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

Optimal. Leaf size=28 \[ a c x+\frac {1}{3} (b c+a d) x^3+\frac {1}{5} b d x^5 \]

[Out]

a*c*x+1/3*(a*d+b*c)*x^3+1/5*b*d*x^5

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Rubi [A]
time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {380} \begin {gather*} \frac {1}{3} x^3 (a d+b c)+a c x+\frac {1}{5} b d x^5 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)*(c + d*x^2),x]

[Out]

a*c*x + ((b*c + a*d)*x^3)/3 + (b*d*x^5)/5

Rule 380

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

Rubi steps

\begin {align*} \int \left (a+b x^2\right ) \left (c+d x^2\right ) \, dx &=\int \left (a c+(b c+a d) x^2+b d x^4\right ) \, dx\\ &=a c x+\frac {1}{3} (b c+a d) x^3+\frac {1}{5} b d x^5\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 28, normalized size = 1.00 \begin {gather*} a c x+\frac {1}{3} (b c+a d) x^3+\frac {1}{5} b d x^5 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)*(c + d*x^2),x]

[Out]

a*c*x + ((b*c + a*d)*x^3)/3 + (b*d*x^5)/5

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Maple [A]
time = 0.08, size = 25, normalized size = 0.89

method	result	size
default	\(a c x +\frac {\left (a d +b c \right ) x^{3}}{3}+\frac {b d \,x^{5}}{5}\)	\(25\)
norman	\(\frac {b d \,x^{5}}{5}+\left (\frac {a d}{3}+\frac {b c}{3}\right ) x^{3}+a c x\)	\(26\)
gosper	\(\frac {1}{5} b d \,x^{5}+\frac {1}{3} x^{3} a d +\frac {1}{3} b c \,x^{3}+a c x\)	\(27\)
risch	\(\frac {1}{5} b d \,x^{5}+\frac {1}{3} x^{3} a d +\frac {1}{3} b c \,x^{3}+a c x\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c*x+1/3*(a*d+b*c)*x^3+1/5*b*d*x^5

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Maxima [A]
time = 0.29, size = 24, normalized size = 0.86 \begin {gather*} \frac {1}{5} \, b d x^{5} + \frac {1}{3} \, {\left (b c + a d\right )} x^{3} + a c x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b*d*x^5 + 1/3*(b*c + a*d)*x^3 + a*c*x

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Fricas [A]
time = 0.85, size = 24, normalized size = 0.86 \begin {gather*} \frac {1}{5} \, b d x^{5} + \frac {1}{3} \, {\left (b c + a d\right )} x^{3} + a c x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b*d*x^5 + 1/3*(b*c + a*d)*x^3 + a*c*x

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Sympy [A]
time = 0.01, size = 26, normalized size = 0.93 \begin {gather*} a c x + \frac {b d x^{5}}{5} + x^{3} \left (\frac {a d}{3} + \frac {b c}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)*(d*x**2+c),x)

[Out]

a*c*x + b*d*x**5/5 + x**3*(a*d/3 + b*c/3)

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Giac [A]
time = 0.92, size = 26, normalized size = 0.93 \begin {gather*} \frac {1}{5} \, b d x^{5} + \frac {1}{3} \, b c x^{3} + \frac {1}{3} \, a d x^{3} + a c x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b*d*x^5 + 1/3*b*c*x^3 + 1/3*a*d*x^3 + a*c*x

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Mupad [B]
time = 0.04, size = 25, normalized size = 0.89 \begin {gather*} \frac {b\,d\,x^5}{5}+\left (\frac {a\,d}{3}+\frac {b\,c}{3}\right )\,x^3+a\,c\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)*(c + d*x^2),x)

[Out]

x^3*((a*d)/3 + (b*c)/3) + a*c*x + (b*d*x^5)/5

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