3.501 \(\int \frac{(d+e x)^5}{\left (a+c x^2\right )^3} \, dx\)

Optimal. Leaf size=198 \[ -\frac{(d+e x)^2 \left (2 a e \left (2 a e^2+c d^2\right )-c d x \left (5 a e^2+3 c d^2\right )\right )}{8 a^2 c^2 \left (a+c x^2\right )}-\frac{d e^2 x \left (7 a e^2+3 c d^2\right )}{8 a^2 c^2}+\frac{d \left (15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{5/2}}+\frac{e^5 \log \left (a+c x^2\right )}{2 c^3}-\frac{(d+e x)^4 (a e-c d x)}{4 a c \left (a+c x^2\right )^2} \]

[Out]

-(d*e^2*(3*c*d^2 + 7*a*e^2)*x)/(8*a^2*c^2) - ((a*e - c*d*x)*(d + e*x)^4)/(4*a*c*
(a + c*x^2)^2) - ((d + e*x)^2*(2*a*e*(c*d^2 + 2*a*e^2) - c*d*(3*c*d^2 + 5*a*e^2)
*x))/(8*a^2*c^2*(a + c*x^2)) + (d*(3*c^2*d^4 + 10*a*c*d^2*e^2 + 15*a^2*e^4)*ArcT
an[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*c^(5/2)) + (e^5*Log[a + c*x^2])/(2*c^3)

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Rubi [A]  time = 0.504413, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ -\frac{(d+e x)^2 \left (2 a e \left (2 a e^2+c d^2\right )-c d x \left (5 a e^2+3 c d^2\right )\right )}{8 a^2 c^2 \left (a+c x^2\right )}-\frac{d e^2 x \left (7 a e^2+3 c d^2\right )}{8 a^2 c^2}+\frac{d \left (15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{5/2}}+\frac{e^5 \log \left (a+c x^2\right )}{2 c^3}-\frac{(d+e x)^4 (a e-c d x)}{4 a c \left (a+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-(d*e^2*(3*c*d^2 + 7*a*e^2)*x)/(8*a^2*c^2) - ((a*e - c*d*x)*(d + e*x)^4)/(4*a*c*
(a + c*x^2)^2) - ((d + e*x)^2*(2*a*e*(c*d^2 + 2*a*e^2) - c*d*(3*c*d^2 + 5*a*e^2)
*x))/(8*a^2*c^2*(a + c*x^2)) + (d*(3*c^2*d^4 + 10*a*c*d^2*e^2 + 15*a^2*e^4)*ArcT
an[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*c^(5/2)) + (e^5*Log[a + c*x^2])/(2*c^3)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{e^{5} \log{\left (a + c x^{2} \right )}}{2 c^{3}} - \frac{\left (d + e x\right )^{4} \left (a e - c d x\right )}{4 a c \left (a + c x^{2}\right )^{2}} - \frac{e^{2} \left (7 a e^{2} + 3 c d^{2}\right ) \int d\, dx}{8 a^{2} c^{2}} - \frac{\left (d + e x\right )^{2} \left (4 a e \left (2 a e^{2} + c d^{2}\right ) - 2 c d x \left (5 a e^{2} + 3 c d^{2}\right )\right )}{16 a^{2} c^{2} \left (a + c x^{2}\right )} + \frac{d \left (15 a^{2} e^{4} + 10 a c d^{2} e^{2} + 3 c^{2} d^{4}\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e**5*log(a + c*x**2)/(2*c**3) - (d + e*x)**4*(a*e - c*d*x)/(4*a*c*(a + c*x**2)**
2) - e**2*(7*a*e**2 + 3*c*d**2)*Integral(d, x)/(8*a**2*c**2) - (d + e*x)**2*(4*a
*e*(2*a*e**2 + c*d**2) - 2*c*d*x*(5*a*e**2 + 3*c*d**2))/(16*a**2*c**2*(a + c*x**
2)) + d*(15*a**2*e**4 + 10*a*c*d**2*e**2 + 3*c**2*d**4)*atan(sqrt(c)*x/sqrt(a))/
(8*a**(5/2)*c**(5/2))

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Mathematica [A]  time = 0.305542, size = 199, normalized size = 1.01 \[ \frac{\frac{\sqrt{c} d \left (15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{5/2}}-\frac{2 \left (a^3 e^5-5 a^2 c d e^3 (2 d+e x)+5 a c^2 d^3 e (d+2 e x)-c^3 d^5 x\right )}{a \left (a+c x^2\right )^2}+\frac{8 a^3 e^5-5 a^2 c d e^3 (8 d+5 e x)+10 a c^2 d^3 e^2 x+3 c^3 d^5 x}{a^2 \left (a+c x^2\right )}+4 e^5 \log \left (a+c x^2\right )}{8 c^3} \]

Antiderivative was successfully verified.

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

[Out]

((-2*(a^3*e^5 - c^3*d^5*x - 5*a^2*c*d*e^3*(2*d + e*x) + 5*a*c^2*d^3*e*(d + 2*e*x
)))/(a*(a + c*x^2)^2) + (8*a^3*e^5 + 3*c^3*d^5*x + 10*a*c^2*d^3*e^2*x - 5*a^2*c*
d*e^3*(8*d + 5*e*x))/(a^2*(a + c*x^2)) + (Sqrt[c]*d*(3*c^2*d^4 + 10*a*c*d^2*e^2
+ 15*a^2*e^4)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(5/2) + 4*e^5*Log[a + c*x^2])/(8*c^
3)

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Maple [A]  time = 0.018, size = 240, normalized size = 1.2 \[{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{2}} \left ( -{\frac{d \left ( 25\,{a}^{2}{e}^{4}-10\,ac{d}^{2}{e}^{2}-3\,{c}^{2}{d}^{4} \right ){x}^{3}}{8\,{a}^{2}c}}+{\frac{{e}^{3} \left ( a{e}^{2}-5\,c{d}^{2} \right ){x}^{2}}{{c}^{2}}}-{\frac{5\,d \left ( 3\,{a}^{2}{e}^{4}+2\,ac{d}^{2}{e}^{2}-{c}^{2}{d}^{4} \right ) x}{8\,a{c}^{2}}}+{\frac{e \left ( 3\,{a}^{2}{e}^{4}-10\,ac{d}^{2}{e}^{2}-5\,{c}^{2}{d}^{4} \right ) }{4\,{c}^{3}}} \right ) }+{\frac{{e}^{5}\ln \left ({a}^{2}{c}^{2} \left ( c{x}^{2}+a \right ) \right ) }{2\,{c}^{3}}}+{\frac{15\,d{e}^{4}}{8\,{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{5\,{d}^{3}{e}^{2}}{4\,ac}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,{d}^{5}}{8\,{a}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^5/(c*x^2+a)^3,x)

[Out]

(-1/8*d*(25*a^2*e^4-10*a*c*d^2*e^2-3*c^2*d^4)/a^2/c*x^3+e^3*(a*e^2-5*c*d^2)/c^2*
x^2-5/8*d*(3*a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/a/c^2*x+1/4*e*(3*a^2*e^4-10*a*c*d^2*
e^2-5*c^2*d^4)/c^3)/(c*x^2+a)^2+1/2*e^5/c^3*ln(a^2*c^2*(c*x^2+a))+15/8/c^2/(a*c)
^(1/2)*arctan(c*x/(a*c)^(1/2))*d*e^4+5/4/a/c/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))
*d^3*e^2+3/8/a^2/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*d^5

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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((e*x + d)^5/(c*x^2 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220954, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (3 \, a^{2} c^{3} d^{5} + 10 \, a^{3} c^{2} d^{3} e^{2} + 15 \, a^{4} c d e^{4} +{\left (3 \, c^{5} d^{5} + 10 \, a c^{4} d^{3} e^{2} + 15 \, a^{2} c^{3} d e^{4}\right )} x^{4} + 2 \,{\left (3 \, a c^{4} d^{5} + 10 \, a^{2} c^{3} d^{3} e^{2} + 15 \, a^{3} c^{2} d e^{4}\right )} x^{2}\right )} \log \left (\frac{2 \, a c x +{\left (c x^{2} - a\right )} \sqrt{-a c}}{c x^{2} + a}\right ) - 2 \,{\left (10 \, a^{2} c^{2} d^{4} e + 20 \, a^{3} c d^{2} e^{3} - 6 \, a^{4} e^{5} -{\left (3 \, c^{4} d^{5} + 10 \, a c^{3} d^{3} e^{2} - 25 \, a^{2} c^{2} d e^{4}\right )} x^{3} + 8 \,{\left (5 \, a^{2} c^{2} d^{2} e^{3} - a^{3} c e^{5}\right )} x^{2} - 5 \,{\left (a c^{3} d^{5} - 2 \, a^{2} c^{2} d^{3} e^{2} - 3 \, a^{3} c d e^{4}\right )} x - 4 \,{\left (a^{2} c^{2} e^{5} x^{4} + 2 \, a^{3} c e^{5} x^{2} + a^{4} e^{5}\right )} \log \left (c x^{2} + a\right )\right )} \sqrt{-a c}}{16 \,{\left (a^{2} c^{5} x^{4} + 2 \, a^{3} c^{4} x^{2} + a^{4} c^{3}\right )} \sqrt{-a c}}, \frac{{\left (3 \, a^{2} c^{3} d^{5} + 10 \, a^{3} c^{2} d^{3} e^{2} + 15 \, a^{4} c d e^{4} +{\left (3 \, c^{5} d^{5} + 10 \, a c^{4} d^{3} e^{2} + 15 \, a^{2} c^{3} d e^{4}\right )} x^{4} + 2 \,{\left (3 \, a c^{4} d^{5} + 10 \, a^{2} c^{3} d^{3} e^{2} + 15 \, a^{3} c^{2} d e^{4}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) -{\left (10 \, a^{2} c^{2} d^{4} e + 20 \, a^{3} c d^{2} e^{3} - 6 \, a^{4} e^{5} -{\left (3 \, c^{4} d^{5} + 10 \, a c^{3} d^{3} e^{2} - 25 \, a^{2} c^{2} d e^{4}\right )} x^{3} + 8 \,{\left (5 \, a^{2} c^{2} d^{2} e^{3} - a^{3} c e^{5}\right )} x^{2} - 5 \,{\left (a c^{3} d^{5} - 2 \, a^{2} c^{2} d^{3} e^{2} - 3 \, a^{3} c d e^{4}\right )} x - 4 \,{\left (a^{2} c^{2} e^{5} x^{4} + 2 \, a^{3} c e^{5} x^{2} + a^{4} e^{5}\right )} \log \left (c x^{2} + a\right )\right )} \sqrt{a c}}{8 \,{\left (a^{2} c^{5} x^{4} + 2 \, a^{3} c^{4} x^{2} + a^{4} c^{3}\right )} \sqrt{a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*((3*a^2*c^3*d^5 + 10*a^3*c^2*d^3*e^2 + 15*a^4*c*d*e^4 + (3*c^5*d^5 + 10*a*
c^4*d^3*e^2 + 15*a^2*c^3*d*e^4)*x^4 + 2*(3*a*c^4*d^5 + 10*a^2*c^3*d^3*e^2 + 15*a
^3*c^2*d*e^4)*x^2)*log((2*a*c*x + (c*x^2 - a)*sqrt(-a*c))/(c*x^2 + a)) - 2*(10*a
^2*c^2*d^4*e + 20*a^3*c*d^2*e^3 - 6*a^4*e^5 - (3*c^4*d^5 + 10*a*c^3*d^3*e^2 - 25
*a^2*c^2*d*e^4)*x^3 + 8*(5*a^2*c^2*d^2*e^3 - a^3*c*e^5)*x^2 - 5*(a*c^3*d^5 - 2*a
^2*c^2*d^3*e^2 - 3*a^3*c*d*e^4)*x - 4*(a^2*c^2*e^5*x^4 + 2*a^3*c*e^5*x^2 + a^4*e
^5)*log(c*x^2 + a))*sqrt(-a*c))/((a^2*c^5*x^4 + 2*a^3*c^4*x^2 + a^4*c^3)*sqrt(-a
*c)), 1/8*((3*a^2*c^3*d^5 + 10*a^3*c^2*d^3*e^2 + 15*a^4*c*d*e^4 + (3*c^5*d^5 + 1
0*a*c^4*d^3*e^2 + 15*a^2*c^3*d*e^4)*x^4 + 2*(3*a*c^4*d^5 + 10*a^2*c^3*d^3*e^2 +
15*a^3*c^2*d*e^4)*x^2)*arctan(sqrt(a*c)*x/a) - (10*a^2*c^2*d^4*e + 20*a^3*c*d^2*
e^3 - 6*a^4*e^5 - (3*c^4*d^5 + 10*a*c^3*d^3*e^2 - 25*a^2*c^2*d*e^4)*x^3 + 8*(5*a
^2*c^2*d^2*e^3 - a^3*c*e^5)*x^2 - 5*(a*c^3*d^5 - 2*a^2*c^2*d^3*e^2 - 3*a^3*c*d*e
^4)*x - 4*(a^2*c^2*e^5*x^4 + 2*a^3*c*e^5*x^2 + a^4*e^5)*log(c*x^2 + a))*sqrt(a*c
))/((a^2*c^5*x^4 + 2*a^3*c^4*x^2 + a^4*c^3)*sqrt(a*c))]

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Sympy [A]  time = 14.6996, size = 520, normalized size = 2.63 \[ \left (\frac{e^{5}}{2 c^{3}} - \frac{d \sqrt{- a^{5} c^{7}} \left (15 a^{2} e^{4} + 10 a c d^{2} e^{2} + 3 c^{2} d^{4}\right )}{16 a^{5} c^{6}}\right ) \log{\left (x + \frac{16 a^{3} c^{3} \left (\frac{e^{5}}{2 c^{3}} - \frac{d \sqrt{- a^{5} c^{7}} \left (15 a^{2} e^{4} + 10 a c d^{2} e^{2} + 3 c^{2} d^{4}\right )}{16 a^{5} c^{6}}\right ) - 8 a^{3} e^{5}}{15 a^{2} c d e^{4} + 10 a c^{2} d^{3} e^{2} + 3 c^{3} d^{5}} \right )} + \left (\frac{e^{5}}{2 c^{3}} + \frac{d \sqrt{- a^{5} c^{7}} \left (15 a^{2} e^{4} + 10 a c d^{2} e^{2} + 3 c^{2} d^{4}\right )}{16 a^{5} c^{6}}\right ) \log{\left (x + \frac{16 a^{3} c^{3} \left (\frac{e^{5}}{2 c^{3}} + \frac{d \sqrt{- a^{5} c^{7}} \left (15 a^{2} e^{4} + 10 a c d^{2} e^{2} + 3 c^{2} d^{4}\right )}{16 a^{5} c^{6}}\right ) - 8 a^{3} e^{5}}{15 a^{2} c d e^{4} + 10 a c^{2} d^{3} e^{2} + 3 c^{3} d^{5}} \right )} - \frac{- 6 a^{4} e^{5} + 20 a^{3} c d^{2} e^{3} + 10 a^{2} c^{2} d^{4} e + x^{3} \left (25 a^{2} c^{2} d e^{4} - 10 a c^{3} d^{3} e^{2} - 3 c^{4} d^{5}\right ) + x^{2} \left (- 8 a^{3} c e^{5} + 40 a^{2} c^{2} d^{2} e^{3}\right ) + x \left (15 a^{3} c d e^{4} + 10 a^{2} c^{2} d^{3} e^{2} - 5 a c^{3} d^{5}\right )}{8 a^{4} c^{3} + 16 a^{3} c^{4} x^{2} + 8 a^{2} c^{5} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(e**5/(2*c**3) - d*sqrt(-a**5*c**7)*(15*a**2*e**4 + 10*a*c*d**2*e**2 + 3*c**2*d*
*4)/(16*a**5*c**6))*log(x + (16*a**3*c**3*(e**5/(2*c**3) - d*sqrt(-a**5*c**7)*(1
5*a**2*e**4 + 10*a*c*d**2*e**2 + 3*c**2*d**4)/(16*a**5*c**6)) - 8*a**3*e**5)/(15
*a**2*c*d*e**4 + 10*a*c**2*d**3*e**2 + 3*c**3*d**5)) + (e**5/(2*c**3) + d*sqrt(-
a**5*c**7)*(15*a**2*e**4 + 10*a*c*d**2*e**2 + 3*c**2*d**4)/(16*a**5*c**6))*log(x
+ (16*a**3*c**3*(e**5/(2*c**3) + d*sqrt(-a**5*c**7)*(15*a**2*e**4 + 10*a*c*d**2
*e**2 + 3*c**2*d**4)/(16*a**5*c**6)) - 8*a**3*e**5)/(15*a**2*c*d*e**4 + 10*a*c**
2*d**3*e**2 + 3*c**3*d**5)) - (-6*a**4*e**5 + 20*a**3*c*d**2*e**3 + 10*a**2*c**2
*d**4*e + x**3*(25*a**2*c**2*d*e**4 - 10*a*c**3*d**3*e**2 - 3*c**4*d**5) + x**2*
(-8*a**3*c*e**5 + 40*a**2*c**2*d**2*e**3) + x*(15*a**3*c*d*e**4 + 10*a**2*c**2*d
**3*e**2 - 5*a*c**3*d**5))/(8*a**4*c**3 + 16*a**3*c**4*x**2 + 8*a**2*c**5*x**4)

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GIAC/XCAS [A]  time = 0.21546, size = 279, normalized size = 1.41 \[ \frac{e^{5}{\rm ln}\left (c x^{2} + a\right )}{2 \, c^{3}} + \frac{{\left (3 \, c^{2} d^{5} + 10 \, a c d^{3} e^{2} + 15 \, a^{2} d e^{4}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{8 \, \sqrt{a c} a^{2} c^{2}} + \frac{{\left (3 \, c^{3} d^{5} + 10 \, a c^{2} d^{3} e^{2} - 25 \, a^{2} c d e^{4}\right )} x^{3} - 8 \,{\left (5 \, a^{2} c d^{2} e^{3} - a^{3} e^{5}\right )} x^{2} + 5 \,{\left (a c^{2} d^{5} - 2 \, a^{2} c d^{3} e^{2} - 3 \, a^{3} d e^{4}\right )} x - \frac{2 \,{\left (5 \, a^{2} c^{2} d^{4} e + 10 \, a^{3} c d^{2} e^{3} - 3 \, a^{4} e^{5}\right )}}{c}}{8 \,{\left (c x^{2} + a\right )}^{2} a^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*e^5*ln(c*x^2 + a)/c^3 + 1/8*(3*c^2*d^5 + 10*a*c*d^3*e^2 + 15*a^2*d*e^4)*arct
an(c*x/sqrt(a*c))/(sqrt(a*c)*a^2*c^2) + 1/8*((3*c^3*d^5 + 10*a*c^2*d^3*e^2 - 25*
a^2*c*d*e^4)*x^3 - 8*(5*a^2*c*d^2*e^3 - a^3*e^5)*x^2 + 5*(a*c^2*d^5 - 2*a^2*c*d^
3*e^2 - 3*a^3*d*e^4)*x - 2*(5*a^2*c^2*d^4*e + 10*a^3*c*d^2*e^3 - 3*a^4*e^5)/c)/(
(c*x^2 + a)^2*a^2*c^2)