3.22 \(\int \frac{\left (a+b x^2\right ) \left (c+d x^2\right )^3}{\left (e+f x^2\right )^4} \, dx\)

Optimal. Leaf size=348 \[ \frac{d x \left (b e \left (-3 c^2 f^2-10 c d e f+105 d^2 e^2\right )-a f \left (15 c^2 f^2+14 c d e f+15 d^2 e^2\right )\right )}{48 e^3 f^4}-\frac{x \left (c+d x^2\right ) \left (b e \left (-3 c^2 f^2-8 c d e f+35 d^2 e^2\right )-a f \left (15 c^2 f^2+4 c d e f+5 d^2 e^2\right )\right )}{48 e^3 f^3 \left (e+f x^2\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (b e \left (-c^3 f^3-3 c^2 d e f^2-15 c d^2 e^2 f+35 d^3 e^3\right )-a f \left (5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right )\right )}{16 e^{7/2} f^{9/2}}-\frac{x \left (c+d x^2\right )^2 (b e (7 d e-c f)-a f (5 c f+d e))}{24 e^2 f^2 \left (e+f x^2\right )^2}-\frac{x \left (c+d x^2\right )^3 (b e-a f)}{6 e f \left (e+f x^2\right )^3} \]

[Out]

(d*(b*e*(105*d^2*e^2 - 10*c*d*e*f - 3*c^2*f^2) - a*f*(15*d^2*e^2 + 14*c*d*e*f +
15*c^2*f^2))*x)/(48*e^3*f^4) - ((b*e - a*f)*x*(c + d*x^2)^3)/(6*e*f*(e + f*x^2)^
3) - ((b*e*(7*d*e - c*f) - a*f*(d*e + 5*c*f))*x*(c + d*x^2)^2)/(24*e^2*f^2*(e +
f*x^2)^2) - ((b*e*(35*d^2*e^2 - 8*c*d*e*f - 3*c^2*f^2) - a*f*(5*d^2*e^2 + 4*c*d*
e*f + 15*c^2*f^2))*x*(c + d*x^2))/(48*e^3*f^3*(e + f*x^2)) - ((b*e*(35*d^3*e^3 -
15*c*d^2*e^2*f - 3*c^2*d*e*f^2 - c^3*f^3) - a*f*(5*d^3*e^3 + 3*c*d^2*e^2*f + 3*
c^2*d*e*f^2 + 5*c^3*f^3))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(16*e^(7/2)*f^(9/2))

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Rubi [A]  time = 1.17798, antiderivative size = 348, 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 \[ \frac{d x \left (b e \left (-3 c^2 f^2-10 c d e f+105 d^2 e^2\right )-a f \left (15 c^2 f^2+14 c d e f+15 d^2 e^2\right )\right )}{48 e^3 f^4}-\frac{x \left (c+d x^2\right ) \left (b e \left (-3 c^2 f^2-8 c d e f+35 d^2 e^2\right )-a f \left (15 c^2 f^2+4 c d e f+5 d^2 e^2\right )\right )}{48 e^3 f^3 \left (e+f x^2\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (b e \left (-c^3 f^3-3 c^2 d e f^2-15 c d^2 e^2 f+35 d^3 e^3\right )-a f \left (5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right )\right )}{16 e^{7/2} f^{9/2}}-\frac{x \left (c+d x^2\right )^2 (b e (7 d e-c f)-a f (5 c f+d e))}{24 e^2 f^2 \left (e+f x^2\right )^2}-\frac{x \left (c+d x^2\right )^3 (b e-a f)}{6 e f \left (e+f x^2\right )^3} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)*(c + d*x^2)^3)/(e + f*x^2)^4,x]

[Out]

(d*(b*e*(105*d^2*e^2 - 10*c*d*e*f - 3*c^2*f^2) - a*f*(15*d^2*e^2 + 14*c*d*e*f +
15*c^2*f^2))*x)/(48*e^3*f^4) - ((b*e - a*f)*x*(c + d*x^2)^3)/(6*e*f*(e + f*x^2)^
3) - ((b*e*(7*d*e - c*f) - a*f*(d*e + 5*c*f))*x*(c + d*x^2)^2)/(24*e^2*f^2*(e +
f*x^2)^2) - ((b*e*(35*d^2*e^2 - 8*c*d*e*f - 3*c^2*f^2) - a*f*(5*d^2*e^2 + 4*c*d*
e*f + 15*c^2*f^2))*x*(c + d*x^2))/(48*e^3*f^3*(e + f*x^2)) - ((b*e*(35*d^3*e^3 -
15*c*d^2*e^2*f - 3*c^2*d*e*f^2 - c^3*f^3) - a*f*(5*d^3*e^3 + 3*c*d^2*e^2*f + 3*
c^2*d*e*f^2 + 5*c^3*f^3))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(16*e^(7/2)*f^(9/2))

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Rubi in Sympy [A]  time = 148.443, size = 359, normalized size = 1.03 \[ - \frac{d x \left (c f \left (3 c f \left (5 a f + b e\right ) - d e \left (a f - 7 b e\right )\right ) + 3 d e \left (c f \left (5 a f + b e\right ) + 5 d e \left (a f - 7 b e\right )\right )\right )}{48 e^{3} f^{4}} + \frac{x \left (c + d x^{2}\right )^{3} \left (a f - b e\right )}{6 e f \left (e + f x^{2}\right )^{3}} + \frac{x \left (c + d x^{2}\right )^{2} \left (c f \left (5 a f + b e\right ) + d e \left (a f - 7 b e\right )\right )}{24 e^{2} f^{2} \left (e + f x^{2}\right )^{2}} + \frac{x \left (c + d x^{2}\right ) \left (c f \left (3 c f \left (5 a f + b e\right ) - d e \left (a f - 7 b e\right )\right ) + d e \left (c f \left (5 a f + b e\right ) + 5 d e \left (a f - 7 b e\right )\right )\right )}{48 e^{3} f^{3} \left (e + f x^{2}\right )} + \frac{\left (5 a c^{3} f^{4} + 3 a c^{2} d e f^{3} + 3 a c d^{2} e^{2} f^{2} + 5 a d^{3} e^{3} f + b c^{3} e f^{3} + 3 b c^{2} d e^{2} f^{2} + 15 b c d^{2} e^{3} f - 35 b d^{3} e^{4}\right ) \operatorname{atan}{\left (\frac{\sqrt{f} x}{\sqrt{e}} \right )}}{16 e^{\frac{7}{2}} f^{\frac{9}{2}}} \]

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)**4,x)

[Out]

-d*x*(c*f*(3*c*f*(5*a*f + b*e) - d*e*(a*f - 7*b*e)) + 3*d*e*(c*f*(5*a*f + b*e) +
5*d*e*(a*f - 7*b*e)))/(48*e**3*f**4) + x*(c + d*x**2)**3*(a*f - b*e)/(6*e*f*(e
+ f*x**2)**3) + x*(c + d*x**2)**2*(c*f*(5*a*f + b*e) + d*e*(a*f - 7*b*e))/(24*e*
*2*f**2*(e + f*x**2)**2) + x*(c + d*x**2)*(c*f*(3*c*f*(5*a*f + b*e) - d*e*(a*f -
7*b*e)) + d*e*(c*f*(5*a*f + b*e) + 5*d*e*(a*f - 7*b*e)))/(48*e**3*f**3*(e + f*x
**2)) + (5*a*c**3*f**4 + 3*a*c**2*d*e*f**3 + 3*a*c*d**2*e**2*f**2 + 5*a*d**3*e**
3*f + b*c**3*e*f**3 + 3*b*c**2*d*e**2*f**2 + 15*b*c*d**2*e**3*f - 35*b*d**3*e**4
)*atan(sqrt(f)*x/sqrt(e))/(16*e**(7/2)*f**(9/2))

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Mathematica [A]  time = 0.42651, size = 295, normalized size = 0.85 \[ \frac{x (d e-c f) \left (b e \left (-c^2 f^2-4 c d e f+29 d^2 e^2\right )-a f \left (5 c^2 f^2+8 c d e f+11 d^2 e^2\right )\right )}{16 e^3 f^4 \left (e+f x^2\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (b e \left (-c^3 f^3-3 c^2 d e f^2-15 c d^2 e^2 f+35 d^3 e^3\right )-a f \left (5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right )\right )}{16 e^{7/2} f^{9/2}}-\frac{x (d e-c f)^2 (b e (19 d e-c f)-a f (5 c f+13 d e))}{24 e^2 f^4 \left (e+f x^2\right )^2}+\frac{x (b e-a f) (d e-c f)^3}{6 e f^4 \left (e+f x^2\right )^3}+\frac{b d^3 x}{f^4} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)*(c + d*x^2)^3)/(e + f*x^2)^4,x]

[Out]

(b*d^3*x)/f^4 + ((b*e - a*f)*(d*e - c*f)^3*x)/(6*e*f^4*(e + f*x^2)^3) - ((d*e -
c*f)^2*(b*e*(19*d*e - c*f) - a*f*(13*d*e + 5*c*f))*x)/(24*e^2*f^4*(e + f*x^2)^2)
+ ((d*e - c*f)*(b*e*(29*d^2*e^2 - 4*c*d*e*f - c^2*f^2) - a*f*(11*d^2*e^2 + 8*c*
d*e*f + 5*c^2*f^2))*x)/(16*e^3*f^4*(e + f*x^2)) - ((b*e*(35*d^3*e^3 - 15*c*d^2*e
^2*f - 3*c^2*d*e*f^2 - c^3*f^3) - a*f*(5*d^3*e^3 + 3*c*d^2*e^2*f + 3*c^2*d*e*f^2
+ 5*c^3*f^3))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(16*e^(7/2)*f^(9/2))

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Maple [B]  time = 0.021, size = 735, normalized size = 2.1 \[ \text{result too large to display} \]

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)^4,x)

[Out]

11/16/(f*x^2+e)^3/e*x*a*c^3-3/16/f/(f*x^2+e)^3*a*c^2*d*x-5/16/f^3/(f*x^2+e)^3*a*
d^3*e^2*x+15/16/f^3/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*b*c*d^2+1/16/f/e^2/(e*f)
^(1/2)*arctan(x*f/(e*f)^(1/2))*b*c^3-35/16/f^4*e/(e*f)^(1/2)*arctan(x*f/(e*f)^(1
/2))*b*d^3+19/16/f^4/(f*x^2+e)^3*b*d^3*e^3*x+1/2/(f*x^2+e)^3/e*x^3*a*c^2*d+3/16/
(f*x^2+e)^3/e*x^5*b*c^2*d-1/2/f/(f*x^2+e)^3*x^3*b*c^2*d+17/6/f^3/(f*x^2+e)^3*x^3
*b*d^3*e^2+3/16/(f*x^2+e)^3/e*x^5*a*c*d^2+5/16*f^2/(f*x^2+e)^3/e^3*x^5*a*c^3+1/1
6*f/(f*x^2+e)^3/e^2*x^5*b*c^3-33/16/f/(f*x^2+e)^3*x^5*b*c*d^2+29/16/f^2/(f*x^2+e
)^3*x^5*b*d^3*e+5/6*f/(f*x^2+e)^3/e^2*x^3*a*c^3-1/2/f/(f*x^2+e)^3*x^3*a*c*d^2-5/
6/f^2/(f*x^2+e)^3*x^3*a*d^3*e+3/16/f/e^2/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*a*c
^2*d-5/2/f^2/(f*x^2+e)^3*x^3*b*c*d^2*e+b*d^3/f^4*x-3/16/f^2/(f*x^2+e)^3*a*c*d^2*
e*x-3/16/f^2/(f*x^2+e)^3*b*c^2*d*e*x+5/16/e^3/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2)
)*a*c^3+5/16/f^3/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*a*d^3+1/6/(f*x^2+e)^3/e*x^3
*b*c^3-11/16/f/(f*x^2+e)^3*x^5*a*d^3-1/16/f/(f*x^2+e)^3*b*c^3*x+3/16/f^2/e/(e*f)
^(1/2)*arctan(x*f/(e*f)^(1/2))*b*c^2*d-15/16/f^3/(f*x^2+e)^3*b*c*d^2*e^2*x+3/16*
f/(f*x^2+e)^3/e^2*x^5*a*c^2*d+3/16/f^2/e/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*a*c
*d^2

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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)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229092, size = 1, normalized size = 0. \[ \text{result too large to display} \]

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)^4,x, algorithm="fricas")

[Out]

[-1/96*(3*(35*b*d^3*e^7 - 5*a*c^3*e^3*f^4 - 5*(3*b*c*d^2 + a*d^3)*e^6*f - 3*(b*c
^2*d + a*c*d^2)*e^5*f^2 - (b*c^3 + 3*a*c^2*d)*e^4*f^3 + (35*b*d^3*e^4*f^3 - 5*a*
c^3*f^7 - 5*(3*b*c*d^2 + a*d^3)*e^3*f^4 - 3*(b*c^2*d + a*c*d^2)*e^2*f^5 - (b*c^3
+ 3*a*c^2*d)*e*f^6)*x^6 + 3*(35*b*d^3*e^5*f^2 - 5*a*c^3*e*f^6 - 5*(3*b*c*d^2 +
a*d^3)*e^4*f^3 - 3*(b*c^2*d + a*c*d^2)*e^3*f^4 - (b*c^3 + 3*a*c^2*d)*e^2*f^5)*x^
4 + 3*(35*b*d^3*e^6*f - 5*a*c^3*e^2*f^5 - 5*(3*b*c*d^2 + a*d^3)*e^5*f^2 - 3*(b*c
^2*d + a*c*d^2)*e^4*f^3 - (b*c^3 + 3*a*c^2*d)*e^3*f^4)*x^2)*log((2*e*f*x + (f*x^
2 - e)*sqrt(-e*f))/(f*x^2 + e)) - 2*(48*b*d^3*e^3*f^3*x^7 + 3*(77*b*d^3*e^4*f^2
+ 5*a*c^3*f^6 - 11*(3*b*c*d^2 + a*d^3)*e^3*f^3 + 3*(b*c^2*d + a*c*d^2)*e^2*f^4 +
(b*c^3 + 3*a*c^2*d)*e*f^5)*x^5 + 8*(35*b*d^3*e^5*f + 5*a*c^3*e*f^5 - 5*(3*b*c*d
^2 + a*d^3)*e^4*f^2 - 3*(b*c^2*d + a*c*d^2)*e^3*f^3 + (b*c^3 + 3*a*c^2*d)*e^2*f^
4)*x^3 + 3*(35*b*d^3*e^6 + 11*a*c^3*e^2*f^4 - 5*(3*b*c*d^2 + a*d^3)*e^5*f - 3*(b
*c^2*d + a*c*d^2)*e^4*f^2 - (b*c^3 + 3*a*c^2*d)*e^3*f^3)*x)*sqrt(-e*f))/((e^3*f^
7*x^6 + 3*e^4*f^6*x^4 + 3*e^5*f^5*x^2 + e^6*f^4)*sqrt(-e*f)), -1/48*(3*(35*b*d^3
*e^7 - 5*a*c^3*e^3*f^4 - 5*(3*b*c*d^2 + a*d^3)*e^6*f - 3*(b*c^2*d + a*c*d^2)*e^5
*f^2 - (b*c^3 + 3*a*c^2*d)*e^4*f^3 + (35*b*d^3*e^4*f^3 - 5*a*c^3*f^7 - 5*(3*b*c*
d^2 + a*d^3)*e^3*f^4 - 3*(b*c^2*d + a*c*d^2)*e^2*f^5 - (b*c^3 + 3*a*c^2*d)*e*f^6
)*x^6 + 3*(35*b*d^3*e^5*f^2 - 5*a*c^3*e*f^6 - 5*(3*b*c*d^2 + a*d^3)*e^4*f^3 - 3*
(b*c^2*d + a*c*d^2)*e^3*f^4 - (b*c^3 + 3*a*c^2*d)*e^2*f^5)*x^4 + 3*(35*b*d^3*e^6
*f - 5*a*c^3*e^2*f^5 - 5*(3*b*c*d^2 + a*d^3)*e^5*f^2 - 3*(b*c^2*d + a*c*d^2)*e^4
*f^3 - (b*c^3 + 3*a*c^2*d)*e^3*f^4)*x^2)*arctan(sqrt(e*f)*x/e) - (48*b*d^3*e^3*f
^3*x^7 + 3*(77*b*d^3*e^4*f^2 + 5*a*c^3*f^6 - 11*(3*b*c*d^2 + a*d^3)*e^3*f^3 + 3*
(b*c^2*d + a*c*d^2)*e^2*f^4 + (b*c^3 + 3*a*c^2*d)*e*f^5)*x^5 + 8*(35*b*d^3*e^5*f
+ 5*a*c^3*e*f^5 - 5*(3*b*c*d^2 + a*d^3)*e^4*f^2 - 3*(b*c^2*d + a*c*d^2)*e^3*f^3
+ (b*c^3 + 3*a*c^2*d)*e^2*f^4)*x^3 + 3*(35*b*d^3*e^6 + 11*a*c^3*e^2*f^4 - 5*(3*
b*c*d^2 + a*d^3)*e^5*f - 3*(b*c^2*d + a*c*d^2)*e^4*f^2 - (b*c^3 + 3*a*c^2*d)*e^3
*f^3)*x)*sqrt(e*f))/((e^3*f^7*x^6 + 3*e^4*f^6*x^4 + 3*e^5*f^5*x^2 + e^6*f^4)*sqr
t(e*f))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.223699, size = 603, normalized size = 1.73 \[ \frac{b d^{3} x}{f^{4}} + \frac{{\left (5 \, a c^{3} f^{4} + b c^{3} f^{3} e + 3 \, a c^{2} d f^{3} e + 3 \, b c^{2} d f^{2} e^{2} + 3 \, a c d^{2} f^{2} e^{2} + 15 \, b c d^{2} f e^{3} + 5 \, a d^{3} f e^{3} - 35 \, b d^{3} e^{4}\right )} \arctan \left (\sqrt{f} x e^{\left (-\frac{1}{2}\right )}\right ) e^{\left (-\frac{7}{2}\right )}}{16 \, f^{\frac{9}{2}}} + \frac{{\left (15 \, a c^{3} f^{6} x^{5} + 3 \, b c^{3} f^{5} x^{5} e + 9 \, a c^{2} d f^{5} x^{5} e + 9 \, b c^{2} d f^{4} x^{5} e^{2} + 9 \, a c d^{2} f^{4} x^{5} e^{2} - 99 \, b c d^{2} f^{3} x^{5} e^{3} - 33 \, a d^{3} f^{3} x^{5} e^{3} + 40 \, a c^{3} f^{5} x^{3} e + 87 \, b d^{3} f^{2} x^{5} e^{4} + 8 \, b c^{3} f^{4} x^{3} e^{2} + 24 \, a c^{2} d f^{4} x^{3} e^{2} - 24 \, b c^{2} d f^{3} x^{3} e^{3} - 24 \, a c d^{2} f^{3} x^{3} e^{3} - 120 \, b c d^{2} f^{2} x^{3} e^{4} - 40 \, a d^{3} f^{2} x^{3} e^{4} + 33 \, a c^{3} f^{4} x e^{2} + 136 \, b d^{3} f x^{3} e^{5} - 3 \, b c^{3} f^{3} x e^{3} - 9 \, a c^{2} d f^{3} x e^{3} - 9 \, b c^{2} d f^{2} x e^{4} - 9 \, a c d^{2} f^{2} x e^{4} - 45 \, b c d^{2} f x e^{5} - 15 \, a d^{3} f x e^{5} + 57 \, b d^{3} x e^{6}\right )} e^{\left (-3\right )}}{48 \,{\left (f x^{2} + e\right )}^{3} f^{4}} \]

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)^4,x, algorithm="giac")

[Out]

b*d^3*x/f^4 + 1/16*(5*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 + 15*b*c*d^2*f*e^3 + 5*a*d^3*f*e^3 - 35*b*d^3*e^4)*arcta
n(sqrt(f)*x*e^(-1/2))*e^(-7/2)/f^(9/2) + 1/48*(15*a*c^3*f^6*x^5 + 3*b*c^3*f^5*x^
5*e + 9*a*c^2*d*f^5*x^5*e + 9*b*c^2*d*f^4*x^5*e^2 + 9*a*c*d^2*f^4*x^5*e^2 - 99*b
*c*d^2*f^3*x^5*e^3 - 33*a*d^3*f^3*x^5*e^3 + 40*a*c^3*f^5*x^3*e + 87*b*d^3*f^2*x^
5*e^4 + 8*b*c^3*f^4*x^3*e^2 + 24*a*c^2*d*f^4*x^3*e^2 - 24*b*c^2*d*f^3*x^3*e^3 -
24*a*c*d^2*f^3*x^3*e^3 - 120*b*c*d^2*f^2*x^3*e^4 - 40*a*d^3*f^2*x^3*e^4 + 33*a*c
^3*f^4*x*e^2 + 136*b*d^3*f*x^3*e^5 - 3*b*c^3*f^3*x*e^3 - 9*a*c^2*d*f^3*x*e^3 - 9
*b*c^2*d*f^2*x*e^4 - 9*a*c*d^2*f^2*x*e^4 - 45*b*c*d^2*f*x*e^5 - 15*a*d^3*f*x*e^5
+ 57*b*d^3*x*e^6)*e^(-3)/((f*x^2 + e)^3*f^4)