3.557 \(\int \frac{1}{(d+e x)^4 \sqrt{a+c x^2}} \, dx\)

Optimal. Leaf size=198 \[ -\frac{c^2 d \left (2 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{7/2}}-\frac{c e \sqrt{a+c x^2} \left (11 c d^2-4 a e^2\right )}{6 (d+e x) \left (a e^2+c d^2\right )^3}-\frac{5 c d e \sqrt{a+c x^2}}{6 (d+e x)^2 \left (a e^2+c d^2\right )^2}-\frac{e \sqrt{a+c x^2}}{3 (d+e x)^3 \left (a e^2+c d^2\right )} \]

[Out]

-(e*Sqrt[a + c*x^2])/(3*(c*d^2 + a*e^2)*(d + e*x)^3) - (5*c*d*e*Sqrt[a + c*x^2])
/(6*(c*d^2 + a*e^2)^2*(d + e*x)^2) - (c*e*(11*c*d^2 - 4*a*e^2)*Sqrt[a + c*x^2])/
(6*(c*d^2 + a*e^2)^3*(d + e*x)) - (c^2*d*(2*c*d^2 - 3*a*e^2)*ArcTanh[(a*e - c*d*
x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(2*(c*d^2 + a*e^2)^(7/2))

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Rubi [A]  time = 0.484666, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{c^2 d \left (2 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{7/2}}-\frac{c e \sqrt{a+c x^2} \left (11 c d^2-4 a e^2\right )}{6 (d+e x) \left (a e^2+c d^2\right )^3}-\frac{5 c d e \sqrt{a+c x^2}}{6 (d+e x)^2 \left (a e^2+c d^2\right )^2}-\frac{e \sqrt{a+c x^2}}{3 (d+e x)^3 \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^4*Sqrt[a + c*x^2]),x]

[Out]

-(e*Sqrt[a + c*x^2])/(3*(c*d^2 + a*e^2)*(d + e*x)^3) - (5*c*d*e*Sqrt[a + c*x^2])
/(6*(c*d^2 + a*e^2)^2*(d + e*x)^2) - (c*e*(11*c*d^2 - 4*a*e^2)*Sqrt[a + c*x^2])/
(6*(c*d^2 + a*e^2)^3*(d + e*x)) - (c^2*d*(2*c*d^2 - 3*a*e^2)*ArcTanh[(a*e - c*d*
x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(2*(c*d^2 + a*e^2)^(7/2))

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Rubi in Sympy [A]  time = 50.5385, size = 180, normalized size = 0.91 \[ \frac{c^{2} d \left (3 a e^{2} - 2 c d^{2}\right ) \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{2 \left (a e^{2} + c d^{2}\right )^{\frac{7}{2}}} - \frac{5 c d e \sqrt{a + c x^{2}}}{6 \left (d + e x\right )^{2} \left (a e^{2} + c d^{2}\right )^{2}} + \frac{c e \sqrt{a + c x^{2}} \left (4 a e^{2} - 11 c d^{2}\right )}{6 \left (d + e x\right ) \left (a e^{2} + c d^{2}\right )^{3}} - \frac{e \sqrt{a + c x^{2}}}{3 \left (d + e x\right )^{3} \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**4/(c*x**2+a)**(1/2),x)

[Out]

c**2*d*(3*a*e**2 - 2*c*d**2)*atanh((a*e - c*d*x)/(sqrt(a + c*x**2)*sqrt(a*e**2 +
c*d**2)))/(2*(a*e**2 + c*d**2)**(7/2)) - 5*c*d*e*sqrt(a + c*x**2)/(6*(d + e*x)*
*2*(a*e**2 + c*d**2)**2) + c*e*sqrt(a + c*x**2)*(4*a*e**2 - 11*c*d**2)/(6*(d + e
*x)*(a*e**2 + c*d**2)**3) - e*sqrt(a + c*x**2)/(3*(d + e*x)**3*(a*e**2 + c*d**2)
)

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Mathematica [A]  time = 0.27212, size = 209, normalized size = 1.06 \[ \frac{-3 c^2 d (d+e x)^3 \left (2 c d^2-3 a e^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )+3 c^2 d (d+e x)^3 \left (2 c d^2-3 a e^2\right ) \log (d+e x)-e \sqrt{a+c x^2} \sqrt{a e^2+c d^2} \left (5 c d (d+e x) \left (a e^2+c d^2\right )+c (d+e x)^2 \left (11 c d^2-4 a e^2\right )+2 \left (a e^2+c d^2\right )^2\right )}{6 (d+e x)^3 \left (a e^2+c d^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^4*Sqrt[a + c*x^2]),x]

[Out]

(-(e*Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]*(2*(c*d^2 + a*e^2)^2 + 5*c*d*(c*d^2 + a
*e^2)*(d + e*x) + c*(11*c*d^2 - 4*a*e^2)*(d + e*x)^2)) + 3*c^2*d*(2*c*d^2 - 3*a*
e^2)*(d + e*x)^3*Log[d + e*x] - 3*c^2*d*(2*c*d^2 - 3*a*e^2)*(d + e*x)^3*Log[a*e
- c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(6*(c*d^2 + a*e^2)^(7/2)*(d + e*
x)^3)

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Maple [B]  time = 0.02, size = 573, normalized size = 2.9 \[ -{\frac{1}{3\,{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) }\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \left ({\frac{d}{e}}+x \right ) ^{-3}}-{\frac{5\,cd}{6\,e \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \left ({\frac{d}{e}}+x \right ) ^{-2}}-{\frac{5\,{c}^{2}{d}^{2}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \left ({\frac{d}{e}}+x \right ) ^{-1}}-{\frac{5\,{c}^{3}{d}^{3}}{2\,e \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{3\,{c}^{2}d}{2\,e \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{2\,c}{3\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \left ({\frac{d}{e}}+x \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^4/(c*x^2+a)^(1/2),x)

[Out]

-1/3/e^2/(a*e^2+c*d^2)/(d/e+x)^3*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)
^(1/2)-5/6/e*c*d/(a*e^2+c*d^2)^2/(d/e+x)^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c
*d^2)/e^2)^(1/2)-5/2*c^2*d^2/(a*e^2+c*d^2)^3/(d/e+x)*(c*(d/e+x)^2-2*c*d/e*(d/e+x
)+(a*e^2+c*d^2)/e^2)^(1/2)-5/2/e*c^3*d^3/(a*e^2+c*d^2)^3/((a*e^2+c*d^2)/e^2)^(1/
2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x
)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))+3/2/e*c^2*d/(a*e^2+c*d^2)
^2/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c
*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))
+2/3/(a*e^2+c*d^2)^2*c/(d/e+x)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(
1/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.710667, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + a)*(e*x + d)^4),x, algorithm="fricas")

[Out]

[-1/12*(2*(18*c^2*d^4*e + 5*a*c*d^2*e^3 + 2*a^2*e^5 + (11*c^2*d^2*e^3 - 4*a*c*e^
5)*x^2 + 3*(9*c^2*d^3*e^2 - a*c*d*e^4)*x)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a) +
3*(2*c^3*d^6 - 3*a*c^2*d^4*e^2 + (2*c^3*d^3*e^3 - 3*a*c^2*d*e^5)*x^3 + 3*(2*c^3*
d^4*e^2 - 3*a*c^2*d^2*e^4)*x^2 + 3*(2*c^3*d^5*e - 3*a*c^2*d^3*e^3)*x)*log(((2*a*
c*d*e*x - a*c*d^2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2)*x^2)*sqrt(c*d^2 + a*e^2) -
2*(a*c*d^2*e + a^2*e^3 - (c^2*d^3 + a*c*d*e^2)*x)*sqrt(c*x^2 + a))/(e^2*x^2 + 2
*d*e*x + d^2)))/((c^3*d^9 + 3*a*c^2*d^7*e^2 + 3*a^2*c*d^5*e^4 + a^3*d^3*e^6 + (c
^3*d^6*e^3 + 3*a*c^2*d^4*e^5 + 3*a^2*c*d^2*e^7 + a^3*e^9)*x^3 + 3*(c^3*d^7*e^2 +
3*a*c^2*d^5*e^4 + 3*a^2*c*d^3*e^6 + a^3*d*e^8)*x^2 + 3*(c^3*d^8*e + 3*a*c^2*d^6
*e^3 + 3*a^2*c*d^4*e^5 + a^3*d^2*e^7)*x)*sqrt(c*d^2 + a*e^2)), -1/6*((18*c^2*d^4
*e + 5*a*c*d^2*e^3 + 2*a^2*e^5 + (11*c^2*d^2*e^3 - 4*a*c*e^5)*x^2 + 3*(9*c^2*d^3
*e^2 - a*c*d*e^4)*x)*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a) - 3*(2*c^3*d^6 - 3*a*c
^2*d^4*e^2 + (2*c^3*d^3*e^3 - 3*a*c^2*d*e^5)*x^3 + 3*(2*c^3*d^4*e^2 - 3*a*c^2*d^
2*e^4)*x^2 + 3*(2*c^3*d^5*e - 3*a*c^2*d^3*e^3)*x)*arctan(sqrt(-c*d^2 - a*e^2)*(c
*d*x - a*e)/((c*d^2 + a*e^2)*sqrt(c*x^2 + a))))/((c^3*d^9 + 3*a*c^2*d^7*e^2 + 3*
a^2*c*d^5*e^4 + a^3*d^3*e^6 + (c^3*d^6*e^3 + 3*a*c^2*d^4*e^5 + 3*a^2*c*d^2*e^7 +
a^3*e^9)*x^3 + 3*(c^3*d^7*e^2 + 3*a*c^2*d^5*e^4 + 3*a^2*c*d^3*e^6 + a^3*d*e^8)*
x^2 + 3*(c^3*d^8*e + 3*a*c^2*d^6*e^3 + 3*a^2*c*d^4*e^5 + a^3*d^2*e^7)*x)*sqrt(-c
*d^2 - a*e^2))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + c x^{2}} \left (d + e x\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**4/(c*x**2+a)**(1/2),x)

[Out]

Integral(1/(sqrt(a + c*x**2)*(d + e*x)**4), x)

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GIAC/XCAS [A]  time = 0.237757, size = 778, normalized size = 3.93 \[ -\frac{1}{3} \, c^{\frac{3}{2}}{\left (\frac{3 \,{\left (2 \, c^{\frac{3}{2}} d^{3} - 3 \, a \sqrt{c} d e^{2}\right )} \arctan \left (\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} - a e^{2}}}\right )}{{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} \sqrt{-c d^{2} - a e^{2}}} + \frac{30 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} c^{2} d^{4} e + 44 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} c^{\frac{5}{2}} d^{5} + 6 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} c^{\frac{3}{2}} d^{3} e^{2} - 102 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} a c^{2} d^{4} e - 82 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} a c^{\frac{3}{2}} d^{3} e^{2} - 45 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} a c d^{2} e^{3} - 9 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} a \sqrt{c} d e^{4} + 60 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} a^{2} c^{\frac{3}{2}} d^{3} e^{2} + 36 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} a^{2} c d^{2} e^{3} + 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} a^{2} \sqrt{c} d e^{4} - 11 \, a^{3} c d^{2} e^{3} - 15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} a^{3} \sqrt{c} d e^{4} - 12 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} a^{3} e^{5} + 4 \, a^{4} e^{5}}{{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} e + 2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} \sqrt{c} d - a e\right )}^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + a)*(e*x + d)^4),x, algorithm="giac")

[Out]

-1/3*c^(3/2)*(3*(2*c^(3/2)*d^3 - 3*a*sqrt(c)*d*e^2)*arctan(((sqrt(c)*x - sqrt(c*
x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))/((c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a
^2*c*d^2*e^4 + a^3*e^6)*sqrt(-c*d^2 - a*e^2)) + (30*(sqrt(c)*x - sqrt(c*x^2 + a)
)^4*c^2*d^4*e + 44*(sqrt(c)*x - sqrt(c*x^2 + a))^3*c^(5/2)*d^5 + 6*(sqrt(c)*x -
sqrt(c*x^2 + a))^5*c^(3/2)*d^3*e^2 - 102*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a*c^2*d
^4*e - 82*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a*c^(3/2)*d^3*e^2 - 45*(sqrt(c)*x - sq
rt(c*x^2 + a))^4*a*c*d^2*e^3 - 9*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*sqrt(c)*d*e^4
+ 60*(sqrt(c)*x - sqrt(c*x^2 + a))*a^2*c^(3/2)*d^3*e^2 + 36*(sqrt(c)*x - sqrt(c
*x^2 + a))^2*a^2*c*d^2*e^3 + 24*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^2*sqrt(c)*d*e^
4 - 11*a^3*c*d^2*e^3 - 15*(sqrt(c)*x - sqrt(c*x^2 + a))*a^3*sqrt(c)*d*e^4 - 12*(
sqrt(c)*x - sqrt(c*x^2 + a))^2*a^3*e^5 + 4*a^4*e^5)/((c^3*d^6 + 3*a*c^2*d^4*e^2
+ 3*a^2*c*d^2*e^4 + a^3*e^6)*((sqrt(c)*x - sqrt(c*x^2 + a))^2*e + 2*(sqrt(c)*x -
sqrt(c*x^2 + a))*sqrt(c)*d - a*e)^3))