3.506 \(\int \frac{1}{(d+e x) \left (a+c x^2\right )^3} \, dx\)

Optimal. Leaf size=212 \[ \frac{4 a^2 e^3+c d x \left (7 a e^2+3 c d^2\right )}{8 a^2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )^2}+\frac{\sqrt{c} d \left (15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} \left (a e^2+c d^2\right )^3}+\frac{a e+c d x}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )}-\frac{e^5 \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )^3}+\frac{e^5 \log (d+e x)}{\left (a e^2+c d^2\right )^3} \]

[Out]

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

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Rubi [A]  time = 0.56224, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ \frac{4 a^2 e^3+c d x \left (7 a e^2+3 c d^2\right )}{8 a^2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )^2}+\frac{\sqrt{c} d \left (15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} \left (a e^2+c d^2\right )^3}+\frac{a e+c d x}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )}-\frac{e^5 \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )^3}+\frac{e^5 \log (d+e x)}{\left (a e^2+c d^2\right )^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)*(a + c*x^2)^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 93.5621, size = 196, normalized size = 0.92 \[ - \frac{e^{5} \log{\left (a + c x^{2} \right )}}{2 \left (a e^{2} + c d^{2}\right )^{3}} + \frac{e^{5} \log{\left (d + e x \right )}}{\left (a e^{2} + c d^{2}\right )^{3}} + \frac{a e + c d x}{4 a \left (a + c x^{2}\right )^{2} \left (a e^{2} + c d^{2}\right )} + \frac{4 a^{2} e^{3} + c d x \left (7 a e^{2} + 3 c d^{2}\right )}{8 a^{2} \left (a + c x^{2}\right ) \left (a e^{2} + c d^{2}\right )^{2}} + \frac{\sqrt{c} d \left (15 a^{2} e^{4} + 10 a c d^{2} e^{2} + 3 c^{2} d^{4}\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}} \left (a e^{2} + c d^{2}\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)/(c*x**2+a)**3,x)

[Out]

-e**5*log(a + c*x**2)/(2*(a*e**2 + c*d**2)**3) + e**5*log(d + e*x)/(a*e**2 + c*d
**2)**3 + (a*e + c*d*x)/(4*a*(a + c*x**2)**2*(a*e**2 + c*d**2)) + (4*a**2*e**3 +
c*d*x*(7*a*e**2 + 3*c*d**2))/(8*a**2*(a + c*x**2)*(a*e**2 + c*d**2)**2) + sqrt(
c)*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)*(a*e**2 + c*d**2)**3)

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Mathematica [A]  time = 0.275862, size = 180, normalized size = 0.85 \[ \frac{\frac{\left (a e^2+c d^2\right ) \left (4 a^2 e^3+7 a c d e^2 x+3 c^2 d^3 x\right )}{a^2 \left (a+c x^2\right )}+\frac{\sqrt{c} d \left (15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{5/2}}+\frac{2 \left (a e^2+c d^2\right )^2 (a e+c d x)}{a \left (a+c x^2\right )^2}-4 e^5 \log \left (a+c x^2\right )+8 e^5 \log (d+e x)}{8 \left (a e^2+c d^2\right )^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)*(a + c*x^2)^3),x]

[Out]

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

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Maple [B]  time = 0.024, size = 536, normalized size = 2.5 \[{\frac{{e}^{5}\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}}+{\frac{7\,{c}^{2}d{x}^{3}{e}^{4}}{8\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3} \left ( c{x}^{2}+a \right ) ^{2}}}+{\frac{5\,{c}^{3}{d}^{3}{x}^{3}{e}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3} \left ( c{x}^{2}+a \right ) ^{2}a}}+{\frac{3\,{c}^{4}{d}^{5}{x}^{3}}{8\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3} \left ( c{x}^{2}+a \right ) ^{2}{a}^{2}}}+{\frac{c{x}^{2}a{e}^{5}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3} \left ( c{x}^{2}+a \right ) ^{2}}}+{\frac{{c}^{2}{x}^{2}{d}^{2}{e}^{3}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3} \left ( c{x}^{2}+a \right ) ^{2}}}+{\frac{9\,acdx{e}^{4}}{8\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3} \left ( c{x}^{2}+a \right ) ^{2}}}+{\frac{7\,{c}^{2}{d}^{3}x{e}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3} \left ( c{x}^{2}+a \right ) ^{2}}}+{\frac{5\,{c}^{3}{d}^{5}x}{8\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3} \left ( c{x}^{2}+a \right ) ^{2}a}}+{\frac{3\,{a}^{2}{e}^{5}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3} \left ( c{x}^{2}+a \right ) ^{2}}}+{\frac{ac{d}^{2}{e}^{3}}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{3} \left ( c{x}^{2}+a \right ) ^{2}}}+{\frac{{c}^{2}{d}^{4}e}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3} \left ( c{x}^{2}+a \right ) ^{2}}}-{\frac{{e}^{5}\ln \left ({a}^{2} \left ( c{x}^{2}+a \right ) \right ) }{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}}+{\frac{15\,d{e}^{4}c}{8\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{5\,{d}^{3}{e}^{2}{c}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}a}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,{c}^{3}{d}^{5}}{8\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}{a}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)/(c*x^2+a)^3,x)

[Out]

e^5*ln(e*x+d)/(a*e^2+c*d^2)^3+7/8*c^2/(a*e^2+c*d^2)^3/(c*x^2+a)^2*d*x^3*e^4+5/4*
c^3/(a*e^2+c*d^2)^3/(c*x^2+a)^2*d^3/a*x^3*e^2+3/8*c^4/(a*e^2+c*d^2)^3/(c*x^2+a)^
2*d^5/a^2*x^3+1/2*c/(a*e^2+c*d^2)^3/(c*x^2+a)^2*x^2*a*e^5+1/2*c^2/(a*e^2+c*d^2)^
3/(c*x^2+a)^2*x^2*d^2*e^3+9/8*c/(a*e^2+c*d^2)^3/(c*x^2+a)^2*d*a*x*e^4+7/4*c^2/(a
*e^2+c*d^2)^3/(c*x^2+a)^2*d^3*x*e^2+5/8*c^3/(a*e^2+c*d^2)^3/(c*x^2+a)^2*d^5/a*x+
3/4/(a*e^2+c*d^2)^3/(c*x^2+a)^2*e^5*a^2+c/(a*e^2+c*d^2)^3/(c*x^2+a)^2*e^3*a*d^2+
1/4*c^2/(a*e^2+c*d^2)^3/(c*x^2+a)^2*e*d^4-1/2/(a*e^2+c*d^2)^3*e^5*ln(a^2*(c*x^2+
a))+15/8*c/(a*e^2+c*d^2)^3/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*d*e^4+5/4*c^2/(a*
e^2+c*d^2)^3/a/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*d^3*e^2+3/8*c^3/(a*e^2+c*d^2)
^3/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(1/((c*x^2 + a)^3*(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.47348, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)^3*(e*x + d)),x, algorithm="fricas")

[Out]

[1/16*(4*a^2*c^2*d^4*e + 16*a^3*c*d^2*e^3 + 12*a^4*e^5 + 2*(3*c^4*d^5 + 10*a*c^3
*d^3*e^2 + 7*a^2*c^2*d*e^4)*x^3 + 8*(a^2*c^2*d^2*e^3 + a^3*c*e^5)*x^2 + (3*a^2*c
^2*d^5 + 10*a^3*c*d^3*e^2 + 15*a^4*d*e^4 + (3*c^4*d^5 + 10*a*c^3*d^3*e^2 + 15*a^
2*c^2*d*e^4)*x^4 + 2*(3*a*c^3*d^5 + 10*a^2*c^2*d^3*e^2 + 15*a^3*c*d*e^4)*x^2)*sq
rt(-c/a)*log((c*x^2 + 2*a*x*sqrt(-c/a) - a)/(c*x^2 + a)) + 2*(5*a*c^3*d^5 + 14*a
^2*c^2*d^3*e^2 + 9*a^3*c*d*e^4)*x - 8*(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) + 16*(a^2*c^2*e^5*x^4 + 2*a^3*c*e^5*x^2 + a^4*e^5)*log(e*x +
d))/(a^4*c^3*d^6 + 3*a^5*c^2*d^4*e^2 + 3*a^6*c*d^2*e^4 + a^7*e^6 + (a^2*c^5*d^6
+ 3*a^3*c^4*d^4*e^2 + 3*a^4*c^3*d^2*e^4 + a^5*c^2*e^6)*x^4 + 2*(a^3*c^4*d^6 + 3*
a^4*c^3*d^4*e^2 + 3*a^5*c^2*d^2*e^4 + a^6*c*e^6)*x^2), 1/8*(2*a^2*c^2*d^4*e + 8*
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 + 7*a^2*c^2*d*e^4)*x^3
+ 4*(a^2*c^2*d^2*e^3 + a^3*c*e^5)*x^2 + (3*a^2*c^2*d^5 + 10*a^3*c*d^3*e^2 + 15*
a^4*d*e^4 + (3*c^4*d^5 + 10*a*c^3*d^3*e^2 + 15*a^2*c^2*d*e^4)*x^4 + 2*(3*a*c^3*d
^5 + 10*a^2*c^2*d^3*e^2 + 15*a^3*c*d*e^4)*x^2)*sqrt(c/a)*arctan(c*x/(a*sqrt(c/a)
)) + (5*a*c^3*d^5 + 14*a^2*c^2*d^3*e^2 + 9*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) + 8*(a^2*c^2*e^5*x^4 + 2*a^3*c*e^5*x^
2 + a^4*e^5)*log(e*x + d))/(a^4*c^3*d^6 + 3*a^5*c^2*d^4*e^2 + 3*a^6*c*d^2*e^4 +
a^7*e^6 + (a^2*c^5*d^6 + 3*a^3*c^4*d^4*e^2 + 3*a^4*c^3*d^2*e^4 + a^5*c^2*e^6)*x^
4 + 2*(a^3*c^4*d^6 + 3*a^4*c^3*d^4*e^2 + 3*a^5*c^2*d^2*e^4 + a^6*c*e^6)*x^2)]

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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(1/(e*x+d)/(c*x**2+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.215141, size = 462, normalized size = 2.18 \[ -\frac{e^{5}{\rm ln}\left (c x^{2} + a\right )}{2 \,{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}} + \frac{e^{6}{\rm ln}\left ({\left | x e + d \right |}\right )}{c^{3} d^{6} e + 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} + a^{3} e^{7}} + \frac{{\left (3 \, c^{3} d^{5} + 10 \, a c^{2} d^{3} e^{2} + 15 \, a^{2} c d e^{4}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{8 \,{\left (a^{2} c^{3} d^{6} + 3 \, a^{3} c^{2} d^{4} e^{2} + 3 \, a^{4} c d^{2} e^{4} + a^{5} e^{6}\right )} \sqrt{a c}} + \frac{2 \, a^{2} c^{2} d^{4} e + 8 \, a^{3} c d^{2} e^{3} + 6 \, a^{4} e^{5} +{\left (3 \, c^{4} d^{5} + 10 \, a c^{3} d^{3} e^{2} + 7 \, a^{2} c^{2} d e^{4}\right )} x^{3} + 4 \,{\left (a^{2} c^{2} d^{2} e^{3} + a^{3} c e^{5}\right )} x^{2} +{\left (5 \, a c^{3} d^{5} + 14 \, a^{2} c^{2} d^{3} e^{2} + 9 \, a^{3} c d e^{4}\right )} x}{8 \,{\left (c d^{2} + a e^{2}\right )}^{3}{\left (c x^{2} + a\right )}^{2} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)^3*(e*x + d)),x, algorithm="giac")

[Out]

-1/2*e^5*ln(c*x^2 + a)/(c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6) +
e^6*ln(abs(x*e + d))/(c^3*d^6*e + 3*a*c^2*d^4*e^3 + 3*a^2*c*d^2*e^5 + a^3*e^7)
+ 1/8*(3*c^3*d^5 + 10*a*c^2*d^3*e^2 + 15*a^2*c*d*e^4)*arctan(c*x/sqrt(a*c))/((a^
2*c^3*d^6 + 3*a^3*c^2*d^4*e^2 + 3*a^4*c*d^2*e^4 + a^5*e^6)*sqrt(a*c)) + 1/8*(2*a
^2*c^2*d^4*e + 8*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 + 7*a
^2*c^2*d*e^4)*x^3 + 4*(a^2*c^2*d^2*e^3 + a^3*c*e^5)*x^2 + (5*a*c^3*d^5 + 14*a^2*
c^2*d^3*e^2 + 9*a^3*c*d*e^4)*x)/((c*d^2 + a*e^2)^3*(c*x^2 + a)^2*a^2)