3.492 \(\int (d+e x)^3 (a+c x^2)^4 \, dx\)

Optimal. Leaf size=209 \[ \frac{3}{4} a^2 c^2 e^3 x^8+\frac{6}{5} a^2 c d x^5 \left (2 a e^2+c d^2\right )+\frac{1}{3} a^3 d x^3 \left (3 a e^2+4 c d^2\right )+\frac{2}{3} a^3 c e^3 x^6+a^4 d^3 x+\frac{1}{4} a^4 e^3 x^4+\frac{1}{9} c^3 d x^9 \left (12 a e^2+c d^2\right )+\frac{2}{7} a c^2 d x^7 \left (9 a e^2+2 c d^2\right )+\frac{2}{5} a c^3 e^3 x^{10}+\frac{3 d^2 e \left (a+c x^2\right )^5}{10 c}+\frac{3}{11} c^4 d e^2 x^{11}+\frac{1}{12} c^4 e^3 x^{12} \]

[Out]

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

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Rubi [A]  time = 0.191729, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {696, 1810} \[ \frac{3}{4} a^2 c^2 e^3 x^8+\frac{6}{5} a^2 c d x^5 \left (2 a e^2+c d^2\right )+\frac{1}{3} a^3 d x^3 \left (3 a e^2+4 c d^2\right )+\frac{2}{3} a^3 c e^3 x^6+a^4 d^3 x+\frac{1}{4} a^4 e^3 x^4+\frac{1}{9} c^3 d x^9 \left (12 a e^2+c d^2\right )+\frac{2}{7} a c^2 d x^7 \left (9 a e^2+2 c d^2\right )+\frac{2}{5} a c^3 e^3 x^{10}+\frac{3 d^2 e \left (a+c x^2\right )^5}{10 c}+\frac{3}{11} c^4 d e^2 x^{11}+\frac{1}{12} c^4 e^3 x^{12} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*(a + c*x^2)^4,x]

[Out]

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

Rule 696

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*m*d^(m - 1)*(a + c*x^2)^(p + 1))
/(2*c*(p + 1)), x] + Int[((d + e*x)^m - e*m*d^(m - 1)*x)*(a + c*x^2)^p, x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*
d^2 + a*e^2, 0] && IGtQ[p, 1] && IGtQ[m, 0] && LeQ[m, p]

Rule 1810

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin{align*} \int (d+e x)^3 \left (a+c x^2\right )^4 \, dx &=\frac{3 d^2 e \left (a+c x^2\right )^5}{10 c}+\int \left (a+c x^2\right )^4 \left (-3 d^2 e x+(d+e x)^3\right ) \, dx\\ &=\frac{3 d^2 e \left (a+c x^2\right )^5}{10 c}+\int \left (a^4 d^3+a^3 d \left (4 c d^2+3 a e^2\right ) x^2+a^4 e^3 x^3+6 a^2 c d \left (c d^2+2 a e^2\right ) x^4+4 a^3 c e^3 x^5+2 a c^2 d \left (2 c d^2+9 a e^2\right ) x^6+6 a^2 c^2 e^3 x^7+c^3 d \left (c d^2+12 a e^2\right ) x^8+4 a c^3 e^3 x^9+3 c^4 d e^2 x^{10}+c^4 e^3 x^{11}\right ) \, dx\\ &=a^4 d^3 x+\frac{1}{3} a^3 d \left (4 c d^2+3 a e^2\right ) x^3+\frac{1}{4} a^4 e^3 x^4+\frac{6}{5} a^2 c d \left (c d^2+2 a e^2\right ) x^5+\frac{2}{3} a^3 c e^3 x^6+\frac{2}{7} a c^2 d \left (2 c d^2+9 a e^2\right ) x^7+\frac{3}{4} a^2 c^2 e^3 x^8+\frac{1}{9} c^3 d \left (c d^2+12 a e^2\right ) x^9+\frac{2}{5} a c^3 e^3 x^{10}+\frac{3}{11} c^4 d e^2 x^{11}+\frac{1}{12} c^4 e^3 x^{12}+\frac{3 d^2 e \left (a+c x^2\right )^5}{10 c}\\ \end{align*}

Mathematica [A]  time = 0.064292, size = 197, normalized size = 0.94 \[ \frac{x \left (297 a^2 c^2 x^4 \left (140 d^2 e x+56 d^3+120 d e^2 x^2+35 e^3 x^3\right )+924 a^3 c x^2 \left (45 d^2 e x+20 d^3+36 d e^2 x^2+10 e^3 x^3\right )+3465 a^4 \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )+66 a c^3 x^6 \left (315 d^2 e x+120 d^3+280 d e^2 x^2+84 e^3 x^3\right )+7 c^4 x^8 \left (594 d^2 e x+220 d^3+540 d e^2 x^2+165 e^3 x^3\right )\right )}{13860} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3*(a + c*x^2)^4,x]

[Out]

(x*(3465*a^4*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 924*a^3*c*x^2*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 +
10*e^3*x^3) + 297*a^2*c^2*x^4*(56*d^3 + 140*d^2*e*x + 120*d*e^2*x^2 + 35*e^3*x^3) + 66*a*c^3*x^6*(120*d^3 + 3
15*d^2*e*x + 280*d*e^2*x^2 + 84*e^3*x^3) + 7*c^4*x^8*(220*d^3 + 594*d^2*e*x + 540*d*e^2*x^2 + 165*e^3*x^3)))/1
3860

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Maple [A]  time = 0.041, size = 247, normalized size = 1.2 \begin{align*}{\frac{{c}^{4}{e}^{3}{x}^{12}}{12}}+{\frac{3\,{c}^{4}d{e}^{2}{x}^{11}}{11}}+{\frac{ \left ( 4\,{e}^{3}a{c}^{3}+3\,{d}^{2}e{c}^{4} \right ){x}^{10}}{10}}+{\frac{ \left ( 12\,d{e}^{2}a{c}^{3}+{d}^{3}{c}^{4} \right ){x}^{9}}{9}}+{\frac{ \left ( 6\,{e}^{3}{a}^{2}{c}^{2}+12\,{d}^{2}ea{c}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( 18\,d{e}^{2}{a}^{2}{c}^{2}+4\,{d}^{3}a{c}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 4\,{e}^{3}{a}^{3}c+18\,{d}^{2}e{a}^{2}{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 12\,d{e}^{2}{a}^{3}c+6\,{d}^{3}{a}^{2}{c}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ({e}^{3}{a}^{4}+12\,{d}^{2}e{a}^{3}c \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,d{e}^{2}{a}^{4}+4\,{a}^{3}c{d}^{3} \right ){x}^{3}}{3}}+{\frac{3\,{d}^{2}e{a}^{4}{x}^{2}}{2}}+{a}^{4}{d}^{3}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(c*x^2+a)^4,x)

[Out]

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

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Maxima [A]  time = 1.13363, size = 329, normalized size = 1.57 \begin{align*} \frac{1}{12} \, c^{4} e^{3} x^{12} + \frac{3}{11} \, c^{4} d e^{2} x^{11} + \frac{1}{10} \,{\left (3 \, c^{4} d^{2} e + 4 \, a c^{3} e^{3}\right )} x^{10} + \frac{1}{9} \,{\left (c^{4} d^{3} + 12 \, a c^{3} d e^{2}\right )} x^{9} + \frac{3}{2} \, a^{4} d^{2} e x^{2} + \frac{3}{4} \,{\left (2 \, a c^{3} d^{2} e + a^{2} c^{2} e^{3}\right )} x^{8} + a^{4} d^{3} x + \frac{2}{7} \,{\left (2 \, a c^{3} d^{3} + 9 \, a^{2} c^{2} d e^{2}\right )} x^{7} + \frac{1}{3} \,{\left (9 \, a^{2} c^{2} d^{2} e + 2 \, a^{3} c e^{3}\right )} x^{6} + \frac{6}{5} \,{\left (a^{2} c^{2} d^{3} + 2 \, a^{3} c d e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (12 \, a^{3} c d^{2} e + a^{4} e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (4 \, a^{3} c d^{3} + 3 \, a^{4} d e^{2}\right )} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(c*x^2+a)^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.53162, size = 548, normalized size = 2.62 \begin{align*} \frac{1}{12} x^{12} e^{3} c^{4} + \frac{3}{11} x^{11} e^{2} d c^{4} + \frac{3}{10} x^{10} e d^{2} c^{4} + \frac{2}{5} x^{10} e^{3} c^{3} a + \frac{1}{9} x^{9} d^{3} c^{4} + \frac{4}{3} x^{9} e^{2} d c^{3} a + \frac{3}{2} x^{8} e d^{2} c^{3} a + \frac{3}{4} x^{8} e^{3} c^{2} a^{2} + \frac{4}{7} x^{7} d^{3} c^{3} a + \frac{18}{7} x^{7} e^{2} d c^{2} a^{2} + 3 x^{6} e d^{2} c^{2} a^{2} + \frac{2}{3} x^{6} e^{3} c a^{3} + \frac{6}{5} x^{5} d^{3} c^{2} a^{2} + \frac{12}{5} x^{5} e^{2} d c a^{3} + 3 x^{4} e d^{2} c a^{3} + \frac{1}{4} x^{4} e^{3} a^{4} + \frac{4}{3} x^{3} d^{3} c a^{3} + x^{3} e^{2} d a^{4} + \frac{3}{2} x^{2} e d^{2} a^{4} + x d^{3} a^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(c*x^2+a)^4,x, algorithm="fricas")

[Out]

1/12*x^12*e^3*c^4 + 3/11*x^11*e^2*d*c^4 + 3/10*x^10*e*d^2*c^4 + 2/5*x^10*e^3*c^3*a + 1/9*x^9*d^3*c^4 + 4/3*x^9
*e^2*d*c^3*a + 3/2*x^8*e*d^2*c^3*a + 3/4*x^8*e^3*c^2*a^2 + 4/7*x^7*d^3*c^3*a + 18/7*x^7*e^2*d*c^2*a^2 + 3*x^6*
e*d^2*c^2*a^2 + 2/3*x^6*e^3*c*a^3 + 6/5*x^5*d^3*c^2*a^2 + 12/5*x^5*e^2*d*c*a^3 + 3*x^4*e*d^2*c*a^3 + 1/4*x^4*e
^3*a^4 + 4/3*x^3*d^3*c*a^3 + x^3*e^2*d*a^4 + 3/2*x^2*e*d^2*a^4 + x*d^3*a^4

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Sympy [A]  time = 0.112578, size = 270, normalized size = 1.29 \begin{align*} a^{4} d^{3} x + \frac{3 a^{4} d^{2} e x^{2}}{2} + \frac{3 c^{4} d e^{2} x^{11}}{11} + \frac{c^{4} e^{3} x^{12}}{12} + x^{10} \left (\frac{2 a c^{3} e^{3}}{5} + \frac{3 c^{4} d^{2} e}{10}\right ) + x^{9} \left (\frac{4 a c^{3} d e^{2}}{3} + \frac{c^{4} d^{3}}{9}\right ) + x^{8} \left (\frac{3 a^{2} c^{2} e^{3}}{4} + \frac{3 a c^{3} d^{2} e}{2}\right ) + x^{7} \left (\frac{18 a^{2} c^{2} d e^{2}}{7} + \frac{4 a c^{3} d^{3}}{7}\right ) + x^{6} \left (\frac{2 a^{3} c e^{3}}{3} + 3 a^{2} c^{2} d^{2} e\right ) + x^{5} \left (\frac{12 a^{3} c d e^{2}}{5} + \frac{6 a^{2} c^{2} d^{3}}{5}\right ) + x^{4} \left (\frac{a^{4} e^{3}}{4} + 3 a^{3} c d^{2} e\right ) + x^{3} \left (a^{4} d e^{2} + \frac{4 a^{3} c d^{3}}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(c*x**2+a)**4,x)

[Out]

a**4*d**3*x + 3*a**4*d**2*e*x**2/2 + 3*c**4*d*e**2*x**11/11 + c**4*e**3*x**12/12 + x**10*(2*a*c**3*e**3/5 + 3*
c**4*d**2*e/10) + x**9*(4*a*c**3*d*e**2/3 + c**4*d**3/9) + x**8*(3*a**2*c**2*e**3/4 + 3*a*c**3*d**2*e/2) + x**
7*(18*a**2*c**2*d*e**2/7 + 4*a*c**3*d**3/7) + x**6*(2*a**3*c*e**3/3 + 3*a**2*c**2*d**2*e) + x**5*(12*a**3*c*d*
e**2/5 + 6*a**2*c**2*d**3/5) + x**4*(a**4*e**3/4 + 3*a**3*c*d**2*e) + x**3*(a**4*d*e**2 + 4*a**3*c*d**3/3)

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Giac [A]  time = 1.35884, size = 327, normalized size = 1.56 \begin{align*} \frac{1}{12} \, c^{4} x^{12} e^{3} + \frac{3}{11} \, c^{4} d x^{11} e^{2} + \frac{3}{10} \, c^{4} d^{2} x^{10} e + \frac{1}{9} \, c^{4} d^{3} x^{9} + \frac{2}{5} \, a c^{3} x^{10} e^{3} + \frac{4}{3} \, a c^{3} d x^{9} e^{2} + \frac{3}{2} \, a c^{3} d^{2} x^{8} e + \frac{4}{7} \, a c^{3} d^{3} x^{7} + \frac{3}{4} \, a^{2} c^{2} x^{8} e^{3} + \frac{18}{7} \, a^{2} c^{2} d x^{7} e^{2} + 3 \, a^{2} c^{2} d^{2} x^{6} e + \frac{6}{5} \, a^{2} c^{2} d^{3} x^{5} + \frac{2}{3} \, a^{3} c x^{6} e^{3} + \frac{12}{5} \, a^{3} c d x^{5} e^{2} + 3 \, a^{3} c d^{2} x^{4} e + \frac{4}{3} \, a^{3} c d^{3} x^{3} + \frac{1}{4} \, a^{4} x^{4} e^{3} + a^{4} d x^{3} e^{2} + \frac{3}{2} \, a^{4} d^{2} x^{2} e + a^{4} d^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(c*x^2+a)^4,x, algorithm="giac")

[Out]

1/12*c^4*x^12*e^3 + 3/11*c^4*d*x^11*e^2 + 3/10*c^4*d^2*x^10*e + 1/9*c^4*d^3*x^9 + 2/5*a*c^3*x^10*e^3 + 4/3*a*c
^3*d*x^9*e^2 + 3/2*a*c^3*d^2*x^8*e + 4/7*a*c^3*d^3*x^7 + 3/4*a^2*c^2*x^8*e^3 + 18/7*a^2*c^2*d*x^7*e^2 + 3*a^2*
c^2*d^2*x^6*e + 6/5*a^2*c^2*d^3*x^5 + 2/3*a^3*c*x^6*e^3 + 12/5*a^3*c*d*x^5*e^2 + 3*a^3*c*d^2*x^4*e + 4/3*a^3*c
*d^3*x^3 + 1/4*a^4*x^4*e^3 + a^4*d*x^3*e^2 + 3/2*a^4*d^2*x^2*e + a^4*d^3*x