3.5.9 \(\int x^{3/2} (a+b x^2)^2 (c+d x^2)^3 \, dx\) [409]

Optimal. Leaf size=139 \[ \frac {2}{5} a^2 c^3 x^{5/2}+\frac {2}{9} a c^2 (2 b c+3 a d) x^{9/2}+\frac {2}{13} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{13/2}+\frac {2}{17} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{17/2}+\frac {2}{21} b d^2 (3 b c+2 a d) x^{21/2}+\frac {2}{25} b^2 d^3 x^{25/2} \]

[Out]

2/5*a^2*c^3*x^(5/2)+2/9*a*c^2*(3*a*d+2*b*c)*x^(9/2)+2/13*c*(3*a^2*d^2+6*a*b*c*d+b^2*c^2)*x^(13/2)+2/17*d*(a^2*
d^2+6*a*b*c*d+3*b^2*c^2)*x^(17/2)+2/21*b*d^2*(2*a*d+3*b*c)*x^(21/2)+2/25*b^2*d^3*x^(25/2)

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Rubi [A]
time = 0.04, 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 = {459} \begin {gather*} \frac {2}{17} d x^{17/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {2}{13} c x^{13/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac {2}{5} a^2 c^3 x^{5/2}+\frac {2}{9} a c^2 x^{9/2} (3 a d+2 b c)+\frac {2}{21} b d^2 x^{21/2} (2 a d+3 b c)+\frac {2}{25} b^2 d^3 x^{25/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^2*c^3*x^(5/2))/5 + (2*a*c^2*(2*b*c + 3*a*d)*x^(9/2))/9 + (2*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(13/2))
/13 + (2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(17/2))/17 + (2*b*d^2*(3*b*c + 2*a*d)*x^(21/2))/21 + (2*b^2*d^3
*x^(25/2))/25

Rule 459

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^{3/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx &=\int \left (a^2 c^3 x^{3/2}+a c^2 (2 b c+3 a d) x^{7/2}+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{11/2}+d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{15/2}+b d^2 (3 b c+2 a d) x^{19/2}+b^2 d^3 x^{23/2}\right ) \, dx\\ &=\frac {2}{5} a^2 c^3 x^{5/2}+\frac {2}{9} a c^2 (2 b c+3 a d) x^{9/2}+\frac {2}{13} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{13/2}+\frac {2}{17} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{17/2}+\frac {2}{21} b d^2 (3 b c+2 a d) x^{21/2}+\frac {2}{25} b^2 d^3 x^{25/2}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 126, normalized size = 0.91 \begin {gather*} \frac {2 x^{5/2} \left (105 a^2 \left (663 c^3+1105 c^2 d x^2+765 c d^2 x^4+195 d^3 x^6\right )+50 a b x^2 \left (1547 c^3+3213 c^2 d x^2+2457 c d^2 x^4+663 d^3 x^6\right )+9 b^2 x^4 \left (2975 c^3+6825 c^2 d x^2+5525 c d^2 x^4+1547 d^3 x^6\right )\right )}{348075} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(5/2)*(105*a^2*(663*c^3 + 1105*c^2*d*x^2 + 765*c*d^2*x^4 + 195*d^3*x^6) + 50*a*b*x^2*(1547*c^3 + 3213*c^2
*d*x^2 + 2457*c*d^2*x^4 + 663*d^3*x^6) + 9*b^2*x^4*(2975*c^3 + 6825*c^2*d*x^2 + 5525*c*d^2*x^4 + 1547*d^3*x^6)
))/348075

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Maple [A]
time = 0.11, size = 128, normalized size = 0.92

method	result	size
derivativedivides	\(\frac {2 b^{2} d^{3} x^{\frac {25}{2}}}{25}+\frac {2 \left (2 a b \,d^{3}+3 b^{2} c \,d^{2}\right ) x^{\frac {21}{2}}}{21}+\frac {2 \left (a^{2} d^{3}+6 a b c \,d^{2}+3 b^{2} c^{2} d \right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (3 a^{2} c \,d^{2}+6 a b \,c^{2} d +b^{2} c^{3}\right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (3 a^{2} c^{2} d +2 a b \,c^{3}\right ) x^{\frac {9}{2}}}{9}+\frac {2 a^{2} c^{3} x^{\frac {5}{2}}}{5}\)	\(128\)
default	\(\frac {2 b^{2} d^{3} x^{\frac {25}{2}}}{25}+\frac {2 \left (2 a b \,d^{3}+3 b^{2} c \,d^{2}\right ) x^{\frac {21}{2}}}{21}+\frac {2 \left (a^{2} d^{3}+6 a b c \,d^{2}+3 b^{2} c^{2} d \right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (3 a^{2} c \,d^{2}+6 a b \,c^{2} d +b^{2} c^{3}\right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (3 a^{2} c^{2} d +2 a b \,c^{3}\right ) x^{\frac {9}{2}}}{9}+\frac {2 a^{2} c^{3} x^{\frac {5}{2}}}{5}\)	\(128\)
gosper	\(\frac {2 x^{\frac {5}{2}} \left (13923 b^{2} d^{3} x^{10}+33150 a b \,d^{3} x^{8}+49725 b^{2} c \,d^{2} x^{8}+20475 a^{2} d^{3} x^{6}+122850 a b c \,d^{2} x^{6}+61425 b^{2} c^{2} d \,x^{6}+80325 a^{2} c \,d^{2} x^{4}+160650 a b \,c^{2} d \,x^{4}+26775 b^{2} c^{3} x^{4}+116025 a^{2} c^{2} d \,x^{2}+77350 a b \,c^{3} x^{2}+69615 a^{2} c^{3}\right )}{348075}\)	\(138\)
trager	\(\frac {2 x^{\frac {5}{2}} \left (13923 b^{2} d^{3} x^{10}+33150 a b \,d^{3} x^{8}+49725 b^{2} c \,d^{2} x^{8}+20475 a^{2} d^{3} x^{6}+122850 a b c \,d^{2} x^{6}+61425 b^{2} c^{2} d \,x^{6}+80325 a^{2} c \,d^{2} x^{4}+160650 a b \,c^{2} d \,x^{4}+26775 b^{2} c^{3} x^{4}+116025 a^{2} c^{2} d \,x^{2}+77350 a b \,c^{3} x^{2}+69615 a^{2} c^{3}\right )}{348075}\)	\(138\)
risch	\(\frac {2 x^{\frac {5}{2}} \left (13923 b^{2} d^{3} x^{10}+33150 a b \,d^{3} x^{8}+49725 b^{2} c \,d^{2} x^{8}+20475 a^{2} d^{3} x^{6}+122850 a b c \,d^{2} x^{6}+61425 b^{2} c^{2} d \,x^{6}+80325 a^{2} c \,d^{2} x^{4}+160650 a b \,c^{2} d \,x^{4}+26775 b^{2} c^{3} x^{4}+116025 a^{2} c^{2} d \,x^{2}+77350 a b \,c^{3} x^{2}+69615 a^{2} c^{3}\right )}{348075}\)	\(138\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/25*b^2*d^3*x^(25/2)+2/21*(2*a*b*d^3+3*b^2*c*d^2)*x^(21/2)+2/17*(a^2*d^3+6*a*b*c*d^2+3*b^2*c^2*d)*x^(17/2)+2/
13*(3*a^2*c*d^2+6*a*b*c^2*d+b^2*c^3)*x^(13/2)+2/9*(3*a^2*c^2*d+2*a*b*c^3)*x^(9/2)+2/5*a^2*c^3*x^(5/2)

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Maxima [A]
time = 0.27, size = 127, normalized size = 0.91 \begin {gather*} \frac {2}{25} \, b^{2} d^{3} x^{\frac {25}{2}} + \frac {2}{21} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{\frac {21}{2}} + \frac {2}{17} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac {17}{2}} + \frac {2}{5} \, a^{2} c^{3} x^{\frac {5}{2}} + \frac {2}{13} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{\frac {13}{2}} + \frac {2}{9} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{\frac {9}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/25*b^2*d^3*x^(25/2) + 2/21*(3*b^2*c*d^2 + 2*a*b*d^3)*x^(21/2) + 2/17*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x
^(17/2) + 2/5*a^2*c^3*x^(5/2) + 2/13*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^(13/2) + 2/9*(2*a*b*c^3 + 3*a^2*c
^2*d)*x^(9/2)

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Fricas [A]
time = 0.50, size = 132, normalized size = 0.95 \begin {gather*} \frac {2}{348075} \, {\left (13923 \, b^{2} d^{3} x^{12} + 16575 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{10} + 20475 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{8} + 69615 \, a^{2} c^{3} x^{2} + 26775 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{6} + 38675 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{4}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/348075*(13923*b^2*d^3*x^12 + 16575*(3*b^2*c*d^2 + 2*a*b*d^3)*x^10 + 20475*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d
^3)*x^8 + 69615*a^2*c^3*x^2 + 26775*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^6 + 38675*(2*a*b*c^3 + 3*a^2*c^2*d
)*x^4)*sqrt(x)

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Sympy [A]
time = 1.19, size = 192, normalized size = 1.38 \begin {gather*} \frac {2 a^{2} c^{3} x^{\frac {5}{2}}}{5} + \frac {2 a^{2} c^{2} d x^{\frac {9}{2}}}{3} + \frac {6 a^{2} c d^{2} x^{\frac {13}{2}}}{13} + \frac {2 a^{2} d^{3} x^{\frac {17}{2}}}{17} + \frac {4 a b c^{3} x^{\frac {9}{2}}}{9} + \frac {12 a b c^{2} d x^{\frac {13}{2}}}{13} + \frac {12 a b c d^{2} x^{\frac {17}{2}}}{17} + \frac {4 a b d^{3} x^{\frac {21}{2}}}{21} + \frac {2 b^{2} c^{3} x^{\frac {13}{2}}}{13} + \frac {6 b^{2} c^{2} d x^{\frac {17}{2}}}{17} + \frac {2 b^{2} c d^{2} x^{\frac {21}{2}}}{7} + \frac {2 b^{2} d^{3} x^{\frac {25}{2}}}{25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**2*c**3*x**(5/2)/5 + 2*a**2*c**2*d*x**(9/2)/3 + 6*a**2*c*d**2*x**(13/2)/13 + 2*a**2*d**3*x**(17/2)/17 + 4*
a*b*c**3*x**(9/2)/9 + 12*a*b*c**2*d*x**(13/2)/13 + 12*a*b*c*d**2*x**(17/2)/17 + 4*a*b*d**3*x**(21/2)/21 + 2*b*
*2*c**3*x**(13/2)/13 + 6*b**2*c**2*d*x**(17/2)/17 + 2*b**2*c*d**2*x**(21/2)/7 + 2*b**2*d**3*x**(25/2)/25

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Giac [A]
time = 0.58, size = 135, normalized size = 0.97 \begin {gather*} \frac {2}{25} \, b^{2} d^{3} x^{\frac {25}{2}} + \frac {2}{7} \, b^{2} c d^{2} x^{\frac {21}{2}} + \frac {4}{21} \, a b d^{3} x^{\frac {21}{2}} + \frac {6}{17} \, b^{2} c^{2} d x^{\frac {17}{2}} + \frac {12}{17} \, a b c d^{2} x^{\frac {17}{2}} + \frac {2}{17} \, a^{2} d^{3} x^{\frac {17}{2}} + \frac {2}{13} \, b^{2} c^{3} x^{\frac {13}{2}} + \frac {12}{13} \, a b c^{2} d x^{\frac {13}{2}} + \frac {6}{13} \, a^{2} c d^{2} x^{\frac {13}{2}} + \frac {4}{9} \, a b c^{3} x^{\frac {9}{2}} + \frac {2}{3} \, a^{2} c^{2} d x^{\frac {9}{2}} + \frac {2}{5} \, a^{2} c^{3} x^{\frac {5}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/25*b^2*d^3*x^(25/2) + 2/7*b^2*c*d^2*x^(21/2) + 4/21*a*b*d^3*x^(21/2) + 6/17*b^2*c^2*d*x^(17/2) + 12/17*a*b*c
*d^2*x^(17/2) + 2/17*a^2*d^3*x^(17/2) + 2/13*b^2*c^3*x^(13/2) + 12/13*a*b*c^2*d*x^(13/2) + 6/13*a^2*c*d^2*x^(1
3/2) + 4/9*a*b*c^3*x^(9/2) + 2/3*a^2*c^2*d*x^(9/2) + 2/5*a^2*c^3*x^(5/2)

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Mupad [B]
time = 0.02, size = 119, normalized size = 0.86 \begin {gather*} x^{13/2}\,\left (\frac {6\,a^2\,c\,d^2}{13}+\frac {12\,a\,b\,c^2\,d}{13}+\frac {2\,b^2\,c^3}{13}\right )+x^{17/2}\,\left (\frac {2\,a^2\,d^3}{17}+\frac {12\,a\,b\,c\,d^2}{17}+\frac {6\,b^2\,c^2\,d}{17}\right )+\frac {2\,a^2\,c^3\,x^{5/2}}{5}+\frac {2\,b^2\,d^3\,x^{25/2}}{25}+\frac {2\,a\,c^2\,x^{9/2}\,\left (3\,a\,d+2\,b\,c\right )}{9}+\frac {2\,b\,d^2\,x^{21/2}\,\left (2\,a\,d+3\,b\,c\right )}{21} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(13/2)*((2*b^2*c^3)/13 + (6*a^2*c*d^2)/13 + (12*a*b*c^2*d)/13) + x^(17/2)*((2*a^2*d^3)/17 + (6*b^2*c^2*d)/17
+ (12*a*b*c*d^2)/17) + (2*a^2*c^3*x^(5/2))/5 + (2*b^2*d^3*x^(25/2))/25 + (2*a*c^2*x^(9/2)*(3*a*d + 2*b*c))/9
+ (2*b*d^2*x^(21/2)*(2*a*d + 3*b*c))/21

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