3.1339 \(\int \frac{(A+B x) (d+e x)^4}{\left (a+c x^2\right )^2} \, dx\)

Optimal. Leaf size=220 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )+4 a B d e \left (c d^2-3 a e^2\right )\right )}{2 a^{3/2} c^{5/2}}+\frac{e^2 \log \left (a+c x^2\right ) \left (-a B e^2+2 A c d e+3 B c d^2\right )}{c^3}-\frac{3 e^2 x \left (-a A e^2-4 a B d e+A c d^2\right )}{2 a c^2}-\frac{e^3 x^2 (A c d-2 a B e)}{2 a c^2}-\frac{(d+e x)^3 (a (A e+B d)-x (A c d-a B e))}{2 a c \left (a+c x^2\right )} \]

[Out]

(-3*e^2*(A*c*d^2 - 4*a*B*d*e - a*A*e^2)*x)/(2*a*c^2) - (e^3*(A*c*d - 2*a*B*e)*x^
2)/(2*a*c^2) - ((d + e*x)^3*(a*(B*d + A*e) - (A*c*d - a*B*e)*x))/(2*a*c*(a + c*x
^2)) + ((4*a*B*d*e*(c*d^2 - 3*a*e^2) + A*(c^2*d^4 + 6*a*c*d^2*e^2 - 3*a^2*e^4))*
ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(3/2)*c^(5/2)) + (e^2*(3*B*c*d^2 + 2*A*c*d*e -
a*B*e^2)*Log[a + c*x^2])/c^3

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Rubi [A]  time = 0.585645, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )+4 a B d e \left (c d^2-3 a e^2\right )\right )}{2 a^{3/2} c^{5/2}}+\frac{e^2 \log \left (a+c x^2\right ) \left (-a B e^2+2 A c d e+3 B c d^2\right )}{c^3}-\frac{3 e^2 x \left (-a A e^2-4 a B d e+A c d^2\right )}{2 a c^2}-\frac{e^3 x^2 (A c d-2 a B e)}{2 a c^2}-\frac{(d+e x)^3 (a (A e+B d)-x (A c d-a B e))}{2 a c \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^4)/(a + c*x^2)^2,x]

[Out]

(-3*e^2*(A*c*d^2 - 4*a*B*d*e - a*A*e^2)*x)/(2*a*c^2) - (e^3*(A*c*d - 2*a*B*e)*x^
2)/(2*a*c^2) - ((d + e*x)^3*(a*(B*d + A*e) - (A*c*d - a*B*e)*x))/(2*a*c*(a + c*x
^2)) + ((4*a*B*d*e*(c*d^2 - 3*a*e^2) + A*(c^2*d^4 + 6*a*c*d^2*e^2 - 3*a^2*e^4))*
ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(3/2)*c^(5/2)) + (e^2*(3*B*c*d^2 + 2*A*c*d*e -
a*B*e^2)*Log[a + c*x^2])/c^3

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{e^{2} \left (- 2 A c d e + B a e^{2} - 3 B c d^{2}\right ) \log{\left (a + c x^{2} \right )}}{c^{3}} - \frac{\left (d + e x\right )^{3} \left (2 a \left (A e + B d\right ) - x \left (2 A c d - 2 B a e\right )\right )}{4 a c \left (a + c x^{2}\right )} - \frac{e^{3} \left (A c d - 2 B a e\right ) \int x\, dx}{a c^{2}} + \frac{3 e^{2} x \left (A a e^{2} - A c d^{2} + 4 B a d e\right )}{2 a c^{2}} - \frac{\left (3 A a^{2} e^{4} - 6 A a c d^{2} e^{2} - A c^{2} d^{4} + 12 B a^{2} d e^{3} - 4 B a c d^{3} e\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**4/(c*x**2+a)**2,x)

[Out]

-e**2*(-2*A*c*d*e + B*a*e**2 - 3*B*c*d**2)*log(a + c*x**2)/c**3 - (d + e*x)**3*(
2*a*(A*e + B*d) - x*(2*A*c*d - 2*B*a*e))/(4*a*c*(a + c*x**2)) - e**3*(A*c*d - 2*
B*a*e)*Integral(x, x)/(a*c**2) + 3*e**2*x*(A*a*e**2 - A*c*d**2 + 4*B*a*d*e)/(2*a
*c**2) - (3*A*a**2*e**4 - 6*A*a*c*d**2*e**2 - A*c**2*d**4 + 12*B*a**2*d*e**3 - 4
*B*a*c*d**3*e)*atan(sqrt(c)*x/sqrt(a))/(2*a**(3/2)*c**(5/2))

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Mathematica [A]  time = 0.442418, size = 231, normalized size = 1.05 \[ \frac{\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )+4 a B d e \left (c d^2-3 a e^2\right )\right )}{a^{3/2}}+\frac{-a^3 B e^4+a^2 c e^2 (A e (4 d+e x)+2 B d (3 d+2 e x))-a c^2 d^2 (2 A e (2 d+3 e x)+B d (d+4 e x))+A c^3 d^4 x}{a \left (a+c x^2\right )}+2 e^2 \log \left (a+c x^2\right ) \left (-a B e^2+2 A c d e+3 B c d^2\right )+2 c e^3 x (A e+4 B d)+B c e^4 x^2}{2 c^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^4)/(a + c*x^2)^2,x]

[Out]

(2*c*e^3*(4*B*d + A*e)*x + B*c*e^4*x^2 + (-(a^3*B*e^4) + A*c^3*d^4*x + a^2*c*e^2
*(A*e*(4*d + e*x) + 2*B*d*(3*d + 2*e*x)) - a*c^2*d^2*(2*A*e*(2*d + 3*e*x) + B*d*
(d + 4*e*x)))/(a*(a + c*x^2)) + (Sqrt[c]*(4*a*B*d*e*(c*d^2 - 3*a*e^2) + A*(c^2*d
^4 + 6*a*c*d^2*e^2 - 3*a^2*e^4))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(3/2) + 2*e^2*(3
*B*c*d^2 + 2*A*c*d*e - a*B*e^2)*Log[a + c*x^2])/(2*c^3)

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Maple [B]  time = 0.015, size = 420, normalized size = 1.9 \[{\frac{B{e}^{4}{x}^{2}}{2\,{c}^{2}}}+{\frac{A{e}^{4}x}{{c}^{2}}}+4\,{\frac{Bd{e}^{3}x}{{c}^{2}}}+{\frac{aAx{e}^{4}}{2\,{c}^{2} \left ( c{x}^{2}+a \right ) }}-3\,{\frac{xA{d}^{2}{e}^{2}}{c \left ( c{x}^{2}+a \right ) }}+{\frac{xA{d}^{4}}{ \left ( 2\,c{x}^{2}+2\,a \right ) a}}+2\,{\frac{aBxd{e}^{3}}{{c}^{2} \left ( c{x}^{2}+a \right ) }}-2\,{\frac{Bx{d}^{3}e}{c \left ( c{x}^{2}+a \right ) }}+2\,{\frac{Aad{e}^{3}}{{c}^{2} \left ( c{x}^{2}+a \right ) }}-2\,{\frac{{d}^{3}Ae}{c \left ( c{x}^{2}+a \right ) }}-{\frac{B{e}^{4}{a}^{2}}{2\,{c}^{3} \left ( c{x}^{2}+a \right ) }}+3\,{\frac{Ba{d}^{2}{e}^{2}}{{c}^{2} \left ( c{x}^{2}+a \right ) }}-{\frac{B{d}^{4}}{2\,c \left ( c{x}^{2}+a \right ) }}+2\,{\frac{\ln \left ( a \left ( c{x}^{2}+a \right ) \right ) Ad{e}^{3}}{{c}^{2}}}-{\frac{a\ln \left ( a \left ( c{x}^{2}+a \right ) \right ) B{e}^{4}}{{c}^{3}}}+3\,{\frac{\ln \left ( a \left ( c{x}^{2}+a \right ) \right ) B{d}^{2}{e}^{2}}{{c}^{2}}}-{\frac{3\,Aa{e}^{4}}{2\,{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+3\,{\frac{A{d}^{2}{e}^{2}}{c\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) }+{\frac{{d}^{4}A}{2\,a}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-6\,{\frac{Bad{e}^{3}}{{c}^{2}\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) }+2\,{\frac{B{d}^{3}e}{c\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^4/(c*x^2+a)^2,x)

[Out]

1/2*B*e^4*x^2/c^2+e^4/c^2*A*x+4*e^3/c^2*B*d*x+1/2/c^2/(c*x^2+a)*a*x*A*e^4-3/c/(c
*x^2+a)*x*A*d^2*e^2+1/2/(c*x^2+a)/a*x*A*d^4+2/c^2/(c*x^2+a)*a*x*B*d*e^3-2/c/(c*x
^2+a)*x*B*d^3*e+2/c^2/(c*x^2+a)*A*a*d*e^3-2/c/(c*x^2+a)*A*d^3*e-1/2/c^3/(c*x^2+a
)*B*e^4*a^2+3/c^2/(c*x^2+a)*B*a*d^2*e^2-1/2/c/(c*x^2+a)*B*d^4+2/c^2*ln(a*(c*x^2+
a))*A*d*e^3-1/c^3*a*ln(a*(c*x^2+a))*B*e^4+3/c^2*ln(a*(c*x^2+a))*B*d^2*e^2-3/2/c^
2*a/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*A*e^4+3/c/(a*c)^(1/2)*arctan(c*x/(a*c)^(
1/2))*A*d^2*e^2+1/2/a/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*A*d^4-6/c^2*a/(a*c)^(1
/2)*arctan(c*x/(a*c)^(1/2))*B*d*e^3+2/c/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*B*d^
3*e

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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 + A)*(e*x + d)^4/(c*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.284241, 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 + A)*(e*x + d)^4/(c*x^2 + a)^2,x, algorithm="fricas")

[Out]

[-1/4*((A*a*c^3*d^4 + 4*B*a^2*c^2*d^3*e + 6*A*a^2*c^2*d^2*e^2 - 12*B*a^3*c*d*e^3
- 3*A*a^3*c*e^4 + (A*c^4*d^4 + 4*B*a*c^3*d^3*e + 6*A*a*c^3*d^2*e^2 - 12*B*a^2*c
^2*d*e^3 - 3*A*a^2*c^2*e^4)*x^2)*log(-(2*a*c*x - (c*x^2 - a)*sqrt(-a*c))/(c*x^2
+ a)) - 2*(B*a*c^2*e^4*x^4 + B*a^2*c*e^4*x^2 - B*a*c^2*d^4 - 4*A*a*c^2*d^3*e + 6
*B*a^2*c*d^2*e^2 + 4*A*a^2*c*d*e^3 - B*a^3*e^4 + 2*(4*B*a*c^2*d*e^3 + A*a*c^2*e^
4)*x^3 + (A*c^3*d^4 - 4*B*a*c^2*d^3*e - 6*A*a*c^2*d^2*e^2 + 12*B*a^2*c*d*e^3 + 3
*A*a^2*c*e^4)*x + 2*(3*B*a^2*c*d^2*e^2 + 2*A*a^2*c*d*e^3 - B*a^3*e^4 + (3*B*a*c^
2*d^2*e^2 + 2*A*a*c^2*d*e^3 - B*a^2*c*e^4)*x^2)*log(c*x^2 + a))*sqrt(-a*c))/((a*
c^4*x^2 + a^2*c^3)*sqrt(-a*c)), 1/2*((A*a*c^3*d^4 + 4*B*a^2*c^2*d^3*e + 6*A*a^2*
c^2*d^2*e^2 - 12*B*a^3*c*d*e^3 - 3*A*a^3*c*e^4 + (A*c^4*d^4 + 4*B*a*c^3*d^3*e +
6*A*a*c^3*d^2*e^2 - 12*B*a^2*c^2*d*e^3 - 3*A*a^2*c^2*e^4)*x^2)*arctan(sqrt(a*c)*
x/a) + (B*a*c^2*e^4*x^4 + B*a^2*c*e^4*x^2 - B*a*c^2*d^4 - 4*A*a*c^2*d^3*e + 6*B*
a^2*c*d^2*e^2 + 4*A*a^2*c*d*e^3 - B*a^3*e^4 + 2*(4*B*a*c^2*d*e^3 + A*a*c^2*e^4)*
x^3 + (A*c^3*d^4 - 4*B*a*c^2*d^3*e - 6*A*a*c^2*d^2*e^2 + 12*B*a^2*c*d*e^3 + 3*A*
a^2*c*e^4)*x + 2*(3*B*a^2*c*d^2*e^2 + 2*A*a^2*c*d*e^3 - B*a^3*e^4 + (3*B*a*c^2*d
^2*e^2 + 2*A*a*c^2*d*e^3 - B*a^2*c*e^4)*x^2)*log(c*x^2 + a))*sqrt(a*c))/((a*c^4*
x^2 + a^2*c^3)*sqrt(a*c))]

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Sympy [A]  time = 42.6613, size = 833, normalized size = 3.79 \[ \frac{B e^{4} x^{2}}{2 c^{2}} + \left (- \frac{e^{2} \left (- 2 A c d e + B a e^{2} - 3 B c d^{2}\right )}{c^{3}} - \frac{\sqrt{- a^{3} c^{7}} \left (3 A a^{2} e^{4} - 6 A a c d^{2} e^{2} - A c^{2} d^{4} + 12 B a^{2} d e^{3} - 4 B a c d^{3} e\right )}{4 a^{3} c^{6}}\right ) \log{\left (x + \frac{8 A a^{2} c d e^{3} - 4 B a^{3} e^{4} + 12 B a^{2} c d^{2} e^{2} - 4 a^{2} c^{3} \left (- \frac{e^{2} \left (- 2 A c d e + B a e^{2} - 3 B c d^{2}\right )}{c^{3}} - \frac{\sqrt{- a^{3} c^{7}} \left (3 A a^{2} e^{4} - 6 A a c d^{2} e^{2} - A c^{2} d^{4} + 12 B a^{2} d e^{3} - 4 B a c d^{3} e\right )}{4 a^{3} c^{6}}\right )}{3 A a^{2} c e^{4} - 6 A a c^{2} d^{2} e^{2} - A c^{3} d^{4} + 12 B a^{2} c d e^{3} - 4 B a c^{2} d^{3} e} \right )} + \left (- \frac{e^{2} \left (- 2 A c d e + B a e^{2} - 3 B c d^{2}\right )}{c^{3}} + \frac{\sqrt{- a^{3} c^{7}} \left (3 A a^{2} e^{4} - 6 A a c d^{2} e^{2} - A c^{2} d^{4} + 12 B a^{2} d e^{3} - 4 B a c d^{3} e\right )}{4 a^{3} c^{6}}\right ) \log{\left (x + \frac{8 A a^{2} c d e^{3} - 4 B a^{3} e^{4} + 12 B a^{2} c d^{2} e^{2} - 4 a^{2} c^{3} \left (- \frac{e^{2} \left (- 2 A c d e + B a e^{2} - 3 B c d^{2}\right )}{c^{3}} + \frac{\sqrt{- a^{3} c^{7}} \left (3 A a^{2} e^{4} - 6 A a c d^{2} e^{2} - A c^{2} d^{4} + 12 B a^{2} d e^{3} - 4 B a c d^{3} e\right )}{4 a^{3} c^{6}}\right )}{3 A a^{2} c e^{4} - 6 A a c^{2} d^{2} e^{2} - A c^{3} d^{4} + 12 B a^{2} c d e^{3} - 4 B a c^{2} d^{3} e} \right )} + \frac{4 A a^{2} c d e^{3} - 4 A a c^{2} d^{3} e - B a^{3} e^{4} + 6 B a^{2} c d^{2} e^{2} - B a c^{2} d^{4} + x \left (A a^{2} c e^{4} - 6 A a c^{2} d^{2} e^{2} + A c^{3} d^{4} + 4 B a^{2} c d e^{3} - 4 B a c^{2} d^{3} e\right )}{2 a^{2} c^{3} + 2 a c^{4} x^{2}} + \frac{x \left (A e^{4} + 4 B d e^{3}\right )}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**4/(c*x**2+a)**2,x)

[Out]

B*e**4*x**2/(2*c**2) + (-e**2*(-2*A*c*d*e + B*a*e**2 - 3*B*c*d**2)/c**3 - sqrt(-
a**3*c**7)*(3*A*a**2*e**4 - 6*A*a*c*d**2*e**2 - A*c**2*d**4 + 12*B*a**2*d*e**3 -
4*B*a*c*d**3*e)/(4*a**3*c**6))*log(x + (8*A*a**2*c*d*e**3 - 4*B*a**3*e**4 + 12*
B*a**2*c*d**2*e**2 - 4*a**2*c**3*(-e**2*(-2*A*c*d*e + B*a*e**2 - 3*B*c*d**2)/c**
3 - sqrt(-a**3*c**7)*(3*A*a**2*e**4 - 6*A*a*c*d**2*e**2 - A*c**2*d**4 + 12*B*a**
2*d*e**3 - 4*B*a*c*d**3*e)/(4*a**3*c**6)))/(3*A*a**2*c*e**4 - 6*A*a*c**2*d**2*e*
*2 - A*c**3*d**4 + 12*B*a**2*c*d*e**3 - 4*B*a*c**2*d**3*e)) + (-e**2*(-2*A*c*d*e
+ B*a*e**2 - 3*B*c*d**2)/c**3 + sqrt(-a**3*c**7)*(3*A*a**2*e**4 - 6*A*a*c*d**2*
e**2 - A*c**2*d**4 + 12*B*a**2*d*e**3 - 4*B*a*c*d**3*e)/(4*a**3*c**6))*log(x + (
8*A*a**2*c*d*e**3 - 4*B*a**3*e**4 + 12*B*a**2*c*d**2*e**2 - 4*a**2*c**3*(-e**2*(
-2*A*c*d*e + B*a*e**2 - 3*B*c*d**2)/c**3 + sqrt(-a**3*c**7)*(3*A*a**2*e**4 - 6*A
*a*c*d**2*e**2 - A*c**2*d**4 + 12*B*a**2*d*e**3 - 4*B*a*c*d**3*e)/(4*a**3*c**6))
)/(3*A*a**2*c*e**4 - 6*A*a*c**2*d**2*e**2 - A*c**3*d**4 + 12*B*a**2*c*d*e**3 - 4
*B*a*c**2*d**3*e)) + (4*A*a**2*c*d*e**3 - 4*A*a*c**2*d**3*e - B*a**3*e**4 + 6*B*
a**2*c*d**2*e**2 - B*a*c**2*d**4 + x*(A*a**2*c*e**4 - 6*A*a*c**2*d**2*e**2 + A*c
**3*d**4 + 4*B*a**2*c*d*e**3 - 4*B*a*c**2*d**3*e))/(2*a**2*c**3 + 2*a*c**4*x**2)
+ x*(A*e**4 + 4*B*d*e**3)/c**2

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GIAC/XCAS [A]  time = 0.286696, size = 352, normalized size = 1.6 \[ \frac{{\left (3 \, B c d^{2} e^{2} + 2 \, A c d e^{3} - B a e^{4}\right )}{\rm ln}\left (c x^{2} + a\right )}{c^{3}} + \frac{{\left (A c^{2} d^{4} + 4 \, B a c d^{3} e + 6 \, A a c d^{2} e^{2} - 12 \, B a^{2} d e^{3} - 3 \, A a^{2} e^{4}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} a c^{2}} + \frac{B c^{2} x^{2} e^{4} + 8 \, B c^{2} d x e^{3} + 2 \, A c^{2} x e^{4}}{2 \, c^{4}} - \frac{B a c^{2} d^{4} + 4 \, A a c^{2} d^{3} e - 6 \, B a^{2} c d^{2} e^{2} - 4 \, A a^{2} c d e^{3} + B a^{3} e^{4} -{\left (A c^{3} d^{4} - 4 \, B a c^{2} d^{3} e - 6 \, A a c^{2} d^{2} e^{2} + 4 \, B a^{2} c d e^{3} + A a^{2} c e^{4}\right )} x}{2 \,{\left (c x^{2} + a\right )} a c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^4/(c*x^2 + a)^2,x, algorithm="giac")

[Out]

(3*B*c*d^2*e^2 + 2*A*c*d*e^3 - B*a*e^4)*ln(c*x^2 + a)/c^3 + 1/2*(A*c^2*d^4 + 4*B
*a*c*d^3*e + 6*A*a*c*d^2*e^2 - 12*B*a^2*d*e^3 - 3*A*a^2*e^4)*arctan(c*x/sqrt(a*c
))/(sqrt(a*c)*a*c^2) + 1/2*(B*c^2*x^2*e^4 + 8*B*c^2*d*x*e^3 + 2*A*c^2*x*e^4)/c^4
- 1/2*(B*a*c^2*d^4 + 4*A*a*c^2*d^3*e - 6*B*a^2*c*d^2*e^2 - 4*A*a^2*c*d*e^3 + B*
a^3*e^4 - (A*c^3*d^4 - 4*B*a*c^2*d^3*e - 6*A*a*c^2*d^2*e^2 + 4*B*a^2*c*d*e^3 + A
*a^2*c*e^4)*x)/((c*x^2 + a)*a*c^3)