3.2.28 \(\int (d+e x^2)^3 (a+c x^4)^2 \, dx\) [128]

Optimal. Leaf size=133 \[ a^2 d^3 x+a^2 d^2 e x^3+\frac {1}{5} a d \left (2 c d^2+3 a e^2\right ) x^5+\frac {1}{7} a e \left (6 c d^2+a e^2\right ) x^7+\frac {1}{9} c d \left (c d^2+6 a e^2\right ) x^9+\frac {1}{11} c e \left (3 c d^2+2 a e^2\right ) x^{11}+\frac {3}{13} c^2 d e^2 x^{13}+\frac {1}{15} c^2 e^3 x^{15} \]

[Out]

a^2*d^3*x+a^2*d^2*e*x^3+1/5*a*d*(3*a*e^2+2*c*d^2)*x^5+1/7*a*e*(a*e^2+6*c*d^2)*x^7+1/9*c*d*(6*a*e^2+c*d^2)*x^9+
1/11*c*e*(2*a*e^2+3*c*d^2)*x^11+3/13*c^2*d*e^2*x^13+1/15*c^2*e^3*x^15

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Rubi [A]
time = 0.07, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {1168} \begin {gather*} a^2 d^3 x+a^2 d^2 e x^3+\frac {1}{11} c e x^{11} \left (2 a e^2+3 c d^2\right )+\frac {1}{9} c d x^9 \left (6 a e^2+c d^2\right )+\frac {1}{7} a e x^7 \left (a e^2+6 c d^2\right )+\frac {1}{5} a d x^5 \left (3 a e^2+2 c d^2\right )+\frac {3}{13} c^2 d e^2 x^{13}+\frac {1}{15} c^2 e^3 x^{15} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)^3*(a + c*x^4)^2,x]

[Out]

a^2*d^3*x + a^2*d^2*e*x^3 + (a*d*(2*c*d^2 + 3*a*e^2)*x^5)/5 + (a*e*(6*c*d^2 + a*e^2)*x^7)/7 + (c*d*(c*d^2 + 6*
a*e^2)*x^9)/9 + (c*e*(3*c*d^2 + 2*a*e^2)*x^11)/11 + (3*c^2*d*e^2*x^13)/13 + (c^2*e^3*x^15)/15

Rule 1168

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a
+ c*x^4)^p, x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rubi steps

\begin {align*} \int \left (d+e x^2\right )^3 \left (a+c x^4\right )^2 \, dx &=\int \left (a^2 d^3+3 a^2 d^2 e x^2+a d \left (2 c d^2+3 a e^2\right ) x^4+a e \left (6 c d^2+a e^2\right ) x^6+c d \left (c d^2+6 a e^2\right ) x^8+c e \left (3 c d^2+2 a e^2\right ) x^{10}+3 c^2 d e^2 x^{12}+c^2 e^3 x^{14}\right ) \, dx\\ &=a^2 d^3 x+a^2 d^2 e x^3+\frac {1}{5} a d \left (2 c d^2+3 a e^2\right ) x^5+\frac {1}{7} a e \left (6 c d^2+a e^2\right ) x^7+\frac {1}{9} c d \left (c d^2+6 a e^2\right ) x^9+\frac {1}{11} c e \left (3 c d^2+2 a e^2\right ) x^{11}+\frac {3}{13} c^2 d e^2 x^{13}+\frac {1}{15} c^2 e^3 x^{15}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 133, normalized size = 1.00 \begin {gather*} a^2 d^3 x+a^2 d^2 e x^3+\frac {1}{5} a d \left (2 c d^2+3 a e^2\right ) x^5+\frac {1}{7} a e \left (6 c d^2+a e^2\right ) x^7+\frac {1}{9} c d \left (c d^2+6 a e^2\right ) x^9+\frac {1}{11} c e \left (3 c d^2+2 a e^2\right ) x^{11}+\frac {3}{13} c^2 d e^2 x^{13}+\frac {1}{15} c^2 e^3 x^{15} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^2)^3*(a + c*x^4)^2,x]

[Out]

a^2*d^3*x + a^2*d^2*e*x^3 + (a*d*(2*c*d^2 + 3*a*e^2)*x^5)/5 + (a*e*(6*c*d^2 + a*e^2)*x^7)/7 + (c*d*(c*d^2 + 6*
a*e^2)*x^9)/9 + (c*e*(3*c*d^2 + 2*a*e^2)*x^11)/11 + (3*c^2*d*e^2*x^13)/13 + (c^2*e^3*x^15)/15

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Maple [A]
time = 0.16, size = 130, normalized size = 0.98

method	result	size
norman	\(a^{2} d^{3} x +a^{2} d^{2} e \,x^{3}+\left (\frac {3}{5} d \,e^{2} a^{2}+\frac {2}{5} d^{3} a c \right ) x^{5}+\left (\frac {1}{7} e^{3} a^{2}+\frac {6}{7} d^{2} e a c \right ) x^{7}+\left (\frac {2}{3} a c d \,e^{2}+\frac {1}{9} c^{2} d^{3}\right ) x^{9}+\left (\frac {2}{11} e^{3} a c +\frac {3}{11} d^{2} e \,c^{2}\right ) x^{11}+\frac {3 c^{2} d \,e^{2} x^{13}}{13}+\frac {c^{2} e^{3} x^{15}}{15}\)	\(128\)
default	\(\frac {c^{2} e^{3} x^{15}}{15}+\frac {3 c^{2} d \,e^{2} x^{13}}{13}+\frac {\left (2 e^{3} a c +3 d^{2} e \,c^{2}\right ) x^{11}}{11}+\frac {\left (6 a c d \,e^{2}+c^{2} d^{3}\right ) x^{9}}{9}+\frac {\left (e^{3} a^{2}+6 d^{2} e a c \right ) x^{7}}{7}+\frac {\left (3 d \,e^{2} a^{2}+2 d^{3} a c \right ) x^{5}}{5}+a^{2} d^{2} e \,x^{3}+a^{2} d^{3} x\)	\(130\)
gosper	\(a^{2} d^{3} x +a^{2} d^{2} e \,x^{3}+\frac {3}{5} x^{5} d \,e^{2} a^{2}+\frac {2}{5} x^{5} d^{3} a c +\frac {1}{7} x^{7} e^{3} a^{2}+\frac {6}{7} x^{7} d^{2} e a c +\frac {2}{3} x^{9} a c d \,e^{2}+\frac {1}{9} x^{9} c^{2} d^{3}+\frac {2}{11} x^{11} e^{3} a c +\frac {3}{11} x^{11} d^{2} e \,c^{2}+\frac {3}{13} c^{2} d \,e^{2} x^{13}+\frac {1}{15} c^{2} e^{3} x^{15}\)	\(132\)
risch	\(a^{2} d^{3} x +a^{2} d^{2} e \,x^{3}+\frac {3}{5} x^{5} d \,e^{2} a^{2}+\frac {2}{5} x^{5} d^{3} a c +\frac {1}{7} x^{7} e^{3} a^{2}+\frac {6}{7} x^{7} d^{2} e a c +\frac {2}{3} x^{9} a c d \,e^{2}+\frac {1}{9} x^{9} c^{2} d^{3}+\frac {2}{11} x^{11} e^{3} a c +\frac {3}{11} x^{11} d^{2} e \,c^{2}+\frac {3}{13} c^{2} d \,e^{2} x^{13}+\frac {1}{15} c^{2} e^{3} x^{15}\)	\(132\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^3*(c*x^4+a)^2,x,method=_RETURNVERBOSE)

[Out]

1/15*c^2*e^3*x^15+3/13*c^2*d*e^2*x^13+1/11*(2*a*c*e^3+3*c^2*d^2*e)*x^11+1/9*(6*a*c*d*e^2+c^2*d^3)*x^9+1/7*(a^2
*e^3+6*a*c*d^2*e)*x^7+1/5*(3*a^2*d*e^2+2*a*c*d^3)*x^5+a^2*d^2*e*x^3+a^2*d^3*x

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Maxima [A]
time = 0.29, size = 126, normalized size = 0.95 \begin {gather*} \frac {1}{15} \, c^{2} x^{15} e^{3} + \frac {3}{13} \, c^{2} d x^{13} e^{2} + \frac {1}{11} \, {\left (3 \, c^{2} d^{2} e + 2 \, a c e^{3}\right )} x^{11} + \frac {1}{9} \, {\left (c^{2} d^{3} + 6 \, a c d e^{2}\right )} x^{9} + \frac {1}{7} \, {\left (6 \, a c d^{2} e + a^{2} e^{3}\right )} x^{7} + a^{2} d^{2} x^{3} e + a^{2} d^{3} x + \frac {1}{5} \, {\left (2 \, a c d^{3} + 3 \, a^{2} d e^{2}\right )} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^3*(c*x^4+a)^2,x, algorithm="maxima")

[Out]

1/15*c^2*x^15*e^3 + 3/13*c^2*d*x^13*e^2 + 1/11*(3*c^2*d^2*e + 2*a*c*e^3)*x^11 + 1/9*(c^2*d^3 + 6*a*c*d*e^2)*x^
9 + 1/7*(6*a*c*d^2*e + a^2*e^3)*x^7 + a^2*d^2*x^3*e + a^2*d^3*x + 1/5*(2*a*c*d^3 + 3*a^2*d*e^2)*x^5

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Fricas [A]
time = 0.38, size = 126, normalized size = 0.95 \begin {gather*} \frac {1}{9} \, c^{2} d^{3} x^{9} + \frac {2}{5} \, a c d^{3} x^{5} + a^{2} d^{3} x + \frac {1}{1155} \, {\left (77 \, c^{2} x^{15} + 210 \, a c x^{11} + 165 \, a^{2} x^{7}\right )} e^{3} + \frac {1}{195} \, {\left (45 \, c^{2} d x^{13} + 130 \, a c d x^{9} + 117 \, a^{2} d x^{5}\right )} e^{2} + \frac {1}{77} \, {\left (21 \, c^{2} d^{2} x^{11} + 66 \, a c d^{2} x^{7} + 77 \, a^{2} d^{2} x^{3}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^3*(c*x^4+a)^2,x, algorithm="fricas")

[Out]

1/9*c^2*d^3*x^9 + 2/5*a*c*d^3*x^5 + a^2*d^3*x + 1/1155*(77*c^2*x^15 + 210*a*c*x^11 + 165*a^2*x^7)*e^3 + 1/195*
(45*c^2*d*x^13 + 130*a*c*d*x^9 + 117*a^2*d*x^5)*e^2 + 1/77*(21*c^2*d^2*x^11 + 66*a*c*d^2*x^7 + 77*a^2*d^2*x^3)
*e

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Sympy [A]
time = 0.02, size = 144, normalized size = 1.08 \begin {gather*} a^{2} d^{3} x + a^{2} d^{2} e x^{3} + \frac {3 c^{2} d e^{2} x^{13}}{13} + \frac {c^{2} e^{3} x^{15}}{15} + x^{11} \cdot \left (\frac {2 a c e^{3}}{11} + \frac {3 c^{2} d^{2} e}{11}\right ) + x^{9} \cdot \left (\frac {2 a c d e^{2}}{3} + \frac {c^{2} d^{3}}{9}\right ) + x^{7} \left (\frac {a^{2} e^{3}}{7} + \frac {6 a c d^{2} e}{7}\right ) + x^{5} \cdot \left (\frac {3 a^{2} d e^{2}}{5} + \frac {2 a c d^{3}}{5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**3*(c*x**4+a)**2,x)

[Out]

a**2*d**3*x + a**2*d**2*e*x**3 + 3*c**2*d*e**2*x**13/13 + c**2*e**3*x**15/15 + x**11*(2*a*c*e**3/11 + 3*c**2*d
**2*e/11) + x**9*(2*a*c*d*e**2/3 + c**2*d**3/9) + x**7*(a**2*e**3/7 + 6*a*c*d**2*e/7) + x**5*(3*a**2*d*e**2/5
+ 2*a*c*d**3/5)

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Giac [A]
time = 3.30, size = 128, normalized size = 0.96 \begin {gather*} \frac {1}{15} \, c^{2} x^{15} e^{3} + \frac {3}{13} \, c^{2} d x^{13} e^{2} + \frac {3}{11} \, c^{2} d^{2} x^{11} e + \frac {1}{9} \, c^{2} d^{3} x^{9} + \frac {2}{11} \, a c x^{11} e^{3} + \frac {2}{3} \, a c d x^{9} e^{2} + \frac {6}{7} \, a c d^{2} x^{7} e + \frac {2}{5} \, a c d^{3} x^{5} + \frac {1}{7} \, a^{2} x^{7} e^{3} + \frac {3}{5} \, a^{2} d x^{5} e^{2} + a^{2} d^{2} x^{3} e + a^{2} d^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^3*(c*x^4+a)^2,x, algorithm="giac")

[Out]

1/15*c^2*x^15*e^3 + 3/13*c^2*d*x^13*e^2 + 3/11*c^2*d^2*x^11*e + 1/9*c^2*d^3*x^9 + 2/11*a*c*x^11*e^3 + 2/3*a*c*
d*x^9*e^2 + 6/7*a*c*d^2*x^7*e + 2/5*a*c*d^3*x^5 + 1/7*a^2*x^7*e^3 + 3/5*a^2*d*x^5*e^2 + a^2*d^2*x^3*e + a^2*d^
3*x

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Mupad [B]
time = 0.06, size = 127, normalized size = 0.95 \begin {gather*} x^5\,\left (\frac {3\,a^2\,d\,e^2}{5}+\frac {2\,c\,a\,d^3}{5}\right )+x^7\,\left (\frac {a^2\,e^3}{7}+\frac {6\,c\,a\,d^2\,e}{7}\right )+x^9\,\left (\frac {c^2\,d^3}{9}+\frac {2\,a\,c\,d\,e^2}{3}\right )+x^{11}\,\left (\frac {3\,c^2\,d^2\,e}{11}+\frac {2\,a\,c\,e^3}{11}\right )+a^2\,d^3\,x+\frac {c^2\,e^3\,x^{15}}{15}+a^2\,d^2\,e\,x^3+\frac {3\,c^2\,d\,e^2\,x^{13}}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^4)^2*(d + e*x^2)^3,x)

[Out]

x^5*((3*a^2*d*e^2)/5 + (2*a*c*d^3)/5) + x^7*((a^2*e^3)/7 + (6*a*c*d^2*e)/7) + x^9*((c^2*d^3)/9 + (2*a*c*d*e^2)
/3) + x^11*((3*c^2*d^2*e)/11 + (2*a*c*e^3)/11) + a^2*d^3*x + (c^2*e^3*x^15)/15 + a^2*d^2*e*x^3 + (3*c^2*d*e^2*
x^13)/13

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