\(\int (d+e x)^4 (a+c x^2) \, dx\) [451]

Optimal result

Integrand size = 15, antiderivative size = 57 \[ \int (d+e x)^4 \left (a+c x^2\right ) \, dx=\frac {\left (c d^2+a e^2\right ) (d+e x)^5}{5 e^3}-\frac {c d (d+e x)^6}{3 e^3}+\frac {c (d+e x)^7}{7 e^3} \]

[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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {711} \[ \int (d+e x)^4 \left (a+c x^2\right ) \, dx=\frac {(d+e x)^5 \left (a e^2+c d^2\right )}{5 e^3}+\frac {c (d+e x)^7}{7 e^3}-\frac {c d (d+e x)^6}{3 e^3} \]

[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 711

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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.77 \[ \int (d+e x)^4 \left (a+c x^2\right ) \, dx=a d^4 x+2 a d^3 e x^2+\frac {1}{3} d^2 \left (c d^2+6 a e^2\right ) x^3+d e \left (c d^2+a e^2\right ) x^4+\frac {1}{5} e^2 \left (6 c d^2+a e^2\right ) x^5+\frac {2}{3} c d e^3 x^6+\frac {1}{7} c e^4 x^7 \]

[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

Maple [A] (verified)

Time = 2.16 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.65

method	result	size
norman	\(\frac {e^{4} c \,x^{7}}{7}+\frac {2 d \,e^{3} c \,x^{6}}{3}+\left (\frac {1}{5} e^{4} a +\frac {6}{5} d^{2} e^{2} c \right ) x^{5}+\left (d \,e^{3} a +d^{3} e c \right ) x^{4}+\left (2 a \,d^{2} e^{2}+\frac {1}{3} c \,d^{4}\right ) x^{3}+2 a \,d^{3} e \,x^{2}+a \,d^{4} x\)	\(94\)
gosper	\(\frac {1}{7} e^{4} c \,x^{7}+\frac {2}{3} d \,e^{3} c \,x^{6}+\frac {1}{5} x^{5} e^{4} a +\frac {6}{5} x^{5} d^{2} e^{2} c +a d \,e^{3} x^{4}+c \,d^{3} e \,x^{4}+2 x^{3} a \,d^{2} e^{2}+\frac {1}{3} c \,d^{4} x^{3}+2 a \,d^{3} e \,x^{2}+a \,d^{4} x\)	\(97\)
default	\(\frac {e^{4} c \,x^{7}}{7}+\frac {2 d \,e^{3} c \,x^{6}}{3}+\frac {\left (e^{4} a +6 d^{2} e^{2} c \right ) x^{5}}{5}+\frac {\left (4 d \,e^{3} a +4 d^{3} e c \right ) x^{4}}{4}+\frac {\left (6 a \,d^{2} e^{2}+c \,d^{4}\right ) x^{3}}{3}+2 a \,d^{3} e \,x^{2}+a \,d^{4} x\)	\(97\)
risch	\(\frac {1}{7} e^{4} c \,x^{7}+\frac {2}{3} d \,e^{3} c \,x^{6}+\frac {1}{5} x^{5} e^{4} a +\frac {6}{5} x^{5} d^{2} e^{2} c +a d \,e^{3} x^{4}+c \,d^{3} e \,x^{4}+2 x^{3} a \,d^{2} e^{2}+\frac {1}{3} c \,d^{4} x^{3}+2 a \,d^{3} e \,x^{2}+a \,d^{4} x\)	\(97\)
parallelrisch	\(\frac {1}{7} e^{4} c \,x^{7}+\frac {2}{3} d \,e^{3} c \,x^{6}+\frac {1}{5} x^{5} e^{4} a +\frac {6}{5} x^{5} d^{2} e^{2} c +a d \,e^{3} x^{4}+c \,d^{3} e \,x^{4}+2 x^{3} a \,d^{2} e^{2}+\frac {1}{3} c \,d^{4} x^{3}+2 a \,d^{3} e \,x^{2}+a \,d^{4} x\)	\(97\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.63 \[ \int (d+e x)^4 \left (a+c x^2\right ) \, dx=\frac {1}{7} \, c e^{4} x^{7} + \frac {2}{3} \, c d e^{3} x^{6} + 2 \, a d^{3} e x^{2} + a d^{4} x + \frac {1}{5} \, {\left (6 \, c d^{2} e^{2} + a e^{4}\right )} x^{5} + {\left (c d^{3} e + a d e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (c d^{4} + 6 \, a d^{2} e^{2}\right )} x^{3} \]

[In]

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

[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

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.75 \[ \int (d+e x)^4 \left (a+c x^2\right ) \, dx=a d^{4} x + 2 a d^{3} e x^{2} + \frac {2 c d e^{3} x^{6}}{3} + \frac {c e^{4} x^{7}}{7} + x^{5} \left (\frac {a e^{4}}{5} + \frac {6 c d^{2} e^{2}}{5}\right ) + x^{4} \left (a d e^{3} + c d^{3} e\right ) + x^{3} \cdot \left (2 a d^{2} e^{2} + \frac {c d^{4}}{3}\right ) \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.63 \[ \int (d+e x)^4 \left (a+c x^2\right ) \, dx=\frac {1}{7} \, c e^{4} x^{7} + \frac {2}{3} \, c d e^{3} x^{6} + 2 \, a d^{3} e x^{2} + a d^{4} x + \frac {1}{5} \, {\left (6 \, c d^{2} e^{2} + a e^{4}\right )} x^{5} + {\left (c d^{3} e + a d e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (c d^{4} + 6 \, a d^{2} e^{2}\right )} x^{3} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.68 \[ \int (d+e x)^4 \left (a+c x^2\right ) \, dx=\frac {1}{7} \, c e^{4} x^{7} + \frac {2}{3} \, c d e^{3} x^{6} + \frac {6}{5} \, c d^{2} e^{2} x^{5} + \frac {1}{5} \, a e^{4} x^{5} + c d^{3} e x^{4} + a d e^{3} x^{4} + \frac {1}{3} \, c d^{4} x^{3} + 2 \, a d^{2} e^{2} x^{3} + 2 \, a d^{3} e x^{2} + a d^{4} x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.63 \[ \int (d+e x)^4 \left (a+c x^2\right ) \, dx=x^3\,\left (\frac {c\,d^4}{3}+2\,a\,d^2\,e^2\right )+x^5\,\left (\frac {6\,c\,d^2\,e^2}{5}+\frac {a\,e^4}{5}\right )+x^4\,\left (c\,d^3\,e+a\,d\,e^3\right )+\frac {c\,e^4\,x^7}{7}+a\,d^4\,x+2\,a\,d^3\,e\,x^2+\frac {2\,c\,d\,e^3\,x^6}{3} \]

[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