3.1070 \(\int \frac{(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=42 \[ \frac{(d+e x) \log (d+e x)}{c^2 e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]

[Out]

((d + e*x)*Log[d + e*x])/(c^2*e*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2])

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Rubi [A]  time = 0.0798306, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094 \[ \frac{(d+e x) \log (d+e x)}{c^2 e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)*Log[d + e*x])/(c^2*e*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2])

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Rubi in Sympy [A]  time = 18.6133, size = 39, normalized size = 0.93 \[ \frac{\left (d + e x\right )^{5} \log{\left (d + e x \right )}}{e \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d + e*x)**5*log(d + e*x)/(e*(c*d**2 + 2*c*d*e*x + c*e**2*x**2)**(5/2))

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Mathematica [A]  time = 0.00504261, size = 31, normalized size = 0.74 \[ \frac{(d+e x) \log (d+e x)}{c^2 e \sqrt{c (d+e x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)*Log[d + e*x])/(c^2*e*Sqrt[c*(d + e*x)^2])

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Maple [A]  time = 0.005, size = 40, normalized size = 1. \[{\frac{ \left ( ex+d \right ) ^{5}\ln \left ( ex+d \right ) }{e} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)*(e*x+d)^5*ln(e*x+d)/e

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Maxima [A]  time = 0.71434, size = 606, normalized size = 14.43 \[ \frac{1}{12} \, e^{4}{\left (\frac{48 \, \sqrt{c} d e^{3} x^{3} + 108 \, \sqrt{c} d^{2} e^{2} x^{2} + 88 \, \sqrt{c} d^{3} e x + 25 \, \sqrt{c} d^{4}}{c^{3} e^{9} x^{4} + 4 \, c^{3} d e^{8} x^{3} + 6 \, c^{3} d^{2} e^{7} x^{2} + 4 \, c^{3} d^{3} e^{6} x + c^{3} d^{4} e^{5}} + \frac{12 \, \log \left (e x + d\right )}{c^{\frac{5}{2}} e^{5}}\right )} - \frac{1}{3} \, d e^{3}{\left (\frac{3 \, c^{2} d^{3} e}{\left (c e^{2}\right )^{\frac{9}{2}}{\left (x + \frac{d}{e}\right )}^{4}} + \frac{12 \, x^{2}}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}} c e^{2}} - \frac{8 \, c d^{2}}{\left (c e^{2}\right )^{\frac{7}{2}}{\left (x + \frac{d}{e}\right )}^{3}} + \frac{8 \, d^{2}}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}} c e^{4}} + \frac{6 \, d}{\left (c e^{2}\right )^{\frac{5}{2}} e{\left (x + \frac{d}{e}\right )}^{2}} - \frac{6 \, d^{3}}{\left (c e^{2}\right )^{\frac{5}{2}} e^{3}{\left (x + \frac{d}{e}\right )}^{4}}\right )} - \frac{1}{2} \, d^{2} e^{2}{\left (\frac{3 \, c^{2} d^{2} e^{2}}{\left (c e^{2}\right )^{\frac{9}{2}}{\left (x + \frac{d}{e}\right )}^{4}} - \frac{8 \, c d e}{\left (c e^{2}\right )^{\frac{7}{2}}{\left (x + \frac{d}{e}\right )}^{3}} + \frac{6}{\left (c e^{2}\right )^{\frac{5}{2}}{\left (x + \frac{d}{e}\right )}^{2}}\right )} - \frac{1}{3} \, d^{3} e{\left (\frac{4}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}} c e^{2}} - \frac{3 \, d}{\left (c e^{2}\right )^{\frac{5}{2}} e{\left (x + \frac{d}{e}\right )}^{4}}\right )} - \frac{d^{4}}{4 \, \left (c e^{2}\right )^{\frac{5}{2}}{\left (x + \frac{d}{e}\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*e^4*((48*sqrt(c)*d*e^3*x^3 + 108*sqrt(c)*d^2*e^2*x^2 + 88*sqrt(c)*d^3*e*x +
25*sqrt(c)*d^4)/(c^3*e^9*x^4 + 4*c^3*d*e^8*x^3 + 6*c^3*d^2*e^7*x^2 + 4*c^3*d^3*
e^6*x + c^3*d^4*e^5) + 12*log(e*x + d)/(c^(5/2)*e^5)) - 1/3*d*e^3*(3*c^2*d^3*e/(
(c*e^2)^(9/2)*(x + d/e)^4) + 12*x^2/((c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2)*c*e^2
) - 8*c*d^2/((c*e^2)^(7/2)*(x + d/e)^3) + 8*d^2/((c*e^2*x^2 + 2*c*d*e*x + c*d^2)
^(3/2)*c*e^4) + 6*d/((c*e^2)^(5/2)*e*(x + d/e)^2) - 6*d^3/((c*e^2)^(5/2)*e^3*(x
+ d/e)^4)) - 1/2*d^2*e^2*(3*c^2*d^2*e^2/((c*e^2)^(9/2)*(x + d/e)^4) - 8*c*d*e/((
c*e^2)^(7/2)*(x + d/e)^3) + 6/((c*e^2)^(5/2)*(x + d/e)^2)) - 1/3*d^3*e*(4/((c*e^
2*x^2 + 2*c*d*e*x + c*d^2)^(3/2)*c*e^2) - 3*d/((c*e^2)^(5/2)*e*(x + d/e)^4)) - 1
/4*d^4/((c*e^2)^(5/2)*(x + d/e)^4)

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Fricas [A]  time = 0.231965, size = 62, normalized size = 1.48 \[ \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} \log \left (e x + d\right )}{c^{3} e^{2} x + c^{3} d e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*log(e*x + d)/(c^3*e^2*x + c^3*d*e)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.32078, size = 149, normalized size = 3.55 \[ \frac{2 \,{\left (C_{0} d^{3} e^{\left (-3\right )} +{\left (3 \, C_{0} d^{2} e^{\left (-2\right )} +{\left (3 \, C_{0} d e^{\left (-1\right )} + C_{0} x\right )} x\right )} x\right )}}{{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{\frac{3}{2}}} - \frac{e^{\left (-1\right )}{\rm ln}\left ({\left | -\sqrt{c} d e^{2} -{\left (\sqrt{c} x e - \sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}}\right )} e^{2} \right |}\right )}{c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(C_0*d^3*e^(-3) + (3*C_0*d^2*e^(-2) + (3*C_0*d*e^(-1) + C_0*x)*x)*x)/(c*x^2*e^
2 + 2*c*d*x*e + c*d^2)^(3/2) - e^(-1)*ln(abs(-sqrt(c)*d*e^2 - (sqrt(c)*x*e - sqr
t(c*x^2*e^2 + 2*c*d*x*e + c*d^2))*e^2))/c^(5/2)