3.1.27 \(\int (d+e x) (a+c x^2)^2 (A+B x+C x^2) \, dx\) [27]

Optimal. Leaf size=128 \[ a^2 A d x+\frac {1}{3} a (2 A c d+a C d+a B e) x^3+\frac {1}{4} a^2 C e x^4+\frac {1}{5} c (A c d+2 a (C d+B e)) x^5+\frac {1}{3} a c C e x^6+\frac {1}{7} c^2 (C d+B e) x^7+\frac {1}{8} c^2 C e x^8+\frac {(B d+A e) \left (a+c x^2\right )^3}{6 c} \]

[Out]

a^2*A*d*x+1/3*a*(2*A*c*d+B*a*e+C*a*d)*x^3+1/4*a^2*C*e*x^4+1/5*c*(A*c*d+2*a*(B*e+C*d))*x^5+1/3*a*c*C*e*x^6+1/7*
c^2*(B*e+C*d)*x^7+1/8*c^2*C*e*x^8+1/6*(A*e+B*d)*(c*x^2+a)^3/c

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Rubi [A]
time = 0.10, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1596, 1824} \begin {gather*} a^2 A d x+\frac {1}{4} a^2 C e x^4+\frac {1}{5} c x^5 (2 a (B e+C d)+A c d)+\frac {1}{3} a x^3 (a B e+a C d+2 A c d)+\frac {\left (a+c x^2\right )^3 (A e+B d)}{6 c}+\frac {1}{3} a c C e x^6+\frac {1}{7} c^2 x^7 (B e+C d)+\frac {1}{8} c^2 C e x^8 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)*(a + c*x^2)^2*(A + B*x + C*x^2),x]

[Out]

a^2*A*d*x + (a*(2*A*c*d + a*C*d + a*B*e)*x^3)/3 + (a^2*C*e*x^4)/4 + (c*(A*c*d + 2*a*(C*d + B*e))*x^5)/5 + (a*c
*C*e*x^6)/3 + (c^2*(C*d + B*e)*x^7)/7 + (c^2*C*e*x^8)/8 + ((B*d + A*e)*(a + c*x^2)^3)/(6*c)

Rule 1596

Int[(Px_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[Coeff[Px, x, n - 1]*((a + b*x^n)^(p + 1)/(b*n*(p +
1))), x] + Int[(Px - Coeff[Px, x, n - 1]*x^(n - 1))*(a + b*x^n)^p, x] /; FreeQ[{a, b}, x] && PolyQ[Px, x] && I
GtQ[p, 1] && IGtQ[n, 1] && NeQ[Coeff[Px, x, n - 1], 0] && NeQ[Px, Coeff[Px, x, n - 1]*x^(n - 1)] &&  !MatchQ[P
x, (Qx_.)*((c_) + (d_.)*x^(m_))^(q_) /; FreeQ[{c, d}, x] && PolyQ[Qx, x] && IGtQ[q, 1] && IGtQ[m, 1] && NeQ[Co
eff[Qx*(a + b*x^n)^p, x, m - 1], 0] && GtQ[m*q, n*p]]

Rule 1824

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

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Mathematica [A]
time = 0.04, size = 144, normalized size = 1.12 \begin {gather*} a^2 A d x+\frac {1}{2} a^2 (B d+A e) x^2+\frac {1}{3} a (2 A c d+a C d+a B e) x^3+\frac {1}{4} a (2 B c d+2 A c e+a C e) x^4+\frac {1}{5} c (A c d+2 a C d+2 a B e) x^5+\frac {1}{6} c (B c d+A c e+2 a C e) x^6+\frac {1}{7} c^2 (C d+B e) x^7+\frac {1}{8} c^2 C e x^8 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)*(a + c*x^2)^2*(A + B*x + C*x^2),x]

[Out]

a^2*A*d*x + (a^2*(B*d + A*e)*x^2)/2 + (a*(2*A*c*d + a*C*d + a*B*e)*x^3)/3 + (a*(2*B*c*d + 2*A*c*e + a*C*e)*x^4
)/4 + (c*(A*c*d + 2*a*C*d + 2*a*B*e)*x^5)/5 + (c*(B*c*d + A*c*e + 2*a*C*e)*x^6)/6 + (c^2*(C*d + B*e)*x^7)/7 +
(c^2*C*e*x^8)/8

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Maple [A]
time = 0.09, size = 151, normalized size = 1.18

method	result	size
default	\(\frac {c^{2} C e \,x^{8}}{8}+\frac {\left (c^{2} e B +c^{2} d C \right ) x^{7}}{7}+\frac {\left (c^{2} e A +c^{2} d B +2 a c e C \right ) x^{6}}{6}+\frac {\left (A \,c^{2} d +2 B a c e +2 a c d C \right ) x^{5}}{5}+\frac {\left (2 a c e A +2 a c d B +e \,a^{2} C \right ) x^{4}}{4}+\frac {\left (2 a c d A +e \,a^{2} B +d \,a^{2} C \right ) x^{3}}{3}+\frac {\left (e \,a^{2} A +d \,a^{2} B \right ) x^{2}}{2}+a^{2} A d x\)	\(151\)
norman	\(\frac {c^{2} C e \,x^{8}}{8}+\left (\frac {1}{7} c^{2} e B +\frac {1}{7} c^{2} d C \right ) x^{7}+\left (\frac {1}{6} c^{2} e A +\frac {1}{6} c^{2} d B +\frac {1}{3} a c e C \right ) x^{6}+\left (\frac {1}{5} A \,c^{2} d +\frac {2}{5} B a c e +\frac {2}{5} a c d C \right ) x^{5}+\left (\frac {1}{2} a c e A +\frac {1}{2} a c d B +\frac {1}{4} e \,a^{2} C \right ) x^{4}+\left (\frac {2}{3} a c d A +\frac {1}{3} e \,a^{2} B +\frac {1}{3} d \,a^{2} C \right ) x^{3}+\left (\frac {1}{2} e \,a^{2} A +\frac {1}{2} d \,a^{2} B \right ) x^{2}+a^{2} A d x\)	\(155\)
gosper	\(\frac {1}{8} c^{2} C e \,x^{8}+\frac {1}{7} x^{7} c^{2} e B +\frac {1}{7} x^{7} c^{2} d C +\frac {1}{6} x^{6} c^{2} e A +\frac {1}{6} x^{6} c^{2} d B +\frac {1}{3} a c C e \,x^{6}+\frac {1}{5} x^{5} A \,c^{2} d +\frac {2}{5} x^{5} B a c e +\frac {2}{5} x^{5} a c d C +\frac {1}{2} x^{4} a c e A +\frac {1}{2} x^{4} a c d B +\frac {1}{4} a^{2} C e \,x^{4}+\frac {2}{3} x^{3} a c d A +\frac {1}{3} x^{3} e \,a^{2} B +\frac {1}{3} x^{3} d \,a^{2} C +\frac {1}{2} x^{2} e \,a^{2} A +\frac {1}{2} x^{2} d \,a^{2} B +a^{2} A d x\)	\(173\)
risch	\(\frac {1}{8} c^{2} C e \,x^{8}+\frac {1}{7} x^{7} c^{2} e B +\frac {1}{7} x^{7} c^{2} d C +\frac {1}{6} x^{6} c^{2} e A +\frac {1}{6} x^{6} c^{2} d B +\frac {1}{3} a c C e \,x^{6}+\frac {1}{5} x^{5} A \,c^{2} d +\frac {2}{5} x^{5} B a c e +\frac {2}{5} x^{5} a c d C +\frac {1}{2} x^{4} a c e A +\frac {1}{2} x^{4} a c d B +\frac {1}{4} a^{2} C e \,x^{4}+\frac {2}{3} x^{3} a c d A +\frac {1}{3} x^{3} e \,a^{2} B +\frac {1}{3} x^{3} d \,a^{2} C +\frac {1}{2} x^{2} e \,a^{2} A +\frac {1}{2} x^{2} d \,a^{2} B +a^{2} A d x\)	\(173\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(c*x^2+a)^2*(C*x^2+B*x+A),x,method=_RETURNVERBOSE)

[Out]

1/8*c^2*C*e*x^8+1/7*(B*c^2*e+C*c^2*d)*x^7+1/6*(A*c^2*e+B*c^2*d+2*C*a*c*e)*x^6+1/5*(A*c^2*d+2*B*a*c*e+2*C*a*c*d
)*x^5+1/4*(2*A*a*c*e+2*B*a*c*d+C*a^2*e)*x^4+1/3*(2*A*a*c*d+B*a^2*e+C*a^2*d)*x^3+1/2*(A*a^2*e+B*a^2*d)*x^2+a^2*
A*d*x

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Maxima [A]
time = 0.27, size = 161, normalized size = 1.26 \begin {gather*} \frac {1}{8} \, C c^{2} x^{8} e + \frac {1}{7} \, {\left (C c^{2} d + B c^{2} e\right )} x^{7} + \frac {1}{6} \, {\left (B c^{2} d + 2 \, C a c e + A c^{2} e\right )} x^{6} + \frac {1}{5} \, {\left (2 \, B a c e + {\left (2 \, C a c + A c^{2}\right )} d\right )} x^{5} + A a^{2} d x + \frac {1}{4} \, {\left (2 \, B a c d + C a^{2} e + 2 \, A a c e\right )} x^{4} + \frac {1}{3} \, {\left (B a^{2} e + {\left (C a^{2} + 2 \, A a c\right )} d\right )} x^{3} + \frac {1}{2} \, {\left (B a^{2} d + A a^{2} e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(c*x^2+a)^2*(C*x^2+B*x+A),x, algorithm="maxima")

[Out]

1/8*C*c^2*x^8*e + 1/7*(C*c^2*d + B*c^2*e)*x^7 + 1/6*(B*c^2*d + 2*C*a*c*e + A*c^2*e)*x^6 + 1/5*(2*B*a*c*e + (2*
C*a*c + A*c^2)*d)*x^5 + A*a^2*d*x + 1/4*(2*B*a*c*d + C*a^2*e + 2*A*a*c*e)*x^4 + 1/3*(B*a^2*e + (C*a^2 + 2*A*a*
c)*d)*x^3 + 1/2*(B*a^2*d + A*a^2*e)*x^2

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Fricas [A]
time = 0.36, size = 162, normalized size = 1.27 \begin {gather*} \frac {1}{7} \, C c^{2} d x^{7} + \frac {1}{6} \, B c^{2} d x^{6} + \frac {1}{2} \, B a c d x^{4} + \frac {1}{5} \, {\left (2 \, C a c + A c^{2}\right )} d x^{5} + \frac {1}{2} \, B a^{2} d x^{2} + A a^{2} d x + \frac {1}{3} \, {\left (C a^{2} + 2 \, A a c\right )} d x^{3} + \frac {1}{840} \, {\left (105 \, C c^{2} x^{8} + 120 \, B c^{2} x^{7} + 336 \, B a c x^{5} + 140 \, {\left (2 \, C a c + A c^{2}\right )} x^{6} + 280 \, B a^{2} x^{3} + 420 \, A a^{2} x^{2} + 210 \, {\left (C a^{2} + 2 \, A a c\right )} x^{4}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(c*x^2+a)^2*(C*x^2+B*x+A),x, algorithm="fricas")

[Out]

1/7*C*c^2*d*x^7 + 1/6*B*c^2*d*x^6 + 1/2*B*a*c*d*x^4 + 1/5*(2*C*a*c + A*c^2)*d*x^5 + 1/2*B*a^2*d*x^2 + A*a^2*d*
x + 1/3*(C*a^2 + 2*A*a*c)*d*x^3 + 1/840*(105*C*c^2*x^8 + 120*B*c^2*x^7 + 336*B*a*c*x^5 + 140*(2*C*a*c + A*c^2)
*x^6 + 280*B*a^2*x^3 + 420*A*a^2*x^2 + 210*(C*a^2 + 2*A*a*c)*x^4)*e

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Sympy [A]
time = 0.02, size = 180, normalized size = 1.41 \begin {gather*} A a^{2} d x + \frac {C c^{2} e x^{8}}{8} + x^{7} \left (\frac {B c^{2} e}{7} + \frac {C c^{2} d}{7}\right ) + x^{6} \left (\frac {A c^{2} e}{6} + \frac {B c^{2} d}{6} + \frac {C a c e}{3}\right ) + x^{5} \left (\frac {A c^{2} d}{5} + \frac {2 B a c e}{5} + \frac {2 C a c d}{5}\right ) + x^{4} \left (\frac {A a c e}{2} + \frac {B a c d}{2} + \frac {C a^{2} e}{4}\right ) + x^{3} \cdot \left (\frac {2 A a c d}{3} + \frac {B a^{2} e}{3} + \frac {C a^{2} d}{3}\right ) + x^{2} \left (\frac {A a^{2} e}{2} + \frac {B a^{2} d}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(c*x**2+a)**2*(C*x**2+B*x+A),x)

[Out]

A*a**2*d*x + C*c**2*e*x**8/8 + x**7*(B*c**2*e/7 + C*c**2*d/7) + x**6*(A*c**2*e/6 + B*c**2*d/6 + C*a*c*e/3) + x
**5*(A*c**2*d/5 + 2*B*a*c*e/5 + 2*C*a*c*d/5) + x**4*(A*a*c*e/2 + B*a*c*d/2 + C*a**2*e/4) + x**3*(2*A*a*c*d/3 +
B*a**2*e/3 + C*a**2*d/3) + x**2*(A*a**2*e/2 + B*a**2*d/2)

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Giac [A]
time = 4.67, size = 181, normalized size = 1.41 \begin {gather*} \frac {1}{8} \, C c^{2} x^{8} e + \frac {1}{7} \, C c^{2} d x^{7} + \frac {1}{7} \, B c^{2} x^{7} e + \frac {1}{6} \, B c^{2} d x^{6} + \frac {1}{3} \, C a c x^{6} e + \frac {1}{6} \, A c^{2} x^{6} e + \frac {2}{5} \, C a c d x^{5} + \frac {1}{5} \, A c^{2} d x^{5} + \frac {2}{5} \, B a c x^{5} e + \frac {1}{2} \, B a c d x^{4} + \frac {1}{4} \, C a^{2} x^{4} e + \frac {1}{2} \, A a c x^{4} e + \frac {1}{3} \, C a^{2} d x^{3} + \frac {2}{3} \, A a c d x^{3} + \frac {1}{3} \, B a^{2} x^{3} e + \frac {1}{2} \, B a^{2} d x^{2} + \frac {1}{2} \, A a^{2} x^{2} e + A a^{2} d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(c*x^2+a)^2*(C*x^2+B*x+A),x, algorithm="giac")

[Out]

1/8*C*c^2*x^8*e + 1/7*C*c^2*d*x^7 + 1/7*B*c^2*x^7*e + 1/6*B*c^2*d*x^6 + 1/3*C*a*c*x^6*e + 1/6*A*c^2*x^6*e + 2/
5*C*a*c*d*x^5 + 1/5*A*c^2*d*x^5 + 2/5*B*a*c*x^5*e + 1/2*B*a*c*d*x^4 + 1/4*C*a^2*x^4*e + 1/2*A*a*c*x^4*e + 1/3*
C*a^2*d*x^3 + 2/3*A*a*c*d*x^3 + 1/3*B*a^2*x^3*e + 1/2*B*a^2*d*x^2 + 1/2*A*a^2*x^2*e + A*a^2*d*x

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Mupad [B]
time = 3.69, size = 140, normalized size = 1.09 \begin {gather*} x^3\,\left (\frac {B\,a^2\,e}{3}+\frac {C\,a^2\,d}{3}+\frac {2\,A\,a\,c\,d}{3}\right )+x^6\,\left (\frac {A\,c^2\,e}{6}+\frac {B\,c^2\,d}{6}+\frac {C\,a\,c\,e}{3}\right )+\frac {c\,x^5\,\left (A\,c\,d+2\,B\,a\,e+2\,C\,a\,d\right )}{5}+\frac {a\,x^4\,\left (2\,A\,c\,e+2\,B\,c\,d+C\,a\,e\right )}{4}+\frac {a^2\,x^2\,\left (A\,e+B\,d\right )}{2}+\frac {c^2\,x^7\,\left (B\,e+C\,d\right )}{7}+A\,a^2\,d\,x+\frac {C\,c^2\,e\,x^8}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^2*(d + e*x)*(A + B*x + C*x^2),x)

[Out]

x^3*((B*a^2*e)/3 + (C*a^2*d)/3 + (2*A*a*c*d)/3) + x^6*((A*c^2*e)/6 + (B*c^2*d)/6 + (C*a*c*e)/3) + (c*x^5*(A*c*
d + 2*B*a*e + 2*C*a*d))/5 + (a*x^4*(2*A*c*e + 2*B*c*d + C*a*e))/4 + (a^2*x^2*(A*e + B*d))/2 + (c^2*x^7*(B*e +
C*d))/7 + A*a^2*d*x + (C*c^2*e*x^8)/8

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