3.625 \(\int x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=114 \[ -\frac{b \left (c+d x^2\right )^{11/2} (3 b c-2 a d)}{11 d^4}+\frac{\left (c+d x^2\right )^{9/2} (b c-a d) (3 b c-a d)}{9 d^4}-\frac{c \left (c+d x^2\right )^{7/2} (b c-a d)^2}{7 d^4}+\frac{b^2 \left (c+d x^2\right )^{13/2}}{13 d^4} \]

[Out]

-(c*(b*c - a*d)^2*(c + d*x^2)^(7/2))/(7*d^4) + ((b*c - a*d)*(3*b*c - a*d)*(c + d
*x^2)^(9/2))/(9*d^4) - (b*(3*b*c - 2*a*d)*(c + d*x^2)^(11/2))/(11*d^4) + (b^2*(c
+ d*x^2)^(13/2))/(13*d^4)

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Rubi [A]  time = 0.265046, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{b \left (c+d x^2\right )^{11/2} (3 b c-2 a d)}{11 d^4}+\frac{\left (c+d x^2\right )^{9/2} (b c-a d) (3 b c-a d)}{9 d^4}-\frac{c \left (c+d x^2\right )^{7/2} (b c-a d)^2}{7 d^4}+\frac{b^2 \left (c+d x^2\right )^{13/2}}{13 d^4} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a + b*x^2)^2*(c + d*x^2)^(5/2),x]

[Out]

-(c*(b*c - a*d)^2*(c + d*x^2)^(7/2))/(7*d^4) + ((b*c - a*d)*(3*b*c - a*d)*(c + d
*x^2)^(9/2))/(9*d^4) - (b*(3*b*c - 2*a*d)*(c + d*x^2)^(11/2))/(11*d^4) + (b^2*(c
+ d*x^2)^(13/2))/(13*d^4)

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Rubi in Sympy [A]  time = 32.6081, size = 100, normalized size = 0.88 \[ \frac{b^{2} \left (c + d x^{2}\right )^{\frac{13}{2}}}{13 d^{4}} + \frac{b \left (c + d x^{2}\right )^{\frac{11}{2}} \left (2 a d - 3 b c\right )}{11 d^{4}} - \frac{c \left (c + d x^{2}\right )^{\frac{7}{2}} \left (a d - b c\right )^{2}}{7 d^{4}} + \frac{\left (c + d x^{2}\right )^{\frac{9}{2}} \left (a d - 3 b c\right ) \left (a d - b c\right )}{9 d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(b*x**2+a)**2*(d*x**2+c)**(5/2),x)

[Out]

b**2*(c + d*x**2)**(13/2)/(13*d**4) + b*(c + d*x**2)**(11/2)*(2*a*d - 3*b*c)/(11
*d**4) - c*(c + d*x**2)**(7/2)*(a*d - b*c)**2/(7*d**4) + (c + d*x**2)**(9/2)*(a*
d - 3*b*c)*(a*d - b*c)/(9*d**4)

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Mathematica [A]  time = 0.164708, size = 99, normalized size = 0.87 \[ \frac{\left (c+d x^2\right )^{7/2} \left (143 a^2 d^2 \left (7 d x^2-2 c\right )+26 a b d \left (8 c^2-28 c d x^2+63 d^2 x^4\right )+b^2 \left (-48 c^3+168 c^2 d x^2-378 c d^2 x^4+693 d^3 x^6\right )\right )}{9009 d^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(a + b*x^2)^2*(c + d*x^2)^(5/2),x]

[Out]

((c + d*x^2)^(7/2)*(143*a^2*d^2*(-2*c + 7*d*x^2) + 26*a*b*d*(8*c^2 - 28*c*d*x^2
+ 63*d^2*x^4) + b^2*(-48*c^3 + 168*c^2*d*x^2 - 378*c*d^2*x^4 + 693*d^3*x^6)))/(9
009*d^4)

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Maple [A]  time = 0.011, size = 108, normalized size = 1. \[ -{\frac{-693\,{b}^{2}{x}^{6}{d}^{3}-1638\,ab{d}^{3}{x}^{4}+378\,{b}^{2}c{d}^{2}{x}^{4}-1001\,{a}^{2}{d}^{3}{x}^{2}+728\,abc{d}^{2}{x}^{2}-168\,{b}^{2}{c}^{2}d{x}^{2}+286\,{a}^{2}c{d}^{2}-208\,ab{c}^{2}d+48\,{b}^{2}{c}^{3}}{9009\,{d}^{4}} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x^2+a)^2*(d*x^2+c)^(5/2),x)

[Out]

-1/9009*(d*x^2+c)^(7/2)*(-693*b^2*d^3*x^6-1638*a*b*d^3*x^4+378*b^2*c*d^2*x^4-100
1*a^2*d^3*x^2+728*a*b*c*d^2*x^2-168*b^2*c^2*d*x^2+286*a^2*c*d^2-208*a*b*c^2*d+48
*b^2*c^3)/d^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^(5/2)*x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232903, size = 292, normalized size = 2.56 \[ \frac{{\left (693 \, b^{2} d^{6} x^{12} + 63 \,{\left (27 \, b^{2} c d^{5} + 26 \, a b d^{6}\right )} x^{10} + 7 \,{\left (159 \, b^{2} c^{2} d^{4} + 598 \, a b c d^{5} + 143 \, a^{2} d^{6}\right )} x^{8} - 48 \, b^{2} c^{6} + 208 \, a b c^{5} d - 286 \, a^{2} c^{4} d^{2} +{\left (15 \, b^{2} c^{3} d^{3} + 2938 \, a b c^{2} d^{4} + 2717 \, a^{2} c d^{5}\right )} x^{6} - 3 \,{\left (6 \, b^{2} c^{4} d^{2} - 26 \, a b c^{3} d^{3} - 715 \, a^{2} c^{2} d^{4}\right )} x^{4} +{\left (24 \, b^{2} c^{5} d - 104 \, a b c^{4} d^{2} + 143 \, a^{2} c^{3} d^{3}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{9009 \, d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^(5/2)*x^3,x, algorithm="fricas")

[Out]

1/9009*(693*b^2*d^6*x^12 + 63*(27*b^2*c*d^5 + 26*a*b*d^6)*x^10 + 7*(159*b^2*c^2*
d^4 + 598*a*b*c*d^5 + 143*a^2*d^6)*x^8 - 48*b^2*c^6 + 208*a*b*c^5*d - 286*a^2*c^
4*d^2 + (15*b^2*c^3*d^3 + 2938*a*b*c^2*d^4 + 2717*a^2*c*d^5)*x^6 - 3*(6*b^2*c^4*
d^2 - 26*a*b*c^3*d^3 - 715*a^2*c^2*d^4)*x^4 + (24*b^2*c^5*d - 104*a*b*c^4*d^2 +
143*a^2*c^3*d^3)*x^2)*sqrt(d*x^2 + c)/d^4

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Sympy [A]  time = 31.5485, size = 468, normalized size = 4.11 \[ \begin{cases} - \frac{2 a^{2} c^{4} \sqrt{c + d x^{2}}}{63 d^{2}} + \frac{a^{2} c^{3} x^{2} \sqrt{c + d x^{2}}}{63 d} + \frac{5 a^{2} c^{2} x^{4} \sqrt{c + d x^{2}}}{21} + \frac{19 a^{2} c d x^{6} \sqrt{c + d x^{2}}}{63} + \frac{a^{2} d^{2} x^{8} \sqrt{c + d x^{2}}}{9} + \frac{16 a b c^{5} \sqrt{c + d x^{2}}}{693 d^{3}} - \frac{8 a b c^{4} x^{2} \sqrt{c + d x^{2}}}{693 d^{2}} + \frac{2 a b c^{3} x^{4} \sqrt{c + d x^{2}}}{231 d} + \frac{226 a b c^{2} x^{6} \sqrt{c + d x^{2}}}{693} + \frac{46 a b c d x^{8} \sqrt{c + d x^{2}}}{99} + \frac{2 a b d^{2} x^{10} \sqrt{c + d x^{2}}}{11} - \frac{16 b^{2} c^{6} \sqrt{c + d x^{2}}}{3003 d^{4}} + \frac{8 b^{2} c^{5} x^{2} \sqrt{c + d x^{2}}}{3003 d^{3}} - \frac{2 b^{2} c^{4} x^{4} \sqrt{c + d x^{2}}}{1001 d^{2}} + \frac{5 b^{2} c^{3} x^{6} \sqrt{c + d x^{2}}}{3003 d} + \frac{53 b^{2} c^{2} x^{8} \sqrt{c + d x^{2}}}{429} + \frac{27 b^{2} c d x^{10} \sqrt{c + d x^{2}}}{143} + \frac{b^{2} d^{2} x^{12} \sqrt{c + d x^{2}}}{13} & \text{for}\: d \neq 0 \\c^{\frac{5}{2}} \left (\frac{a^{2} x^{4}}{4} + \frac{a b x^{6}}{3} + \frac{b^{2} x^{8}}{8}\right ) & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(b*x**2+a)**2*(d*x**2+c)**(5/2),x)

[Out]

Piecewise((-2*a**2*c**4*sqrt(c + d*x**2)/(63*d**2) + a**2*c**3*x**2*sqrt(c + d*x
**2)/(63*d) + 5*a**2*c**2*x**4*sqrt(c + d*x**2)/21 + 19*a**2*c*d*x**6*sqrt(c + d
*x**2)/63 + a**2*d**2*x**8*sqrt(c + d*x**2)/9 + 16*a*b*c**5*sqrt(c + d*x**2)/(69
3*d**3) - 8*a*b*c**4*x**2*sqrt(c + d*x**2)/(693*d**2) + 2*a*b*c**3*x**4*sqrt(c +
d*x**2)/(231*d) + 226*a*b*c**2*x**6*sqrt(c + d*x**2)/693 + 46*a*b*c*d*x**8*sqrt
(c + d*x**2)/99 + 2*a*b*d**2*x**10*sqrt(c + d*x**2)/11 - 16*b**2*c**6*sqrt(c + d
*x**2)/(3003*d**4) + 8*b**2*c**5*x**2*sqrt(c + d*x**2)/(3003*d**3) - 2*b**2*c**4
*x**4*sqrt(c + d*x**2)/(1001*d**2) + 5*b**2*c**3*x**6*sqrt(c + d*x**2)/(3003*d)
+ 53*b**2*c**2*x**8*sqrt(c + d*x**2)/429 + 27*b**2*c*d*x**10*sqrt(c + d*x**2)/14
3 + b**2*d**2*x**12*sqrt(c + d*x**2)/13, Ne(d, 0)), (c**(5/2)*(a**2*x**4/4 + a*b
*x**6/3 + b**2*x**8/8), True))

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GIAC/XCAS [A]  time = 0.239024, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^(5/2)*x^3,x, algorithm="giac")

[Out]

Done