3.5.62 \(\int (d+e x)^2 (a+c x^2)^2 \, dx\) [462]

Optimal. Leaf size=80 \[ a^2 d^2 x+\frac {1}{3} a \left (2 c d^2+a e^2\right ) x^3+\frac {1}{5} c \left (c d^2+2 a e^2\right ) x^5+\frac {1}{7} c^2 e^2 x^7+\frac {d e \left (a+c x^2\right )^3}{3 c} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 80, 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^2 d^2 x+\frac {1}{5} c x^5 \left (2 a e^2+c d^2\right )+\frac {1}{3} a x^3 \left (a e^2+2 c d^2\right )+\frac {d e \left (a+c x^2\right )^3}{3 c}+\frac {1}{7} c^2 e^2 x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2*(a + c*x^2)^2,x]

[Out]

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

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Mathematica [A]
time = 0.01, size = 91, normalized size = 1.14 \begin {gather*} a^2 d^2 x+a^2 d e x^2+\frac {1}{3} a \left (2 c d^2+a e^2\right ) x^3+a c d e x^4+\frac {1}{5} c \left (c d^2+2 a e^2\right ) x^5+\frac {1}{3} c^2 d e x^6+\frac {1}{7} c^2 e^2 x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.43, size = 88, normalized size = 1.10

method	result	size
default	\(\frac {c^{2} e^{2} x^{7}}{7}+\frac {c^{2} d e \,x^{6}}{3}+\frac {\left (2 a c \,e^{2}+d^{2} c^{2}\right ) x^{5}}{5}+a c d e \,x^{4}+\frac {\left (a^{2} e^{2}+2 a c \,d^{2}\right ) x^{3}}{3}+d e \,a^{2} x^{2}+a^{2} d^{2} x\)	\(88\)
norman	\(\frac {c^{2} e^{2} x^{7}}{7}+\frac {c^{2} d e \,x^{6}}{3}+\left (\frac {2}{5} a c \,e^{2}+\frac {1}{5} d^{2} c^{2}\right ) x^{5}+a c d e \,x^{4}+\left (\frac {1}{3} a^{2} e^{2}+\frac {2}{3} a c \,d^{2}\right ) x^{3}+d e \,a^{2} x^{2}+a^{2} d^{2} x\)	\(88\)
gosper	\(\frac {1}{7} c^{2} e^{2} x^{7}+\frac {1}{3} c^{2} d e \,x^{6}+\frac {2}{5} x^{5} a c \,e^{2}+\frac {1}{5} x^{5} d^{2} c^{2}+a c d e \,x^{4}+\frac {1}{3} x^{3} a^{2} e^{2}+\frac {2}{3} d^{2} a c \,x^{3}+d e \,a^{2} x^{2}+a^{2} d^{2} x\)	\(90\)
risch	\(\frac {1}{7} c^{2} e^{2} x^{7}+\frac {1}{3} c^{2} d e \,x^{6}+\frac {2}{5} x^{5} a c \,e^{2}+\frac {1}{5} x^{5} d^{2} c^{2}+a c d e \,x^{4}+\frac {1}{3} x^{3} a^{2} e^{2}+\frac {2}{3} d^{2} a c \,x^{3}+d e \,a^{2} x^{2}+a^{2} d^{2} x\)	\(90\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 87, normalized size = 1.09 \begin {gather*} \frac {1}{7} \, c^{2} x^{7} e^{2} + \frac {1}{3} \, c^{2} d x^{6} e + a c d x^{4} e + \frac {1}{5} \, {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} x^{5} + a^{2} d x^{2} e + a^{2} d^{2} x + \frac {1}{3} \, {\left (2 \, a c d^{2} + a^{2} 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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.84, size = 88, normalized size = 1.10 \begin {gather*} \frac {1}{5} \, c^{2} d^{2} x^{5} + \frac {2}{3} \, a c d^{2} x^{3} + a^{2} d^{2} x + \frac {1}{105} \, {\left (15 \, c^{2} x^{7} + 42 \, a c x^{5} + 35 \, a^{2} x^{3}\right )} e^{2} + \frac {1}{3} \, {\left (c^{2} d x^{6} + 3 \, a c d x^{4} + 3 \, a^{2} 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)^2,x, algorithm="fricas")

[Out]

1/5*c^2*d^2*x^5 + 2/3*a*c*d^2*x^3 + a^2*d^2*x + 1/105*(15*c^2*x^7 + 42*a*c*x^5 + 35*a^2*x^3)*e^2 + 1/3*(c^2*d*
x^6 + 3*a*c*d*x^4 + 3*a^2*d*x^2)*e

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Sympy [A]
time = 0.01, size = 95, normalized size = 1.19 \begin {gather*} a^{2} d^{2} x + a^{2} d e x^{2} + a c d e x^{4} + \frac {c^{2} d e x^{6}}{3} + \frac {c^{2} e^{2} x^{7}}{7} + x^{5} \cdot \left (\frac {2 a c e^{2}}{5} + \frac {c^{2} d^{2}}{5}\right ) + x^{3} \left (\frac {a^{2} e^{2}}{3} + \frac {2 a 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)**2,x)

[Out]

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

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Giac [A]
time = 1.22, size = 89, normalized size = 1.11 \begin {gather*} \frac {1}{7} \, c^{2} x^{7} e^{2} + \frac {1}{3} \, c^{2} d x^{6} e + \frac {1}{5} \, c^{2} d^{2} x^{5} + \frac {2}{5} \, a c x^{5} e^{2} + a c d x^{4} e + \frac {2}{3} \, a c d^{2} x^{3} + \frac {1}{3} \, a^{2} x^{3} e^{2} + a^{2} d x^{2} e + a^{2} 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)^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.27, size = 87, normalized size = 1.09 \begin {gather*} x^3\,\left (\frac {a^2\,e^2}{3}+\frac {2\,c\,a\,d^2}{3}\right )+x^5\,\left (\frac {c^2\,d^2}{5}+\frac {2\,a\,c\,e^2}{5}\right )+a^2\,d^2\,x+\frac {c^2\,e^2\,x^7}{7}+a^2\,d\,e\,x^2+\frac {c^2\,d\,e\,x^6}{3}+a\,c\,d\,e\,x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^2*(d + e*x)^2,x)

[Out]

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

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