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

Optimal. Leaf size=57 \[ \frac {(d+e x)^5 \left (a e^2+c d^2\right )}{5 e^3}+\frac {c (d+e x)^7}{7 e^3}-\frac {c d (d+e x)^6}{3 e^3} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {697} \[ \frac {(d+e x)^5 \left (a e^2+c d^2\right )}{5 e^3}+\frac {c (d+e x)^7}{7 e^3}-\frac {c d (d+e x)^6}{3 e^3} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^4*(a + c*x^2),x]

[Out]

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

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

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Mathematica [A]  time = 0.02, size = 101, normalized size = 1.77 \[ \frac {1}{5} e^2 x^5 \left (a e^2+6 c d^2\right )+d e x^4 \left (a e^2+c d^2\right )+\frac {1}{3} d^2 x^3 \left (6 a e^2+c d^2\right )+a d^4 x+2 a d^3 e x^2+\frac {2}{3} c d e^3 x^6+\frac {1}{7} c e^4 x^7 \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^4*(a + c*x^2),x]

[Out]

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

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fricas [A]  time = 0.98, size = 96, normalized size = 1.68 \[ \frac {1}{7} x^{7} e^{4} c + \frac {2}{3} x^{6} e^{3} d c + \frac {6}{5} x^{5} e^{2} d^{2} c + \frac {1}{5} x^{5} e^{4} a + x^{4} e d^{3} c + x^{4} e^{3} d a + \frac {1}{3} x^{3} d^{4} c + 2 x^{3} e^{2} d^{2} a + 2 x^{2} e d^{3} a + x d^{4} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.15, size = 92, normalized size = 1.61 \[ \frac {1}{7} \, c x^{7} e^{4} + \frac {2}{3} \, c d x^{6} e^{3} + \frac {6}{5} \, c d^{2} x^{5} e^{2} + c d^{3} x^{4} e + \frac {1}{3} \, c d^{4} x^{3} + \frac {1}{5} \, a x^{5} e^{4} + a d x^{4} e^{3} + 2 \, a d^{2} x^{3} e^{2} + 2 \, a d^{3} x^{2} e + a d^{4} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 97, normalized size = 1.70 \[ \frac {c \,e^{4} x^{7}}{7}+\frac {2 c d \,e^{3} x^{6}}{3}+2 a \,d^{3} e \,x^{2}+a \,d^{4} x +\frac {\left (e^{4} a +6 d^{2} e^{2} c \right ) x^{5}}{5}+\frac {\left (4 d \,e^{3} a +4 d^{3} e c \right ) x^{4}}{4}+\frac {\left (6 d^{2} e^{2} a +d^{4} c \right ) x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.32, size = 93, normalized size = 1.63 \[ \frac {1}{7} \, c e^{4} x^{7} + \frac {2}{3} \, c d e^{3} x^{6} + 2 \, a d^{3} e x^{2} + a d^{4} x + \frac {1}{5} \, {\left (6 \, c d^{2} e^{2} + a e^{4}\right )} x^{5} + {\left (c d^{3} e + a d e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (c d^{4} + 6 \, a d^{2} e^{2}\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.27, size = 93, normalized size = 1.63 \[ x^3\,\left (\frac {c\,d^4}{3}+2\,a\,d^2\,e^2\right )+x^5\,\left (\frac {6\,c\,d^2\,e^2}{5}+\frac {a\,e^4}{5}\right )+x^4\,\left (c\,d^3\,e+a\,d\,e^3\right )+\frac {c\,e^4\,x^7}{7}+a\,d^4\,x+2\,a\,d^3\,e\,x^2+\frac {2\,c\,d\,e^3\,x^6}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)*(d + e*x)^4,x)

[Out]

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

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sympy [A]  time = 0.09, size = 100, normalized size = 1.75 \[ a d^{4} x + 2 a d^{3} e x^{2} + \frac {2 c d e^{3} x^{6}}{3} + \frac {c e^{4} x^{7}}{7} + x^{5} \left (\frac {a e^{4}}{5} + \frac {6 c d^{2} e^{2}}{5}\right ) + x^{4} \left (a d e^{3} + c d^{3} e\right ) + x^{3} \left (2 a d^{2} e^{2} + \frac {c d^{4}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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