3.407 \(\int x^{7/2} (a+b x^2)^2 (c+d x^2)^3 \, dx\)

Optimal. Leaf size=139 \[ \frac {2}{21} d x^{21/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {2}{17} c x^{17/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac {2}{9} a^2 c^3 x^{9/2}+\frac {2}{13} a c^2 x^{13/2} (3 a d+2 b c)+\frac {2}{25} b d^2 x^{25/2} (2 a d+3 b c)+\frac {2}{29} b^2 d^3 x^{29/2} \]

[Out]

2/9*a^2*c^3*x^(9/2)+2/13*a*c^2*(3*a*d+2*b*c)*x^(13/2)+2/17*c*(3*a^2*d^2+6*a*b*c*d+b^2*c^2)*x^(17/2)+2/21*d*(a^
2*d^2+6*a*b*c*d+3*b^2*c^2)*x^(21/2)+2/25*b*d^2*(2*a*d+3*b*c)*x^(25/2)+2/29*b^2*d^3*x^(29/2)

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Rubi [A]  time = 0.06, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {448} \[ \frac {2}{21} d x^{21/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {2}{17} c x^{17/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac {2}{9} a^2 c^3 x^{9/2}+\frac {2}{13} a c^2 x^{13/2} (3 a d+2 b c)+\frac {2}{25} b d^2 x^{25/2} (2 a d+3 b c)+\frac {2}{29} b^2 d^3 x^{29/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(7/2)*(a + b*x^2)^2*(c + d*x^2)^3,x]

[Out]

(2*a^2*c^3*x^(9/2))/9 + (2*a*c^2*(2*b*c + 3*a*d)*x^(13/2))/13 + (2*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(17/2
))/17 + (2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(21/2))/21 + (2*b*d^2*(3*b*c + 2*a*d)*x^(25/2))/25 + (2*b^2*d
^3*x^(29/2))/29

Rule 448

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

Rubi steps

\begin {align*} \int x^{7/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx &=\int \left (a^2 c^3 x^{7/2}+a c^2 (2 b c+3 a d) x^{11/2}+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{15/2}+d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{19/2}+b d^2 (3 b c+2 a d) x^{23/2}+b^2 d^3 x^{27/2}\right ) \, dx\\ &=\frac {2}{9} a^2 c^3 x^{9/2}+\frac {2}{13} a c^2 (2 b c+3 a d) x^{13/2}+\frac {2}{17} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{17/2}+\frac {2}{21} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{21/2}+\frac {2}{25} b d^2 (3 b c+2 a d) x^{25/2}+\frac {2}{29} b^2 d^3 x^{29/2}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 139, normalized size = 1.00 \[ \frac {2}{21} d x^{21/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {2}{17} c x^{17/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac {2}{9} a^2 c^3 x^{9/2}+\frac {2}{13} a c^2 x^{13/2} (3 a d+2 b c)+\frac {2}{25} b d^2 x^{25/2} (2 a d+3 b c)+\frac {2}{29} b^2 d^3 x^{29/2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(7/2)*(a + b*x^2)^2*(c + d*x^2)^3,x]

[Out]

(2*a^2*c^3*x^(9/2))/9 + (2*a*c^2*(2*b*c + 3*a*d)*x^(13/2))/13 + (2*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(17/2
))/17 + (2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(21/2))/21 + (2*b*d^2*(3*b*c + 2*a*d)*x^(25/2))/25 + (2*b^2*d
^3*x^(29/2))/29

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fricas [A]  time = 0.43, size = 132, normalized size = 0.95 \[ \frac {2}{10094175} \, {\left (348075 \, b^{2} d^{3} x^{14} + 403767 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{12} + 480675 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{10} + 1121575 \, a^{2} c^{3} x^{4} + 593775 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{8} + 776475 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{6}\right )} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(b*x^2+a)^2*(d*x^2+c)^3,x, algorithm="fricas")

[Out]

2/10094175*(348075*b^2*d^3*x^14 + 403767*(3*b^2*c*d^2 + 2*a*b*d^3)*x^12 + 480675*(3*b^2*c^2*d + 6*a*b*c*d^2 +
a^2*d^3)*x^10 + 1121575*a^2*c^3*x^4 + 593775*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^8 + 776475*(2*a*b*c^3 + 3
*a^2*c^2*d)*x^6)*sqrt(x)

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giac [A]  time = 0.35, size = 135, normalized size = 0.97 \[ \frac {2}{29} \, b^{2} d^{3} x^{\frac {29}{2}} + \frac {6}{25} \, b^{2} c d^{2} x^{\frac {25}{2}} + \frac {4}{25} \, a b d^{3} x^{\frac {25}{2}} + \frac {2}{7} \, b^{2} c^{2} d x^{\frac {21}{2}} + \frac {4}{7} \, a b c d^{2} x^{\frac {21}{2}} + \frac {2}{21} \, a^{2} d^{3} x^{\frac {21}{2}} + \frac {2}{17} \, b^{2} c^{3} x^{\frac {17}{2}} + \frac {12}{17} \, a b c^{2} d x^{\frac {17}{2}} + \frac {6}{17} \, a^{2} c d^{2} x^{\frac {17}{2}} + \frac {4}{13} \, a b c^{3} x^{\frac {13}{2}} + \frac {6}{13} \, a^{2} c^{2} d x^{\frac {13}{2}} + \frac {2}{9} \, a^{2} c^{3} x^{\frac {9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(b*x^2+a)^2*(d*x^2+c)^3,x, algorithm="giac")

[Out]

2/29*b^2*d^3*x^(29/2) + 6/25*b^2*c*d^2*x^(25/2) + 4/25*a*b*d^3*x^(25/2) + 2/7*b^2*c^2*d*x^(21/2) + 4/7*a*b*c*d
^2*x^(21/2) + 2/21*a^2*d^3*x^(21/2) + 2/17*b^2*c^3*x^(17/2) + 12/17*a*b*c^2*d*x^(17/2) + 6/17*a^2*c*d^2*x^(17/
2) + 4/13*a*b*c^3*x^(13/2) + 6/13*a^2*c^2*d*x^(13/2) + 2/9*a^2*c^3*x^(9/2)

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maple [A]  time = 0.01, size = 138, normalized size = 0.99 \[ \frac {2 \left (348075 b^{2} d^{3} x^{10}+807534 a b \,d^{3} x^{8}+1211301 b^{2} c \,d^{2} x^{8}+480675 a^{2} d^{3} x^{6}+2884050 a b c \,d^{2} x^{6}+1442025 b^{2} c^{2} d \,x^{6}+1781325 a^{2} c \,d^{2} x^{4}+3562650 a b \,c^{2} d \,x^{4}+593775 b^{2} c^{3} x^{4}+2329425 a^{2} c^{2} d \,x^{2}+1552950 a b \,c^{3} x^{2}+1121575 a^{2} c^{3}\right ) x^{\frac {9}{2}}}{10094175} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(b*x^2+a)^2*(d*x^2+c)^3,x)

[Out]

2/10094175*x^(9/2)*(348075*b^2*d^3*x^10+807534*a*b*d^3*x^8+1211301*b^2*c*d^2*x^8+480675*a^2*d^3*x^6+2884050*a*
b*c*d^2*x^6+1442025*b^2*c^2*d*x^6+1781325*a^2*c*d^2*x^4+3562650*a*b*c^2*d*x^4+593775*b^2*c^3*x^4+2329425*a^2*c
^2*d*x^2+1552950*a*b*c^3*x^2+1121575*a^2*c^3)

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maxima [A]  time = 1.06, size = 127, normalized size = 0.91 \[ \frac {2}{29} \, b^{2} d^{3} x^{\frac {29}{2}} + \frac {2}{25} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{\frac {25}{2}} + \frac {2}{21} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac {21}{2}} + \frac {2}{9} \, a^{2} c^{3} x^{\frac {9}{2}} + \frac {2}{17} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{\frac {17}{2}} + \frac {2}{13} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{\frac {13}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(b*x^2+a)^2*(d*x^2+c)^3,x, algorithm="maxima")

[Out]

2/29*b^2*d^3*x^(29/2) + 2/25*(3*b^2*c*d^2 + 2*a*b*d^3)*x^(25/2) + 2/21*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x
^(21/2) + 2/9*a^2*c^3*x^(9/2) + 2/17*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^(17/2) + 2/13*(2*a*b*c^3 + 3*a^2*
c^2*d)*x^(13/2)

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mupad [B]  time = 0.21, size = 119, normalized size = 0.86 \[ x^{17/2}\,\left (\frac {6\,a^2\,c\,d^2}{17}+\frac {12\,a\,b\,c^2\,d}{17}+\frac {2\,b^2\,c^3}{17}\right )+x^{21/2}\,\left (\frac {2\,a^2\,d^3}{21}+\frac {4\,a\,b\,c\,d^2}{7}+\frac {2\,b^2\,c^2\,d}{7}\right )+\frac {2\,a^2\,c^3\,x^{9/2}}{9}+\frac {2\,b^2\,d^3\,x^{29/2}}{29}+\frac {2\,a\,c^2\,x^{13/2}\,\left (3\,a\,d+2\,b\,c\right )}{13}+\frac {2\,b\,d^2\,x^{25/2}\,\left (2\,a\,d+3\,b\,c\right )}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(a + b*x^2)^2*(c + d*x^2)^3,x)

[Out]

x^(17/2)*((2*b^2*c^3)/17 + (6*a^2*c*d^2)/17 + (12*a*b*c^2*d)/17) + x^(21/2)*((2*a^2*d^3)/21 + (2*b^2*c^2*d)/7
+ (4*a*b*c*d^2)/7) + (2*a^2*c^3*x^(9/2))/9 + (2*b^2*d^3*x^(29/2))/29 + (2*a*c^2*x^(13/2)*(3*a*d + 2*b*c))/13 +
(2*b*d^2*x^(25/2)*(2*a*d + 3*b*c))/25

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sympy [A]  time = 53.08, size = 192, normalized size = 1.38 \[ \frac {2 a^{2} c^{3} x^{\frac {9}{2}}}{9} + \frac {6 a^{2} c^{2} d x^{\frac {13}{2}}}{13} + \frac {6 a^{2} c d^{2} x^{\frac {17}{2}}}{17} + \frac {2 a^{2} d^{3} x^{\frac {21}{2}}}{21} + \frac {4 a b c^{3} x^{\frac {13}{2}}}{13} + \frac {12 a b c^{2} d x^{\frac {17}{2}}}{17} + \frac {4 a b c d^{2} x^{\frac {21}{2}}}{7} + \frac {4 a b d^{3} x^{\frac {25}{2}}}{25} + \frac {2 b^{2} c^{3} x^{\frac {17}{2}}}{17} + \frac {2 b^{2} c^{2} d x^{\frac {21}{2}}}{7} + \frac {6 b^{2} c d^{2} x^{\frac {25}{2}}}{25} + \frac {2 b^{2} d^{3} x^{\frac {29}{2}}}{29} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(7/2)*(b*x**2+a)**2*(d*x**2+c)**3,x)

[Out]

2*a**2*c**3*x**(9/2)/9 + 6*a**2*c**2*d*x**(13/2)/13 + 6*a**2*c*d**2*x**(17/2)/17 + 2*a**2*d**3*x**(21/2)/21 +
4*a*b*c**3*x**(13/2)/13 + 12*a*b*c**2*d*x**(17/2)/17 + 4*a*b*c*d**2*x**(21/2)/7 + 4*a*b*d**3*x**(25/2)/25 + 2*
b**2*c**3*x**(17/2)/17 + 2*b**2*c**2*d*x**(21/2)/7 + 6*b**2*c*d**2*x**(25/2)/25 + 2*b**2*d**3*x**(29/2)/29

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