3.5.93 \(\int (d+e x)^2 (a+c x^2)^4 \, dx\) [493]

Optimal. Leaf size=132 \[ a^4 d^2 x+\frac {1}{3} a^3 \left (4 c d^2+a e^2\right ) x^3+\frac {2}{5} a^2 c \left (3 c d^2+2 a e^2\right ) x^5+\frac {2}{7} a c^2 \left (2 c d^2+3 a e^2\right ) x^7+\frac {1}{9} c^3 \left (c d^2+4 a e^2\right ) x^9+\frac {1}{11} c^4 e^2 x^{11}+\frac {d e \left (a+c x^2\right )^5}{5 c} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 132, normalized size of antiderivative = 1.00, 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 = {710, 1824} \begin {gather*} a^4 d^2 x+\frac {1}{3} a^3 x^3 \left (a e^2+4 c d^2\right )+\frac {2}{5} a^2 c x^5 \left (2 a e^2+3 c d^2\right )+\frac {1}{9} c^3 x^9 \left (4 a e^2+c d^2\right )+\frac {2}{7} a c^2 x^7 \left (3 a e^2+2 c d^2\right )+\frac {d e \left (a+c x^2\right )^5}{5 c}+\frac {1}{11} c^4 e^2 x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2*(a + c*x^2)^4,x]

[Out]

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

Rule 710

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

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Mathematica [A]
time = 0.02, size = 148, normalized size = 1.12 \begin {gather*} \frac {2}{15} a^3 c x^3 \left (10 d^2+15 d e x+6 e^2 x^2\right )+\frac {2}{35} a^2 c^2 x^5 \left (21 d^2+35 d e x+15 e^2 x^2\right )+\frac {1}{63} a c^3 x^7 \left (36 d^2+63 d e x+28 e^2 x^2\right )+\frac {1}{495} c^4 x^9 \left (55 d^2+99 d e x+45 e^2 x^2\right )+a^4 \left (d^2 x+d e x^2+\frac {e^2 x^3}{3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^3*c*x^3*(10*d^2 + 15*d*e*x + 6*e^2*x^2))/15 + (2*a^2*c^2*x^5*(21*d^2 + 35*d*e*x + 15*e^2*x^2))/35 + (a*c^
3*x^7*(36*d^2 + 63*d*e*x + 28*e^2*x^2))/63 + (c^4*x^9*(55*d^2 + 99*d*e*x + 45*e^2*x^2))/495 + a^4*(d^2*x + d*e
*x^2 + (e^2*x^3)/3)

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Maple [A]
time = 0.43, size = 170, normalized size = 1.29

method	result	size
norman	\(\frac {c^{4} e^{2} x^{11}}{11}+\frac {d e \,c^{4} x^{10}}{5}+\left (\frac {4}{9} e^{2} c^{3} a +\frac {1}{9} c^{4} d^{2}\right ) x^{9}+d e \,c^{3} a \,x^{8}+\left (\frac {6}{7} a^{2} c^{2} e^{2}+\frac {4}{7} a \,c^{3} d^{2}\right ) x^{7}+2 a^{2} c^{2} d e \,x^{6}+\left (\frac {4}{5} e^{2} c \,a^{3}+\frac {6}{5} a^{2} c^{2} d^{2}\right ) x^{5}+2 d e c \,a^{3} x^{4}+\left (\frac {1}{3} a^{4} e^{2}+\frac {4}{3} a^{3} c \,d^{2}\right ) x^{3}+d e \,a^{4} x^{2}+a^{4} d^{2} x\)	\(168\)
default	\(\frac {c^{4} e^{2} x^{11}}{11}+\frac {d e \,c^{4} x^{10}}{5}+\frac {\left (4 e^{2} c^{3} a +c^{4} d^{2}\right ) x^{9}}{9}+d e \,c^{3} a \,x^{8}+\frac {\left (6 a^{2} c^{2} e^{2}+4 a \,c^{3} d^{2}\right ) x^{7}}{7}+2 a^{2} c^{2} d e \,x^{6}+\frac {\left (4 e^{2} c \,a^{3}+6 a^{2} c^{2} d^{2}\right ) x^{5}}{5}+2 d e c \,a^{3} x^{4}+\frac {\left (a^{4} e^{2}+4 a^{3} c \,d^{2}\right ) x^{3}}{3}+d e \,a^{4} x^{2}+a^{4} d^{2} x\)	\(170\)
gosper	\(\frac {1}{11} c^{4} e^{2} x^{11}+\frac {1}{5} d e \,c^{4} x^{10}+\frac {4}{9} x^{9} e^{2} c^{3} a +\frac {1}{9} x^{9} c^{4} d^{2}+d e \,c^{3} a \,x^{8}+\frac {6}{7} x^{7} a^{2} c^{2} e^{2}+\frac {4}{7} x^{7} a \,c^{3} d^{2}+2 a^{2} c^{2} d e \,x^{6}+\frac {4}{5} x^{5} e^{2} c \,a^{3}+\frac {6}{5} x^{5} a^{2} c^{2} d^{2}+2 d e c \,a^{3} x^{4}+\frac {1}{3} x^{3} a^{4} e^{2}+\frac {4}{3} d^{2} a^{3} c \,x^{3}+d e \,a^{4} x^{2}+a^{4} d^{2} x\)	\(172\)
risch	\(\frac {1}{11} c^{4} e^{2} x^{11}+\frac {1}{5} d e \,c^{4} x^{10}+\frac {4}{9} x^{9} e^{2} c^{3} a +\frac {1}{9} x^{9} c^{4} d^{2}+d e \,c^{3} a \,x^{8}+\frac {6}{7} x^{7} a^{2} c^{2} e^{2}+\frac {4}{7} x^{7} a \,c^{3} d^{2}+2 a^{2} c^{2} d e \,x^{6}+\frac {4}{5} x^{5} e^{2} c \,a^{3}+\frac {6}{5} x^{5} a^{2} c^{2} d^{2}+2 d e c \,a^{3} x^{4}+\frac {1}{3} x^{3} a^{4} e^{2}+\frac {4}{3} d^{2} a^{3} c \,x^{3}+d e \,a^{4} x^{2}+a^{4} d^{2} x\)	\(172\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.31, size = 169, normalized size = 1.28 \begin {gather*} \frac {1}{11} \, c^{4} x^{11} e^{2} + \frac {1}{5} \, c^{4} d x^{10} e + a c^{3} d x^{8} e + 2 \, a^{2} c^{2} d x^{6} e + \frac {1}{9} \, {\left (c^{4} d^{2} + 4 \, a c^{3} e^{2}\right )} x^{9} + 2 \, a^{3} c d x^{4} e + \frac {2}{7} \, {\left (2 \, a c^{3} d^{2} + 3 \, a^{2} c^{2} e^{2}\right )} x^{7} + a^{4} d x^{2} e + a^{4} d^{2} x + \frac {2}{5} \, {\left (3 \, a^{2} c^{2} d^{2} + 2 \, a^{3} c e^{2}\right )} x^{5} + \frac {1}{3} \, {\left (4 \, a^{3} c d^{2} + a^{4} e^{2}\right )} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 2.77, size = 162, normalized size = 1.23 \begin {gather*} \frac {1}{9} \, c^{4} d^{2} x^{9} + \frac {4}{7} \, a c^{3} d^{2} x^{7} + \frac {6}{5} \, a^{2} c^{2} d^{2} x^{5} + \frac {4}{3} \, a^{3} c d^{2} x^{3} + a^{4} d^{2} x + \frac {1}{3465} \, {\left (315 \, c^{4} x^{11} + 1540 \, a c^{3} x^{9} + 2970 \, a^{2} c^{2} x^{7} + 2772 \, a^{3} c x^{5} + 1155 \, a^{4} x^{3}\right )} e^{2} + \frac {1}{5} \, {\left (c^{4} d x^{10} + 5 \, a c^{3} d x^{8} + 10 \, a^{2} c^{2} d x^{6} + 10 \, a^{3} c d x^{4} + 5 \, a^{4} d x^{2}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*c^4*d^2*x^9 + 4/7*a*c^3*d^2*x^7 + 6/5*a^2*c^2*d^2*x^5 + 4/3*a^3*c*d^2*x^3 + a^4*d^2*x + 1/3465*(315*c^4*x^
11 + 1540*a*c^3*x^9 + 2970*a^2*c^2*x^7 + 2772*a^3*c*x^5 + 1155*a^4*x^3)*e^2 + 1/5*(c^4*d*x^10 + 5*a*c^3*d*x^8
+ 10*a^2*c^2*d*x^6 + 10*a^3*c*d*x^4 + 5*a^4*d*x^2)*e

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Sympy [A]
time = 0.02, size = 187, normalized size = 1.42 \begin {gather*} a^{4} d^{2} x + a^{4} d e x^{2} + 2 a^{3} c d e x^{4} + 2 a^{2} c^{2} d e x^{6} + a c^{3} d e x^{8} + \frac {c^{4} d e x^{10}}{5} + \frac {c^{4} e^{2} x^{11}}{11} + x^{9} \cdot \left (\frac {4 a c^{3} e^{2}}{9} + \frac {c^{4} d^{2}}{9}\right ) + x^{7} \cdot \left (\frac {6 a^{2} c^{2} e^{2}}{7} + \frac {4 a c^{3} d^{2}}{7}\right ) + x^{5} \cdot \left (\frac {4 a^{3} c e^{2}}{5} + \frac {6 a^{2} c^{2} d^{2}}{5}\right ) + x^{3} \left (\frac {a^{4} e^{2}}{3} + \frac {4 a^{3} c d^{2}}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.28, size = 171, normalized size = 1.30 \begin {gather*} \frac {1}{11} \, c^{4} x^{11} e^{2} + \frac {1}{5} \, c^{4} d x^{10} e + \frac {1}{9} \, c^{4} d^{2} x^{9} + \frac {4}{9} \, a c^{3} x^{9} e^{2} + a c^{3} d x^{8} e + \frac {4}{7} \, a c^{3} d^{2} x^{7} + \frac {6}{7} \, a^{2} c^{2} x^{7} e^{2} + 2 \, a^{2} c^{2} d x^{6} e + \frac {6}{5} \, a^{2} c^{2} d^{2} x^{5} + \frac {4}{5} \, a^{3} c x^{5} e^{2} + 2 \, a^{3} c d x^{4} e + \frac {4}{3} \, a^{3} c d^{2} x^{3} + \frac {1}{3} \, a^{4} x^{3} e^{2} + a^{4} d x^{2} e + a^{4} d^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.07, size = 161, normalized size = 1.22 \begin {gather*} x^3\,\left (\frac {a^4\,e^2}{3}+\frac {4\,c\,a^3\,d^2}{3}\right )+x^9\,\left (\frac {c^4\,d^2}{9}+\frac {4\,a\,c^3\,e^2}{9}\right )+a^4\,d^2\,x+\frac {c^4\,e^2\,x^{11}}{11}+a^4\,d\,e\,x^2+\frac {c^4\,d\,e\,x^{10}}{5}+\frac {2\,a^2\,c\,x^5\,\left (3\,c\,d^2+2\,a\,e^2\right )}{5}+\frac {2\,a\,c^2\,x^7\,\left (2\,c\,d^2+3\,a\,e^2\right )}{7}+2\,a^3\,c\,d\,e\,x^4+a\,c^3\,d\,e\,x^8+2\,a^2\,c^2\,d\,e\,x^6 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^4*(d + e*x)^2,x)

[Out]

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

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