3.2.30 \(\int (d+e x^2) (a+c x^4)^2 \, dx\) [130]

Optimal. Leaf size=60 \[ a^2 d x+\frac {1}{3} a^2 e x^3+\frac {2}{5} a c d x^5+\frac {2}{7} a c e x^7+\frac {1}{9} c^2 d x^9+\frac {1}{11} c^2 e x^{11} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {1168} \begin {gather*} a^2 d x+\frac {1}{3} a^2 e x^3+\frac {2}{5} a c d x^5+\frac {2}{7} a c e x^7+\frac {1}{9} c^2 d x^9+\frac {1}{11} c^2 e x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)*(a + c*x^4)^2,x]

[Out]

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

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

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Mathematica [A]
time = 0.00, size = 60, normalized size = 1.00 \begin {gather*} a^2 d x+\frac {1}{3} a^2 e x^3+\frac {2}{5} a c d x^5+\frac {2}{7} a c e x^7+\frac {1}{9} c^2 d x^9+\frac {1}{11} c^2 e x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.15, size = 51, normalized size = 0.85

method	result	size
gosper	\(a^{2} d x +\frac {1}{3} a^{2} e \,x^{3}+\frac {2}{5} a c d \,x^{5}+\frac {2}{7} a c e \,x^{7}+\frac {1}{9} c^{2} d \,x^{9}+\frac {1}{11} c^{2} e \,x^{11}\)	\(51\)
default	\(a^{2} d x +\frac {1}{3} a^{2} e \,x^{3}+\frac {2}{5} a c d \,x^{5}+\frac {2}{7} a c e \,x^{7}+\frac {1}{9} c^{2} d \,x^{9}+\frac {1}{11} c^{2} e \,x^{11}\)	\(51\)
norman	\(a^{2} d x +\frac {1}{3} a^{2} e \,x^{3}+\frac {2}{5} a c d \,x^{5}+\frac {2}{7} a c e \,x^{7}+\frac {1}{9} c^{2} d \,x^{9}+\frac {1}{11} c^{2} e \,x^{11}\)	\(51\)
risch	\(a^{2} d x +\frac {1}{3} a^{2} e \,x^{3}+\frac {2}{5} a c d \,x^{5}+\frac {2}{7} a c e \,x^{7}+\frac {1}{9} c^{2} d \,x^{9}+\frac {1}{11} c^{2} e \,x^{11}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 53, normalized size = 0.88 \begin {gather*} \frac {1}{11} \, c^{2} x^{11} e + \frac {1}{9} \, c^{2} d x^{9} + \frac {2}{7} \, a c x^{7} e + \frac {2}{5} \, a c d x^{5} + \frac {1}{3} \, a^{2} x^{3} e + a^{2} d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.33, size = 52, normalized size = 0.87 \begin {gather*} \frac {1}{9} \, c^{2} d x^{9} + \frac {2}{5} \, a c d x^{5} + a^{2} d x + \frac {1}{231} \, {\left (21 \, c^{2} x^{11} + 66 \, a c x^{7} + 77 \, a^{2} x^{3}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*c^2*d*x^9 + 2/5*a*c*d*x^5 + a^2*d*x + 1/231*(21*c^2*x^11 + 66*a*c*x^7 + 77*a^2*x^3)*e

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Sympy [A]
time = 0.01, size = 60, normalized size = 1.00 \begin {gather*} a^{2} d x + \frac {a^{2} e x^{3}}{3} + \frac {2 a c d x^{5}}{5} + \frac {2 a c e x^{7}}{7} + \frac {c^{2} d x^{9}}{9} + \frac {c^{2} e x^{11}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*d*x + a**2*e*x**3/3 + 2*a*c*d*x**5/5 + 2*a*c*e*x**7/7 + c**2*d*x**9/9 + c**2*e*x**11/11

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Giac [A]
time = 4.97, size = 53, normalized size = 0.88 \begin {gather*} \frac {1}{11} \, c^{2} x^{11} e + \frac {1}{9} \, c^{2} d x^{9} + \frac {2}{7} \, a c x^{7} e + \frac {2}{5} \, a c d x^{5} + \frac {1}{3} \, a^{2} x^{3} e + a^{2} d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.03, size = 50, normalized size = 0.83 \begin {gather*} \frac {e\,a^2\,x^3}{3}+d\,a^2\,x+\frac {2\,e\,a\,c\,x^7}{7}+\frac {2\,d\,a\,c\,x^5}{5}+\frac {e\,c^2\,x^{11}}{11}+\frac {d\,c^2\,x^9}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^4)^2*(d + e*x^2),x)

[Out]

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

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