3.567 \(\int \frac{(d+e x)^4}{\left (a+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=161 \[ -\frac{d e \sqrt{a+c x^2} \left (5 a e^2+2 c d^2\right )}{3 a^2 c^2}-\frac{(d+e x) \left (a e \left (3 a e^2+c d^2\right )-2 c d x \left (2 a e^2+c d^2\right )\right )}{3 a^2 c^2 \sqrt{a+c x^2}}+\frac{e^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{5/2}}-\frac{(d+e x)^3 (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.341431, antiderivative size = 161, 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 \[ -\frac{d e \sqrt{a+c x^2} \left (5 a e^2+2 c d^2\right )}{3 a^2 c^2}-\frac{(d+e x) \left (a e \left (3 a e^2+c d^2\right )-2 c d x \left (2 a e^2+c d^2\right )\right )}{3 a^2 c^2 \sqrt{a+c x^2}}+\frac{e^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{5/2}}-\frac{(d+e x)^3 (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^4/(a + c*x^2)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 43.4229, size = 151, normalized size = 0.94 \[ \frac{e^{4} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{c^{\frac{5}{2}}} - \frac{\left (d + e x\right )^{3} \left (a e - c d x\right )}{3 a c \left (a + c x^{2}\right )^{\frac{3}{2}}} - \frac{d e \sqrt{a + c x^{2}} \left (5 a e^{2} + 2 c d^{2}\right )}{3 a^{2} c^{2}} - \frac{\left (d + e x\right ) \left (2 a e \left (3 a e^{2} + c d^{2}\right ) - 4 c d x \left (2 a e^{2} + c d^{2}\right )\right )}{6 a^{2} c^{2} \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**4/(c*x**2+a)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.284987, size = 130, normalized size = 0.81 \[ \frac{-a^3 e^3 (8 d+3 e x)-4 a^2 c e \left (d^3+3 d e^2 x^2+e^3 x^3\right )+3 a c^2 d^2 x \left (d^2+2 e^2 x^2\right )+2 c^3 d^4 x^3}{3 a^2 c^2 \left (a+c x^2\right )^{3/2}}+\frac{e^4 \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{c^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^4/(a + c*x^2)^(5/2),x]

[Out]

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

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Maple [A]  time = 0.011, size = 202, normalized size = 1.3 \[{\frac{{d}^{4}x}{3\,a} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,{d}^{4}x}{3\,{a}^{2}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{{e}^{4}{x}^{3}}{3\,c} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{{e}^{4}x}{{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{{e}^{4}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}}-4\,{\frac{d{e}^{3}{x}^{2}}{c \left ( c{x}^{2}+a \right ) ^{3/2}}}-{\frac{8\,d{e}^{3}a}{3\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-2\,{\frac{{d}^{2}{e}^{2}x}{c \left ( c{x}^{2}+a \right ) ^{3/2}}}+2\,{\frac{{d}^{2}{e}^{2}x}{ac\sqrt{c{x}^{2}+a}}}-{\frac{4\,{d}^{3}e}{3\,c} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4/(c*x^2+a)^(5/2),x)

[Out]

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

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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((e*x + d)^4/(c*x^2 + a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238005, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (12 \, a^{2} c d e^{3} x^{2} + 4 \, a^{2} c d^{3} e + 8 \, a^{3} d e^{3} - 2 \,{\left (c^{3} d^{4} + 3 \, a c^{2} d^{2} e^{2} - 2 \, a^{2} c e^{4}\right )} x^{3} - 3 \,{\left (a c^{2} d^{4} - a^{3} e^{4}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{c} - 3 \,{\left (a^{2} c^{2} e^{4} x^{4} + 2 \, a^{3} c e^{4} x^{2} + a^{4} e^{4}\right )} \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right )}{6 \,{\left (a^{2} c^{4} x^{4} + 2 \, a^{3} c^{3} x^{2} + a^{4} c^{2}\right )} \sqrt{c}}, -\frac{{\left (12 \, a^{2} c d e^{3} x^{2} + 4 \, a^{2} c d^{3} e + 8 \, a^{3} d e^{3} - 2 \,{\left (c^{3} d^{4} + 3 \, a c^{2} d^{2} e^{2} - 2 \, a^{2} c e^{4}\right )} x^{3} - 3 \,{\left (a c^{2} d^{4} - a^{3} e^{4}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{-c} - 3 \,{\left (a^{2} c^{2} e^{4} x^{4} + 2 \, a^{3} c e^{4} x^{2} + a^{4} e^{4}\right )} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right )}{3 \,{\left (a^{2} c^{4} x^{4} + 2 \, a^{3} c^{3} x^{2} + a^{4} c^{2}\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(c*x^2 + a)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{4}}{\left (a + c x^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**4/(c*x**2+a)**(5/2),x)

[Out]

Integral((d + e*x)**4/(a + c*x**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.223218, size = 203, normalized size = 1.26 \[ -\frac{{\left (2 \, x{\left (\frac{6 \, d e^{3}}{c} - \frac{{\left (c^{5} d^{4} + 3 \, a c^{4} d^{2} e^{2} - 2 \, a^{2} c^{3} e^{4}\right )} x}{a^{2} c^{4}}\right )} - \frac{3 \,{\left (a c^{4} d^{4} - a^{3} c^{2} e^{4}\right )}}{a^{2} c^{4}}\right )} x + \frac{4 \,{\left (a^{2} c^{3} d^{3} e + 2 \, a^{3} c^{2} d e^{3}\right )}}{a^{2} c^{4}}}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}}} - \frac{e^{4}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(c*x^2 + a)^(5/2),x, algorithm="giac")

[Out]

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