3.19 \(\int (d+e x)^2 (a+c x^2) (A+B x+C x^2) \, dx\)

Optimal. Leaf size=175 \[ \frac {(d+e x)^5 \left (a C e^2+c \left (6 C d^2-e (3 B d-A e)\right )\right )}{5 e^5}-\frac {(d+e x)^4 \left (a e^2 (2 C d-B e)+c d \left (4 C d^2-e (3 B d-2 A e)\right )\right )}{4 e^5}+\frac {(d+e x)^3 \left (a e^2+c d^2\right ) \left (A e^2-B d e+C d^2\right )}{3 e^5}-\frac {c (d+e x)^6 (4 C d-B e)}{6 e^5}+\frac {c C (d+e x)^7}{7 e^5} \]

[Out]

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

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Rubi [A]  time = 0.22, antiderivative size = 173, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {1628} \[ \frac {(d+e x)^5 \left (a C e^2-c e (3 B d-A e)+6 c C d^2\right )}{5 e^5}-\frac {(d+e x)^4 \left (a e^2 (2 C d-B e)-c d e (3 B d-2 A e)+4 c C d^3\right )}{4 e^5}+\frac {(d+e x)^3 \left (a e^2+c d^2\right ) \left (A e^2-B d e+C d^2\right )}{3 e^5}-\frac {c (d+e x)^6 (4 C d-B e)}{6 e^5}+\frac {c C (d+e x)^7}{7 e^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 150, normalized size = 0.86 \[ \frac {1}{5} x^5 \left (a C e^2+A c e^2+2 B c d e+c C d^2\right )+\frac {1}{4} x^4 \left (a B e^2+2 a C d e+2 A c d e+B c d^2\right )+\frac {1}{3} x^3 \left (a A e^2+2 a B d e+a C d^2+A c d^2\right )+\frac {1}{2} a d x^2 (2 A e+B d)+a A d^2 x+\frac {1}{6} c e x^6 (B e+2 C d)+\frac {1}{7} c C e^2 x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.74, size = 171, normalized size = 0.98 \[ \frac {1}{7} x^{7} e^{2} c C + \frac {1}{3} x^{6} e d c C + \frac {1}{6} x^{6} e^{2} c B + \frac {1}{5} x^{5} d^{2} c C + \frac {1}{5} x^{5} e^{2} a C + \frac {2}{5} x^{5} e d c B + \frac {1}{5} x^{5} e^{2} c A + \frac {1}{2} x^{4} e d a C + \frac {1}{4} x^{4} d^{2} c B + \frac {1}{4} x^{4} e^{2} a B + \frac {1}{2} x^{4} e d c A + \frac {1}{3} x^{3} d^{2} a C + \frac {2}{3} x^{3} e d a B + \frac {1}{3} x^{3} d^{2} c A + \frac {1}{3} x^{3} e^{2} a A + \frac {1}{2} x^{2} d^{2} a B + x^{2} e d a A + x d^{2} a A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.15, size = 171, normalized size = 0.98 \[ \frac {1}{7} \, C c x^{7} e^{2} + \frac {1}{3} \, C c d x^{6} e + \frac {1}{5} \, C c d^{2} x^{5} + \frac {1}{6} \, B c x^{6} e^{2} + \frac {2}{5} \, B c d x^{5} e + \frac {1}{4} \, B c d^{2} x^{4} + \frac {1}{5} \, C a x^{5} e^{2} + \frac {1}{5} \, A c x^{5} e^{2} + \frac {1}{2} \, C a d x^{4} e + \frac {1}{2} \, A c d x^{4} e + \frac {1}{3} \, C a d^{2} x^{3} + \frac {1}{3} \, A c d^{2} x^{3} + \frac {1}{4} \, B a x^{4} e^{2} + \frac {2}{3} \, B a d x^{3} e + \frac {1}{2} \, B a d^{2} x^{2} + \frac {1}{3} \, A a x^{3} e^{2} + A a d x^{2} e + A a d^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 148, normalized size = 0.85 \[ \frac {C c \,e^{2} x^{7}}{7}+\frac {\left (c \,e^{2} B +2 d e c C \right ) x^{6}}{6}+A a \,d^{2} x +\frac {\left (A c \,e^{2}+2 B c d e +\left (a \,e^{2}+c \,d^{2}\right ) C \right ) x^{5}}{5}+\frac {\left (2 A c d e +2 C a d e +\left (a \,e^{2}+c \,d^{2}\right ) B \right ) x^{4}}{4}+\frac {\left (2 B a d e +C a \,d^{2}+\left (a \,e^{2}+c \,d^{2}\right ) A \right ) x^{3}}{3}+\frac {\left (2 d e a A +d^{2} a B \right ) x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.45, size = 141, normalized size = 0.81 \[ \frac {1}{7} \, C c e^{2} x^{7} + \frac {1}{6} \, {\left (2 \, C c d e + B c e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (C c d^{2} + 2 \, B c d e + {\left (C a + A c\right )} e^{2}\right )} x^{5} + A a d^{2} x + \frac {1}{4} \, {\left (B c d^{2} + B a e^{2} + 2 \, {\left (C a + A c\right )} d e\right )} x^{4} + \frac {1}{3} \, {\left (2 \, B a d e + A a e^{2} + {\left (C a + A c\right )} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a d^{2} + 2 \, A a d e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.61, size = 143, normalized size = 0.82 \[ x^3\,\left (\frac {A\,a\,e^2}{3}+\frac {A\,c\,d^2}{3}+\frac {C\,a\,d^2}{3}+\frac {2\,B\,a\,d\,e}{3}\right )+x^5\,\left (\frac {A\,c\,e^2}{5}+\frac {C\,a\,e^2}{5}+\frac {C\,c\,d^2}{5}+\frac {2\,B\,c\,d\,e}{5}\right )+x^4\,\left (\frac {B\,a\,e^2}{4}+\frac {B\,c\,d^2}{4}+\frac {A\,c\,d\,e}{2}+\frac {C\,a\,d\,e}{2}\right )+A\,a\,d^2\,x+\frac {a\,d\,x^2\,\left (2\,A\,e+B\,d\right )}{2}+\frac {c\,e\,x^6\,\left (B\,e+2\,C\,d\right )}{6}+\frac {C\,c\,e^2\,x^7}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.10, size = 173, normalized size = 0.99 \[ A a d^{2} x + \frac {C c e^{2} x^{7}}{7} + x^{6} \left (\frac {B c e^{2}}{6} + \frac {C c d e}{3}\right ) + x^{5} \left (\frac {A c e^{2}}{5} + \frac {2 B c d e}{5} + \frac {C a e^{2}}{5} + \frac {C c d^{2}}{5}\right ) + x^{4} \left (\frac {A c d e}{2} + \frac {B a e^{2}}{4} + \frac {B c d^{2}}{4} + \frac {C a d e}{2}\right ) + x^{3} \left (\frac {A a e^{2}}{3} + \frac {A c d^{2}}{3} + \frac {2 B a d e}{3} + \frac {C a d^{2}}{3}\right ) + x^{2} \left (A a d e + \frac {B a d^{2}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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