3.19 \(\int \frac{\left (a+b x^2\right ) \left (c+d x^2\right )^3}{e+f x^2} \, dx\)

Optimal. Leaf size=227 \[ -\frac{x \left (c+d x^2\right ) \left (7 a d f (5 d e-9 c f)-b \left (24 c^2 f^2-63 c d e f+35 d^2 e^2\right )\right )}{105 f^3}+\frac{x \left (7 a d f \left (33 c^2 f^2-40 c d e f+15 d^2 e^2\right )-b \left (-48 c^3 f^3+231 c^2 d e f^2-280 c d^2 e^2 f+105 d^3 e^3\right )\right )}{105 f^4}+\frac{(b e-a f) (d e-c f)^3 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e} f^{9/2}}-\frac{x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{35 f^2}+\frac{b x \left (c+d x^2\right )^3}{7 f} \]

[Out]

((7*a*d*f*(15*d^2*e^2 - 40*c*d*e*f + 33*c^2*f^2) - b*(105*d^3*e^3 - 280*c*d^2*e^
2*f + 231*c^2*d*e*f^2 - 48*c^3*f^3))*x)/(105*f^4) - ((7*a*d*f*(5*d*e - 9*c*f) -
b*(35*d^2*e^2 - 63*c*d*e*f + 24*c^2*f^2))*x*(c + d*x^2))/(105*f^3) - ((7*b*d*e -
6*b*c*f - 7*a*d*f)*x*(c + d*x^2)^2)/(35*f^2) + (b*x*(c + d*x^2)^3)/(7*f) + ((b*
e - a*f)*(d*e - c*f)^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*f^(9/2))

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Rubi [A]  time = 0.918329, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{x \left (c+d x^2\right ) \left (7 a d f (5 d e-9 c f)-b \left (24 c^2 f^2-63 c d e f+35 d^2 e^2\right )\right )}{105 f^3}+\frac{x \left (7 a d f \left (33 c^2 f^2-40 c d e f+15 d^2 e^2\right )-b \left (-48 c^3 f^3+231 c^2 d e f^2-280 c d^2 e^2 f+105 d^3 e^3\right )\right )}{105 f^4}+\frac{(b e-a f) (d e-c f)^3 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e} f^{9/2}}-\frac{x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{35 f^2}+\frac{b x \left (c+d x^2\right )^3}{7 f} \]

Antiderivative was successfully verified.

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

[Out]

((7*a*d*f*(15*d^2*e^2 - 40*c*d*e*f + 33*c^2*f^2) - b*(105*d^3*e^3 - 280*c*d^2*e^
2*f + 231*c^2*d*e*f^2 - 48*c^3*f^3))*x)/(105*f^4) - ((7*a*d*f*(5*d*e - 9*c*f) -
b*(35*d^2*e^2 - 63*c*d*e*f + 24*c^2*f^2))*x*(c + d*x^2))/(105*f^3) - ((7*b*d*e -
6*b*c*f - 7*a*d*f)*x*(c + d*x^2)^2)/(35*f^2) + (b*x*(c + d*x^2)^3)/(7*f) + ((b*
e - a*f)*(d*e - c*f)^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*f^(9/2))

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Rubi in Sympy [A]  time = 120.378, size = 279, normalized size = 1.23 \[ \frac{b x \left (c + d x^{2}\right )^{3}}{7 f} + \frac{d x \left (c \left (35 a c f^{2} - 7 a d e f - 11 b c e f + 7 b d e^{2}\right ) + x^{2} \left (63 a c d f^{2} - 35 a d^{2} e f + 24 b c^{2} f^{2} - 63 b c d e f + 35 b d^{2} e^{2}\right )\right )}{105 f^{3}} + \frac{x \left (c + d x^{2}\right )^{2} \left (7 a d f + 6 b c f - 7 b d e\right )}{35 f^{2}} + \frac{x \left (259 a c^{2} d f^{3} - 308 a c d^{2} e f^{2} + 105 a d^{3} e^{2} f + 72 b c^{3} f^{3} - 283 b c^{2} d e f^{2} + 308 b c d^{2} e^{2} f - 105 b d^{3} e^{3}\right )}{105 f^{4}} + \frac{\left (a f - b e\right ) \left (c f - d e\right )^{3} \operatorname{atan}{\left (\frac{\sqrt{f} x}{\sqrt{e}} \right )}}{\sqrt{e} f^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x*(c + d*x**2)**3/(7*f) + d*x*(c*(35*a*c*f**2 - 7*a*d*e*f - 11*b*c*e*f + 7*b*d
*e**2) + x**2*(63*a*c*d*f**2 - 35*a*d**2*e*f + 24*b*c**2*f**2 - 63*b*c*d*e*f + 3
5*b*d**2*e**2))/(105*f**3) + x*(c + d*x**2)**2*(7*a*d*f + 6*b*c*f - 7*b*d*e)/(35
*f**2) + x*(259*a*c**2*d*f**3 - 308*a*c*d**2*e*f**2 + 105*a*d**3*e**2*f + 72*b*c
**3*f**3 - 283*b*c**2*d*e*f**2 + 308*b*c*d**2*e**2*f - 105*b*d**3*e**3)/(105*f**
4) + (a*f - b*e)*(c*f - d*e)**3*atan(sqrt(f)*x/sqrt(e))/(sqrt(e)*f**(9/2))

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Mathematica [A]  time = 0.176666, size = 179, normalized size = 0.79 \[ \frac{x \left (a d f \left (3 c^2 f^2-3 c d e f+d^2 e^2\right )-b (d e-c f)^3\right )}{f^4}+\frac{d x^3 \left (a d f (3 c f-d e)+b \left (3 c^2 f^2-3 c d e f+d^2 e^2\right )\right )}{3 f^3}+\frac{d^2 x^5 (a d f+3 b c f-b d e)}{5 f^2}+\frac{(b e-a f) (d e-c f)^3 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e} f^{9/2}}+\frac{b d^3 x^7}{7 f} \]

Antiderivative was successfully verified.

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

[Out]

((-(b*(d*e - c*f)^3) + a*d*f*(d^2*e^2 - 3*c*d*e*f + 3*c^2*f^2))*x)/f^4 + (d*(a*d
*f*(-(d*e) + 3*c*f) + b*(d^2*e^2 - 3*c*d*e*f + 3*c^2*f^2))*x^3)/(3*f^3) + (d^2*(
-(b*d*e) + 3*b*c*f + a*d*f)*x^5)/(5*f^2) + (b*d^3*x^7)/(7*f) + ((b*e - a*f)*(d*e
- c*f)^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*f^(9/2))

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Maple [A]  time = 0.007, size = 401, normalized size = 1.8 \[{\frac{b{d}^{3}{x}^{7}}{7\,f}}+{\frac{{x}^{5}a{d}^{3}}{5\,f}}+{\frac{3\,{x}^{5}bc{d}^{2}}{5\,f}}-{\frac{{x}^{5}b{d}^{3}e}{5\,{f}^{2}}}+{\frac{{x}^{3}ac{d}^{2}}{f}}-{\frac{{x}^{3}a{d}^{3}e}{3\,{f}^{2}}}+{\frac{{x}^{3}b{c}^{2}d}{f}}-{\frac{{x}^{3}bc{d}^{2}e}{{f}^{2}}}+{\frac{{x}^{3}b{d}^{3}{e}^{2}}{3\,{f}^{3}}}+3\,{\frac{a{c}^{2}dx}{f}}-3\,{\frac{ac{d}^{2}ex}{{f}^{2}}}+{\frac{a{d}^{3}{e}^{2}x}{{f}^{3}}}+{\frac{b{c}^{3}x}{f}}-3\,{\frac{b{c}^{2}dex}{{f}^{2}}}+3\,{\frac{bc{d}^{2}{e}^{2}x}{{f}^{3}}}-{\frac{b{d}^{3}{e}^{3}x}{{f}^{4}}}+{a{c}^{3}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}-3\,{\frac{a{c}^{2}de}{f\sqrt{ef}}\arctan \left ({\frac{fx}{\sqrt{ef}}} \right ) }+3\,{\frac{ac{d}^{2}{e}^{2}}{{f}^{2}\sqrt{ef}}\arctan \left ({\frac{fx}{\sqrt{ef}}} \right ) }-{\frac{a{d}^{3}{e}^{3}}{{f}^{3}}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}-{\frac{b{c}^{3}e}{f}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}+3\,{\frac{b{c}^{2}d{e}^{2}}{{f}^{2}\sqrt{ef}}\arctan \left ({\frac{fx}{\sqrt{ef}}} \right ) }-3\,{\frac{bc{d}^{2}{e}^{3}}{{f}^{3}\sqrt{ef}}\arctan \left ({\frac{fx}{\sqrt{ef}}} \right ) }+{\frac{b{d}^{3}{e}^{4}}{{f}^{4}}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7/f*b*d^3*x^7+1/5/f*x^5*a*d^3+3/5/f*x^5*b*c*d^2-1/5/f^2*x^5*b*d^3*e+1/f*x^3*a*
c*d^2-1/3/f^2*x^3*a*d^3*e+1/f*x^3*b*c^2*d-1/f^2*x^3*b*c*d^2*e+1/3/f^3*x^3*b*d^3*
e^2+3/f*a*c^2*d*x-3/f^2*a*c*d^2*e*x+1/f^3*a*d^3*e^2*x+1/f*b*c^3*x-3/f^2*b*c^2*d*
e*x+3/f^3*b*c*d^2*e^2*x-1/f^4*b*d^3*e^3*x+1/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*
a*c^3-3/f/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*a*c^2*d*e+3/f^2/(e*f)^(1/2)*arctan
(x*f/(e*f)^(1/2))*a*c*d^2*e^2-1/f^3/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*a*d^3*e^
3-1/f/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*b*c^3*e+3/f^2/(e*f)^(1/2)*arctan(x*f/(
e*f)^(1/2))*b*c^2*d*e^2-3/f^3/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*b*c*d^2*e^3+1/
f^4/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*b*d^3*e^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)*(d*x^2 + c)^3/(f*x^2 + e),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218182, size = 1, normalized size = 0. \[ \left [\frac{105 \,{\left (b d^{3} e^{4} + a c^{3} f^{4} -{\left (3 \, b c d^{2} + a d^{3}\right )} e^{3} f + 3 \,{\left (b c^{2} d + a c d^{2}\right )} e^{2} f^{2} -{\left (b c^{3} + 3 \, a c^{2} d\right )} e f^{3}\right )} \log \left (\frac{2 \, e f x +{\left (f x^{2} - e\right )} \sqrt{-e f}}{f x^{2} + e}\right ) + 2 \,{\left (15 \, b d^{3} f^{3} x^{7} - 21 \,{\left (b d^{3} e f^{2} -{\left (3 \, b c d^{2} + a d^{3}\right )} f^{3}\right )} x^{5} + 35 \,{\left (b d^{3} e^{2} f -{\left (3 \, b c d^{2} + a d^{3}\right )} e f^{2} + 3 \,{\left (b c^{2} d + a c d^{2}\right )} f^{3}\right )} x^{3} - 105 \,{\left (b d^{3} e^{3} -{\left (3 \, b c d^{2} + a d^{3}\right )} e^{2} f + 3 \,{\left (b c^{2} d + a c d^{2}\right )} e f^{2} -{\left (b c^{3} + 3 \, a c^{2} d\right )} f^{3}\right )} x\right )} \sqrt{-e f}}{210 \, \sqrt{-e f} f^{4}}, \frac{105 \,{\left (b d^{3} e^{4} + a c^{3} f^{4} -{\left (3 \, b c d^{2} + a d^{3}\right )} e^{3} f + 3 \,{\left (b c^{2} d + a c d^{2}\right )} e^{2} f^{2} -{\left (b c^{3} + 3 \, a c^{2} d\right )} e f^{3}\right )} \arctan \left (\frac{\sqrt{e f} x}{e}\right ) +{\left (15 \, b d^{3} f^{3} x^{7} - 21 \,{\left (b d^{3} e f^{2} -{\left (3 \, b c d^{2} + a d^{3}\right )} f^{3}\right )} x^{5} + 35 \,{\left (b d^{3} e^{2} f -{\left (3 \, b c d^{2} + a d^{3}\right )} e f^{2} + 3 \,{\left (b c^{2} d + a c d^{2}\right )} f^{3}\right )} x^{3} - 105 \,{\left (b d^{3} e^{3} -{\left (3 \, b c d^{2} + a d^{3}\right )} e^{2} f + 3 \,{\left (b c^{2} d + a c d^{2}\right )} e f^{2} -{\left (b c^{3} + 3 \, a c^{2} d\right )} f^{3}\right )} x\right )} \sqrt{e f}}{105 \, \sqrt{e f} f^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/210*(105*(b*d^3*e^4 + a*c^3*f^4 - (3*b*c*d^2 + a*d^3)*e^3*f + 3*(b*c^2*d + a*
c*d^2)*e^2*f^2 - (b*c^3 + 3*a*c^2*d)*e*f^3)*log((2*e*f*x + (f*x^2 - e)*sqrt(-e*f
))/(f*x^2 + e)) + 2*(15*b*d^3*f^3*x^7 - 21*(b*d^3*e*f^2 - (3*b*c*d^2 + a*d^3)*f^
3)*x^5 + 35*(b*d^3*e^2*f - (3*b*c*d^2 + a*d^3)*e*f^2 + 3*(b*c^2*d + a*c*d^2)*f^3
)*x^3 - 105*(b*d^3*e^3 - (3*b*c*d^2 + a*d^3)*e^2*f + 3*(b*c^2*d + a*c*d^2)*e*f^2
- (b*c^3 + 3*a*c^2*d)*f^3)*x)*sqrt(-e*f))/(sqrt(-e*f)*f^4), 1/105*(105*(b*d^3*e
^4 + a*c^3*f^4 - (3*b*c*d^2 + a*d^3)*e^3*f + 3*(b*c^2*d + a*c*d^2)*e^2*f^2 - (b*
c^3 + 3*a*c^2*d)*e*f^3)*arctan(sqrt(e*f)*x/e) + (15*b*d^3*f^3*x^7 - 21*(b*d^3*e*
f^2 - (3*b*c*d^2 + a*d^3)*f^3)*x^5 + 35*(b*d^3*e^2*f - (3*b*c*d^2 + a*d^3)*e*f^2
+ 3*(b*c^2*d + a*c*d^2)*f^3)*x^3 - 105*(b*d^3*e^3 - (3*b*c*d^2 + a*d^3)*e^2*f +
3*(b*c^2*d + a*c*d^2)*e*f^2 - (b*c^3 + 3*a*c^2*d)*f^3)*x)*sqrt(e*f))/(sqrt(e*f)
*f^4)]

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Sympy [A]  time = 3.70016, size = 508, normalized size = 2.24 \[ \frac{b d^{3} x^{7}}{7 f} - \frac{\sqrt{- \frac{1}{e f^{9}}} \left (a f - b e\right ) \left (c f - d e\right )^{3} \log{\left (- \frac{e f^{4} \sqrt{- \frac{1}{e f^{9}}} \left (a f - b e\right ) \left (c f - d e\right )^{3}}{a c^{3} f^{4} - 3 a c^{2} d e f^{3} + 3 a c d^{2} e^{2} f^{2} - a d^{3} e^{3} f - b c^{3} e f^{3} + 3 b c^{2} d e^{2} f^{2} - 3 b c d^{2} e^{3} f + b d^{3} e^{4}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{e f^{9}}} \left (a f - b e\right ) \left (c f - d e\right )^{3} \log{\left (\frac{e f^{4} \sqrt{- \frac{1}{e f^{9}}} \left (a f - b e\right ) \left (c f - d e\right )^{3}}{a c^{3} f^{4} - 3 a c^{2} d e f^{3} + 3 a c d^{2} e^{2} f^{2} - a d^{3} e^{3} f - b c^{3} e f^{3} + 3 b c^{2} d e^{2} f^{2} - 3 b c d^{2} e^{3} f + b d^{3} e^{4}} + x \right )}}{2} + \frac{x^{5} \left (a d^{3} f + 3 b c d^{2} f - b d^{3} e\right )}{5 f^{2}} + \frac{x^{3} \left (3 a c d^{2} f^{2} - a d^{3} e f + 3 b c^{2} d f^{2} - 3 b c d^{2} e f + b d^{3} e^{2}\right )}{3 f^{3}} + \frac{x \left (3 a c^{2} d f^{3} - 3 a c d^{2} e f^{2} + a d^{3} e^{2} f + b c^{3} f^{3} - 3 b c^{2} d e f^{2} + 3 b c d^{2} e^{2} f - b d^{3} e^{3}\right )}{f^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*d**3*x**7/(7*f) - sqrt(-1/(e*f**9))*(a*f - b*e)*(c*f - d*e)**3*log(-e*f**4*sqr
t(-1/(e*f**9))*(a*f - b*e)*(c*f - d*e)**3/(a*c**3*f**4 - 3*a*c**2*d*e*f**3 + 3*a
*c*d**2*e**2*f**2 - a*d**3*e**3*f - b*c**3*e*f**3 + 3*b*c**2*d*e**2*f**2 - 3*b*c
*d**2*e**3*f + b*d**3*e**4) + x)/2 + sqrt(-1/(e*f**9))*(a*f - b*e)*(c*f - d*e)**
3*log(e*f**4*sqrt(-1/(e*f**9))*(a*f - b*e)*(c*f - d*e)**3/(a*c**3*f**4 - 3*a*c**
2*d*e*f**3 + 3*a*c*d**2*e**2*f**2 - a*d**3*e**3*f - b*c**3*e*f**3 + 3*b*c**2*d*e
**2*f**2 - 3*b*c*d**2*e**3*f + b*d**3*e**4) + x)/2 + x**5*(a*d**3*f + 3*b*c*d**2
*f - b*d**3*e)/(5*f**2) + x**3*(3*a*c*d**2*f**2 - a*d**3*e*f + 3*b*c**2*d*f**2 -
3*b*c*d**2*e*f + b*d**3*e**2)/(3*f**3) + x*(3*a*c**2*d*f**3 - 3*a*c*d**2*e*f**2
+ a*d**3*e**2*f + b*c**3*f**3 - 3*b*c**2*d*e*f**2 + 3*b*c*d**2*e**2*f - b*d**3*
e**3)/f**4

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GIAC/XCAS [A]  time = 0.229311, size = 414, normalized size = 1.82 \[ \frac{{\left (a c^{3} f^{4} - b c^{3} f^{3} e - 3 \, a c^{2} d f^{3} e + 3 \, b c^{2} d f^{2} e^{2} + 3 \, a c d^{2} f^{2} e^{2} - 3 \, b c d^{2} f e^{3} - a d^{3} f e^{3} + b d^{3} e^{4}\right )} \arctan \left (\sqrt{f} x e^{\left (-\frac{1}{2}\right )}\right ) e^{\left (-\frac{1}{2}\right )}}{f^{\frac{9}{2}}} + \frac{15 \, b d^{3} f^{6} x^{7} + 63 \, b c d^{2} f^{6} x^{5} + 21 \, a d^{3} f^{6} x^{5} - 21 \, b d^{3} f^{5} x^{5} e + 105 \, b c^{2} d f^{6} x^{3} + 105 \, a c d^{2} f^{6} x^{3} - 105 \, b c d^{2} f^{5} x^{3} e - 35 \, a d^{3} f^{5} x^{3} e + 35 \, b d^{3} f^{4} x^{3} e^{2} + 105 \, b c^{3} f^{6} x + 315 \, a c^{2} d f^{6} x - 315 \, b c^{2} d f^{5} x e - 315 \, a c d^{2} f^{5} x e + 315 \, b c d^{2} f^{4} x e^{2} + 105 \, a d^{3} f^{4} x e^{2} - 105 \, b d^{3} f^{3} x e^{3}}{105 \, f^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a*c^3*f^4 - b*c^3*f^3*e - 3*a*c^2*d*f^3*e + 3*b*c^2*d*f^2*e^2 + 3*a*c*d^2*f^2*e
^2 - 3*b*c*d^2*f*e^3 - a*d^3*f*e^3 + b*d^3*e^4)*arctan(sqrt(f)*x*e^(-1/2))*e^(-1
/2)/f^(9/2) + 1/105*(15*b*d^3*f^6*x^7 + 63*b*c*d^2*f^6*x^5 + 21*a*d^3*f^6*x^5 -
21*b*d^3*f^5*x^5*e + 105*b*c^2*d*f^6*x^3 + 105*a*c*d^2*f^6*x^3 - 105*b*c*d^2*f^5
*x^3*e - 35*a*d^3*f^5*x^3*e + 35*b*d^3*f^4*x^3*e^2 + 105*b*c^3*f^6*x + 315*a*c^2
*d*f^6*x - 315*b*c^2*d*f^5*x*e - 315*a*c*d^2*f^5*x*e + 315*b*c*d^2*f^4*x*e^2 + 1
05*a*d^3*f^4*x*e^2 - 105*b*d^3*f^3*x*e^3)/f^7