3.1035 \(\int \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^4} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0803519, 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 \[ \frac{c^2 (d+e x) \log (d+e x)}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.3354, size = 39, normalized size = 0.93 \[ \frac{\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{3}{2}} \log{\left (d + e x \right )}}{e \left (d + e x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0170087, size = 31, normalized size = 0.74 \[ \frac{c^2 (d+e x) \log (d+e x)}{e \sqrt{c (d+e x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 40, normalized size = 1. \[{\frac{\ln \left ( ex+d \right ) }{ \left ( ex+d \right ) ^{3}e} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0.2231, size = 55, normalized size = 1.31 \[ \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c \log \left (e x + d\right )}{e^{2} x + d e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError