*c**2*d**6/5) + x**4*(5*a**3*d**3*e**3 + 9*a**2*
c*d**5*e/2) + x**3*(5*a**3*d**4*e**2 + a**2*c*d**6)

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Giac [A]  time = 1.34576, size = 467, normalized size = 2.46 \begin{align*} \frac{1}{13} \, c^{3} x^{13} e^{6} + \frac{1}{2} \, c^{3} d x^{12} e^{5} + \frac{15}{11} \, c^{3} d^{2} x^{11} e^{4} + 2 \, c^{3} d^{3} x^{10} e^{3} + \frac{5}{3} \, c^{3} d^{4} x^{9} e^{2} + \frac{3}{4} \, c^{3} d^{5} x^{8} e + \frac{1}{7} \, c^{3} d^{6} x^{7} + \frac{3}{11} \, a c^{2} x^{11} e^{6} + \frac{9}{5} \, a c^{2} d x^{10} e^{5} + 5 \, a c^{2} d^{2} x^{9} e^{4} + \frac{15}{2} \, a c^{2} d^{3} x^{8} e^{3} + \frac{45}{7} \, a c^{2} d^{4} x^{7} e^{2} + 3 \, a c^{2} d^{5} x^{6} e + \frac{3}{5} \, a c^{2} d^{6} x^{5} + \frac{1}{3} \, a^{2} c x^{9} e^{6} + \frac{9}{4} \, a^{2} c d x^{8} e^{5} + \frac{45}{7} \, a^{2} c d^{2} x^{7} e^{4} + 10 \, a^{2} c d^{3} x^{6} e^{3} + 9 \, a^{2} c d^{4} x^{5} e^{2} + \frac{9}{2} \, a^{2} c d^{5} x^{4} e + a^{2} c d^{6} x^{3} + \frac{1}{7} \, a^{3} x^{7} e^{6} + a^{3} d x^{6} e^{5} + 3 \, a^{3} d^{2} x^{5} e^{4} + 5 \, a^{3} d^{3} x^{4} e^{3} + 5 \, a^{3} d^{4} x^{3} e^{2} + 3 \, a^{3} d^{5} x^{2} e + a^{3} d^{6} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6*(c*x^2+a)^3,x, algorithm="giac")

[Out]

1/13*c^3*x^13*e^6 + 1/2*c^3*d*x^12*e^5 + 15/11*c^3*d^2*x^11*e^4 + 2*c^3*d^3*x^10*e^3 + 5/3*c^3*d^4*x^9*e^2 + 3
/4*c^3*d^5*x^8*e + 1/7*c^3*d^6*x^7 + 3/11*a*c^2*x^11*e^6 + 9/5*a*c^2*d*x^10*e^5 + 5*a*c^2*d^2*x^9*e^4 + 15/2*a
*c^2*d^3*x^8*e^3 + 45/7*a*c^2*d^4*x^7*e^2 + 3*a*c^2*d^5*x^6*e + 3/5*a*c^2*d^6*x^5 + 1/3*a^2*c*x^9*e^6 + 9/4*a^
2*c*d*x^8*e^5 + 45/7*a^2*c*d^2*x^7*e^4 + 10*a^2*c*d^3*x^6*e^3 + 9*a^2*c*d^4*x^5*e^2 + 9/2*a^2*c*d^5*x^4*e + a^
2*c*d^6*x^3 + 1/7*a^3*x^7*e^6 + a^3*d*x^6*e^5 + 3*a^3*d^2*x^5*e^4 + 5*a^3*d^3*x^4*e^3 + 5*a^3*d^4*x^3*e^2 + 3*
a^3*d^5*x^2*e + a^3*d^6*x