3.1.8 \(\int (a+b x^2)^2 (c+d x^2)^3 \, dx\)

Optimal. Leaf size=122 \[ \frac {1}{7} d x^7 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {1}{5} c x^5 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a^2 c^3 x+\frac {1}{3} a c^2 x^3 (3 a d+2 b c)+\frac {1}{9} b d^2 x^9 (2 a d+3 b c)+\frac {1}{11} b^2 d^3 x^{11} \]

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Rubi [A]  time = 0.07, antiderivative size = 122, 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 = {373} \begin {gather*} \frac {1}{7} d x^7 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {1}{5} c x^5 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a^2 c^3 x+\frac {1}{3} a c^2 x^3 (3 a d+2 b c)+\frac {1}{9} b d^2 x^9 (2 a d+3 b c)+\frac {1}{11} b^2 d^3 x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2*(c + d*x^2)^3,x]

[Out]

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

Rule 373

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

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Mathematica [A]  time = 0.02, size = 122, normalized size = 1.00 \begin {gather*} \frac {1}{7} d x^7 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {1}{5} c x^5 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a^2 c^3 x+\frac {1}{3} a c^2 x^3 (3 a d+2 b c)+\frac {1}{9} b d^2 x^9 (2 a d+3 b c)+\frac {1}{11} b^2 d^3 x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2)^2*(c + d*x^2)^3,x]

[Out]

IntegrateAlgebraic[(a + b*x^2)^2*(c + d*x^2)^3, x]

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fricas [A]  time = 0.74, size = 131, normalized size = 1.07 \begin {gather*} \frac {1}{11} x^{11} d^{3} b^{2} + \frac {1}{3} x^{9} d^{2} c b^{2} + \frac {2}{9} x^{9} d^{3} b a + \frac {3}{7} x^{7} d c^{2} b^{2} + \frac {6}{7} x^{7} d^{2} c b a + \frac {1}{7} x^{7} d^{3} a^{2} + \frac {1}{5} x^{5} c^{3} b^{2} + \frac {6}{5} x^{5} d c^{2} b a + \frac {3}{5} x^{5} d^{2} c a^{2} + \frac {2}{3} x^{3} c^{3} b a + x^{3} d c^{2} a^{2} + x c^{3} a^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.56, size = 131, normalized size = 1.07 \begin {gather*} \frac {1}{11} \, b^{2} d^{3} x^{11} + \frac {1}{3} \, b^{2} c d^{2} x^{9} + \frac {2}{9} \, a b d^{3} x^{9} + \frac {3}{7} \, b^{2} c^{2} d x^{7} + \frac {6}{7} \, a b c d^{2} x^{7} + \frac {1}{7} \, a^{2} d^{3} x^{7} + \frac {1}{5} \, b^{2} c^{3} x^{5} + \frac {6}{5} \, a b c^{2} d x^{5} + \frac {3}{5} \, a^{2} c d^{2} x^{5} + \frac {2}{3} \, a b c^{3} x^{3} + a^{2} c^{2} d x^{3} + a^{2} c^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 125, normalized size = 1.02 \begin {gather*} \frac {b^{2} d^{3} x^{11}}{11}+\frac {\left (2 a b \,d^{3}+3 b^{2} c \,d^{2}\right ) x^{9}}{9}+\frac {\left (a^{2} d^{3}+6 a b c \,d^{2}+3 b^{2} c^{2} d \right ) x^{7}}{7}+a^{2} c^{3} x +\frac {\left (3 a^{2} c \,d^{2}+6 a b \,c^{2} d +b^{2} c^{3}\right ) x^{5}}{5}+\frac {\left (3 a^{2} c^{2} d +2 a b \,c^{3}\right ) x^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2*(d*x^2+c)^3,x)

[Out]

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

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maxima [A]  time = 1.39, size = 124, normalized size = 1.02 \begin {gather*} \frac {1}{11} \, b^{2} d^{3} x^{11} + \frac {1}{9} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{9} + \frac {1}{7} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{7} + a^{2} c^{3} x + \frac {1}{5} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{5} + \frac {1}{3} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.95, size = 116, normalized size = 0.95 \begin {gather*} x^5\,\left (\frac {3\,a^2\,c\,d^2}{5}+\frac {6\,a\,b\,c^2\,d}{5}+\frac {b^2\,c^3}{5}\right )+x^7\,\left (\frac {a^2\,d^3}{7}+\frac {6\,a\,b\,c\,d^2}{7}+\frac {3\,b^2\,c^2\,d}{7}\right )+a^2\,c^3\,x+\frac {b^2\,d^3\,x^{11}}{11}+\frac {a\,c^2\,x^3\,\left (3\,a\,d+2\,b\,c\right )}{3}+\frac {b\,d^2\,x^9\,\left (2\,a\,d+3\,b\,c\right )}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^2*(c + d*x^2)^3,x)

[Out]

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

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sympy [A]  time = 0.09, size = 136, normalized size = 1.11 \begin {gather*} a^{2} c^{3} x + \frac {b^{2} d^{3} x^{11}}{11} + x^{9} \left (\frac {2 a b d^{3}}{9} + \frac {b^{2} c d^{2}}{3}\right ) + x^{7} \left (\frac {a^{2} d^{3}}{7} + \frac {6 a b c d^{2}}{7} + \frac {3 b^{2} c^{2} d}{7}\right ) + x^{5} \left (\frac {3 a^{2} c d^{2}}{5} + \frac {6 a b c^{2} d}{5} + \frac {b^{2} c^{3}}{5}\right ) + x^{3} \left (a^{2} c^{2} d + \frac {2 a b c^{3}}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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