3.1.15 \(\int (a+b x^2)^3 (c+d x^2)^2 \, dx\) [15]

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

[Out]

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

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Rubi [A]
time = 0.05, 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 = {380} \begin {gather*} a^3 c^2 x+\frac {1}{7} b x^7 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac {1}{5} a x^5 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {1}{3} a^2 c x^3 (2 a d+3 b c)+\frac {1}{9} b^2 d x^9 (3 a d+2 b c)+\frac {1}{11} b^3 d^2 x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^3*(c + d*x^2)^2,x]

[Out]

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

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

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Mathematica [A]
time = 0.01, size = 122, normalized size = 1.00 \begin {gather*} a^3 c^2 x+\frac {1}{3} a^2 c (3 b c+2 a d) x^3+\frac {1}{5} a \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^5+\frac {1}{7} b \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^7+\frac {1}{9} b^2 d (2 b c+3 a d) x^9+\frac {1}{11} b^3 d^2 x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.11, size = 125, normalized size = 1.02

method	result	size
norman	\(\frac {b^{3} d^{2} x^{11}}{11}+\left (\frac {1}{3} a \,b^{2} d^{2}+\frac {2}{9} b^{3} c d \right ) x^{9}+\left (\frac {3}{7} a^{2} b \,d^{2}+\frac {6}{7} a \,b^{2} c d +\frac {1}{7} b^{3} c^{2}\right ) x^{7}+\left (\frac {1}{5} a^{3} d^{2}+\frac {6}{5} a^{2} b c d +\frac {3}{5} a \,b^{2} c^{2}\right ) x^{5}+\left (\frac {2}{3} a^{3} c d +a^{2} b \,c^{2}\right ) x^{3}+a^{3} c^{2} x\)	\(122\)
default	\(\frac {b^{3} d^{2} x^{11}}{11}+\frac {\left (3 a \,b^{2} d^{2}+2 b^{3} c d \right ) x^{9}}{9}+\frac {\left (3 a^{2} b \,d^{2}+6 a \,b^{2} c d +b^{3} c^{2}\right ) x^{7}}{7}+\frac {\left (a^{3} d^{2}+6 a^{2} b c d +3 a \,b^{2} c^{2}\right ) x^{5}}{5}+\frac {\left (2 a^{3} c d +3 a^{2} b \,c^{2}\right ) x^{3}}{3}+a^{3} c^{2} x\)	\(125\)
gosper	\(\frac {1}{11} b^{3} d^{2} x^{11}+\frac {1}{3} x^{9} a \,b^{2} d^{2}+\frac {2}{9} x^{9} b^{3} c d +\frac {3}{7} x^{7} a^{2} b \,d^{2}+\frac {6}{7} x^{7} a \,b^{2} c d +\frac {1}{7} x^{7} b^{3} c^{2}+\frac {1}{5} x^{5} a^{3} d^{2}+\frac {6}{5} x^{5} a^{2} b c d +\frac {3}{5} x^{5} a \,b^{2} c^{2}+\frac {2}{3} x^{3} a^{3} c d +x^{3} a^{2} b \,c^{2}+a^{3} c^{2} x\)	\(132\)
risch	\(\frac {1}{11} b^{3} d^{2} x^{11}+\frac {1}{3} x^{9} a \,b^{2} d^{2}+\frac {2}{9} x^{9} b^{3} c d +\frac {3}{7} x^{7} a^{2} b \,d^{2}+\frac {6}{7} x^{7} a \,b^{2} c d +\frac {1}{7} x^{7} b^{3} c^{2}+\frac {1}{5} x^{5} a^{3} d^{2}+\frac {6}{5} x^{5} a^{2} b c d +\frac {3}{5} x^{5} a \,b^{2} c^{2}+\frac {2}{3} x^{3} a^{3} c d +x^{3} a^{2} b \,c^{2}+a^{3} c^{2} x\)	\(132\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.35, size = 124, normalized size = 1.02 \begin {gather*} \frac {1}{11} \, b^{3} d^{2} x^{11} + \frac {1}{9} \, {\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} x^{9} + \frac {1}{7} \, {\left (b^{3} c^{2} + 6 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x^{7} + a^{3} c^{2} x + \frac {1}{5} \, {\left (3 \, a b^{2} c^{2} + 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{5} + \frac {1}{3} \, {\left (3 \, a^{2} b c^{2} + 2 \, a^{3} c d\right )} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.79, size = 124, normalized size = 1.02 \begin {gather*} \frac {1}{11} \, b^{3} d^{2} x^{11} + \frac {1}{9} \, {\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} x^{9} + \frac {1}{7} \, {\left (b^{3} c^{2} + 6 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x^{7} + a^{3} c^{2} x + \frac {1}{5} \, {\left (3 \, a b^{2} c^{2} + 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{5} + \frac {1}{3} \, {\left (3 \, a^{2} b c^{2} + 2 \, a^{3} c d\right )} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.02, size = 136, normalized size = 1.11 \begin {gather*} a^{3} c^{2} x + \frac {b^{3} d^{2} x^{11}}{11} + x^{9} \left (\frac {a b^{2} d^{2}}{3} + \frac {2 b^{3} c d}{9}\right ) + x^{7} \cdot \left (\frac {3 a^{2} b d^{2}}{7} + \frac {6 a b^{2} c d}{7} + \frac {b^{3} c^{2}}{7}\right ) + x^{5} \left (\frac {a^{3} d^{2}}{5} + \frac {6 a^{2} b c d}{5} + \frac {3 a b^{2} c^{2}}{5}\right ) + x^{3} \cdot \left (\frac {2 a^{3} c d}{3} + a^{2} b c^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 2.98, size = 131, normalized size = 1.07 \begin {gather*} \frac {1}{11} \, b^{3} d^{2} x^{11} + \frac {2}{9} \, b^{3} c d x^{9} + \frac {1}{3} \, a b^{2} d^{2} x^{9} + \frac {1}{7} \, b^{3} c^{2} x^{7} + \frac {6}{7} \, a b^{2} c d x^{7} + \frac {3}{7} \, a^{2} b d^{2} x^{7} + \frac {3}{5} \, a b^{2} c^{2} x^{5} + \frac {6}{5} \, a^{2} b c d x^{5} + \frac {1}{5} \, a^{3} d^{2} x^{5} + a^{2} b c^{2} x^{3} + \frac {2}{3} \, a^{3} c d x^{3} + a^{3} c^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 4.87, size = 116, normalized size = 0.95 \begin {gather*} x^5\,\left (\frac {a^3\,d^2}{5}+\frac {6\,a^2\,b\,c\,d}{5}+\frac {3\,a\,b^2\,c^2}{5}\right )+x^7\,\left (\frac {3\,a^2\,b\,d^2}{7}+\frac {6\,a\,b^2\,c\,d}{7}+\frac {b^3\,c^2}{7}\right )+a^3\,c^2\,x+\frac {b^3\,d^2\,x^{11}}{11}+\frac {a^2\,c\,x^3\,\left (2\,a\,d+3\,b\,c\right )}{3}+\frac {b^2\,d\,x^9\,\left (3\,a\,d+2\,b\,c\right )}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^3*(c + d*x^2)^2,x)

[Out]

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

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