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

Optimal. Leaf size=162 \[ \frac{3 e^2 \left (4 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{5/2}}-\frac{e \sqrt{a+c x^2} \left (e x \left (2 c d^2-3 a e^2\right )+4 d \left (c d^2-4 a e^2\right )\right )}{2 a c^2}-\frac{(d+e x)^3 (a e-c d x)}{a c \sqrt{a+c x^2}}-\frac{d e \sqrt{a+c x^2} (d+e x)^2}{a c} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.427964, antiderivative size = 162, 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 \[ \frac{3 e^2 \left (4 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{5/2}}-\frac{e \sqrt{a+c x^2} \left (e x \left (2 c d^2-3 a e^2\right )+4 d \left (c d^2-4 a e^2\right )\right )}{2 a c^2}-\frac{(d+e x)^3 (a e-c d x)}{a c \sqrt{a+c x^2}}-\frac{d e \sqrt{a+c x^2} (d+e x)^2}{a c} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 46.0674, size = 146, normalized size = 0.9 \[ - \frac{3 e^{2} \left (a e^{2} - 4 c d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{2 c^{\frac{5}{2}}} - \frac{d e \sqrt{a + c x^{2}} \left (d + e x\right )^{2}}{a c} - \frac{\left (d + e x\right )^{3} \left (a e - c d x\right )}{a c \sqrt{a + c x^{2}}} + \frac{e \sqrt{a + c x^{2}} \left (12 d \left (4 a e^{2} - c d^{2}\right ) + 3 e x \left (3 a e^{2} - 2 c d^{2}\right )\right )}{6 a c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**4/(c*x**2+a)**(3/2),x)

[Out]

-3*e**2*(a*e**2 - 4*c*d**2)*atanh(sqrt(c)*x/sqrt(a + c*x**2))/(2*c**(5/2)) - d*e
*sqrt(a + c*x**2)*(d + e*x)**2/(a*c) - (d + e*x)**3*(a*e - c*d*x)/(a*c*sqrt(a +
c*x**2)) + e*sqrt(a + c*x**2)*(12*d*(4*a*e**2 - c*d**2) + 3*e*x*(3*a*e**2 - 2*c*
d**2))/(6*a*c**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.230327, size = 127, normalized size = 0.78 \[ \frac{a^2 e^3 (16 d+3 e x)+a c e \left (-8 d^3-12 d^2 e x+8 d e^2 x^2+e^3 x^3\right )+2 c^2 d^4 x}{2 a c^2 \sqrt{a+c x^2}}+\frac{3 \left (4 c d^2 e^2-a e^4\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{2 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*c^2*d^4*x + a^2*e^3*(16*d + 3*e*x) + a*c*e*(-8*d^3 - 12*d^2*e*x + 8*d*e^2*x^2
+ e^3*x^3))/(2*a*c^2*Sqrt[a + c*x^2]) + (3*(4*c*d^2*e^2 - a*e^4)*Log[c*x + Sqrt
[c]*Sqrt[a + c*x^2]])/(2*c^(5/2))

_______________________________________________________________________________________

Maple [A]  time = 0.017, size = 189, normalized size = 1.2 \[{\frac{{d}^{4}x}{a}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{{e}^{4}{x}^{3}}{2\,c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{3\,a{e}^{4}x}{2\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{3\,a{e}^{4}}{2}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}}+4\,{\frac{d{e}^{3}{x}^{2}}{c\sqrt{c{x}^{2}+a}}}+8\,{\frac{d{e}^{3}a}{{c}^{2}\sqrt{c{x}^{2}+a}}}-6\,{\frac{{d}^{2}{e}^{2}x}{c\sqrt{c{x}^{2}+a}}}+6\,{\frac{{d}^{2}{e}^{2}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ) }{{c}^{3/2}}}-4\,{\frac{{d}^{3}e}{c\sqrt{c{x}^{2}+a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d^4*x/a/(c*x^2+a)^(1/2)+1/2*e^4*x^3/c/(c*x^2+a)^(1/2)+3/2*e^4*a/c^2*x/(c*x^2+a)^
(1/2)-3/2*e^4*a/c^(5/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2))+4*d*e^3*x^2/c/(c*x^2+a)^(1
/2)+8*d*e^3*a/c^2/(c*x^2+a)^(1/2)-6*d^2*e^2*x/c/(c*x^2+a)^(1/2)+6*d^2*e^2/c^(3/2
)*ln(c^(1/2)*x+(c*x^2+a)^(1/2))-4*d^3*e/c/(c*x^2+a)^(1/2)

_______________________________________________________________________________________

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)^4/(c*x^2 + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.236264, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (a c e^{4} x^{3} + 8 \, a c d e^{3} x^{2} - 8 \, a c d^{3} e + 16 \, a^{2} d e^{3} +{\left (2 \, c^{2} d^{4} - 12 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{c} + 3 \,{\left (4 \, a^{2} c d^{2} e^{2} - a^{3} e^{4} +{\left (4 \, a c^{2} d^{2} e^{2} - a^{2} c e^{4}\right )} x^{2}\right )} \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right )}{4 \,{\left (a c^{3} x^{2} + a^{2} c^{2}\right )} \sqrt{c}}, \frac{{\left (a c e^{4} x^{3} + 8 \, a c d e^{3} x^{2} - 8 \, a c d^{3} e + 16 \, a^{2} d e^{3} +{\left (2 \, c^{2} d^{4} - 12 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{-c} + 3 \,{\left (4 \, a^{2} c d^{2} e^{2} - a^{3} e^{4} +{\left (4 \, a c^{2} d^{2} e^{2} - a^{2} c e^{4}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right )}{2 \,{\left (a c^{3} x^{2} + a^{2} c^{2}\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(c*x^2 + a)^(3/2),x, algorithm="fricas")

[Out]

[1/4*(2*(a*c*e^4*x^3 + 8*a*c*d*e^3*x^2 - 8*a*c*d^3*e + 16*a^2*d*e^3 + (2*c^2*d^4
- 12*a*c*d^2*e^2 + 3*a^2*e^4)*x)*sqrt(c*x^2 + a)*sqrt(c) + 3*(4*a^2*c*d^2*e^2 -
a^3*e^4 + (4*a*c^2*d^2*e^2 - a^2*c*e^4)*x^2)*log(-2*sqrt(c*x^2 + a)*c*x - (2*c*
x^2 + a)*sqrt(c)))/((a*c^3*x^2 + a^2*c^2)*sqrt(c)), 1/2*((a*c*e^4*x^3 + 8*a*c*d*
e^3*x^2 - 8*a*c*d^3*e + 16*a^2*d*e^3 + (2*c^2*d^4 - 12*a*c*d^2*e^2 + 3*a^2*e^4)*
x)*sqrt(c*x^2 + a)*sqrt(-c) + 3*(4*a^2*c*d^2*e^2 - a^3*e^4 + (4*a*c^2*d^2*e^2 -
a^2*c*e^4)*x^2)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)))/((a*c^3*x^2 + a^2*c^2)*sqrt(
-c))]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{4}}{\left (a + c x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**4/(c*x**2+a)**(3/2),x)

[Out]

Integral((d + e*x)**4/(a + c*x**2)**(3/2), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.220574, size = 186, normalized size = 1.15 \[ \frac{{\left (x{\left (\frac{x e^{4}}{c} + \frac{8 \, d e^{3}}{c}\right )} + \frac{2 \, c^{4} d^{4} - 12 \, a c^{3} d^{2} e^{2} + 3 \, a^{2} c^{2} e^{4}}{a c^{4}}\right )} x - \frac{8 \,{\left (a c^{3} d^{3} e - 2 \, a^{2} c^{2} d e^{3}\right )}}{a c^{4}}}{2 \, \sqrt{c x^{2} + a}} - \frac{3 \,{\left (4 \, c d^{2} e^{2} - a e^{4}\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{2 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(c*x^2 + a)^(3/2),x, algorithm="giac")

[Out]

1/2*((x*(x*e^4/c + 8*d*e^3/c) + (2*c^4*d^4 - 12*a*c^3*d^2*e^2 + 3*a^2*c^2*e^4)/(
a*c^4))*x - 8*(a*c^3*d^3*e - 2*a^2*c^2*d*e^3)/(a*c^4))/sqrt(c*x^2 + a) - 3/2*(4*
c*d^2*e^2 - a*e^4)*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(5/2)