3.1057 \(\int \frac {(c d^2+2 c d e x+c e^2 x^2)^{5/2}}{(d+e x)^4} \, dx\)
Optimal. Leaf size=39 \[ \frac {c^2 (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 e} \]
[Out]
1/2*c^2*(e*x+d)*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2)/e
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Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used =
{642, 609} \[ \frac {c^2 (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 e} \]
Antiderivative was successfully verified.
[In]
Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)/(d + e*x)^4,x]
[Out]
(c^2*(d + e*x)*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2])/(2*e)
Rule 609
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p + 1
)), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && NeQ[p, -2^(-1)]
Rule 642
Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[e^m/c^(m/2), Int[(a +
b*x + c*x^2)^(p + m/2), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p] && EqQ[
2*c*d - b*e, 0] && IntegerQ[m/2]
Rubi steps
\begin {align*} \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx &=c^2 \int \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx\\ &=\frac {c^2 (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 e}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 33, normalized size = 0.85 \[ \frac {c^3 x (d+e x) (2 d+e x)}{2 \sqrt {c (d+e x)^2}} \]
Antiderivative was successfully verified.
[In]
Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)/(d + e*x)^4,x]
[Out]
(c^3*x*(d + e*x)*(2*d + e*x))/(2*Sqrt[c*(d + e*x)^2])
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fricas [A] time = 1.08, size = 47, normalized size = 1.21 \[ \frac {{\left (c^{2} e x^{2} + 2 \, c^{2} d x\right )} \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{2 \, {\left (e x + d\right )}} \]
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)^(5/2)/(e*x+d)^4,x, algorithm="fricas")
[Out]
1/2*(c^2*e*x^2 + 2*c^2*d*x)*sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)/(e*x + d)
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
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)^(5/2)/(e*x+d)^4,x, algorithm="giac")
[Out]
Exception raised: NotImplementedError >> Unable to parse Giac output: 2*(2*c^2*exp(1)*1/8/exp(1)*x+2*c^2*d*1/8
/exp(1))*sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))+2*((33*c^3*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*ex
p(2))*x)^5*d^3*exp(1)^8-126*c^3*exp(2)*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^5*d^3*exp(1)
^6+153*c^3*exp(2)^2*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^5*d^3*exp(1)^4-60*c^3*exp(2)^3*
(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^5*d^3*exp(1)^2-93*c^3*sqrt(c*exp(2))*(sqrt(c*d^2+2*
c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^4*d^4*exp(1)^7+396*c^3*exp(2)*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*
exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^4*d^4*exp(1)^5-513*c^3*exp(2)^2*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(
1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^4*d^4*exp(1)^3+210*c^3*exp(2)^3*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c
*x^2*exp(2))-sqrt(c*exp(2))*x)^4*d^4*exp(1)+40*c^4*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^
3*d^5*exp(1)^8-130*c^4*exp(2)*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^3*d^5*exp(1)^6-48*c^4
*exp(2)^2*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^3*d^5*exp(1)^4+326*c^4*exp(2)^3*(sqrt(c*d
^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^3*d^5*exp(1)^2-188*c^4*exp(2)^4*(sqrt(c*d^2+2*c*d*x*exp(1)+c
*x^2*exp(2))-sqrt(c*exp(2))*x)^3*d^5-48*c^4*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp
(2))*x)^2*d^6*exp(1)^7+306*c^4*exp(2)*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x
)^2*d^6*exp(1)^5-468*c^4*exp(2)^2*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^2*
d^6*exp(1)^3+210*c^4*exp(2)^3*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^2*d^6*
exp(1)+15*c^5*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)*d^7*exp(1)^8-108*c^5*exp(2)*(sqrt(c*d
^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)*d^7*exp(1)^6+171*c^5*exp(2)^2*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x
^2*exp(2))-sqrt(c*exp(2))*x)*d^7*exp(1)^4-78*c^5*exp(2)^3*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(
2))*x)*d^7*exp(1)^2+9*c^5*sqrt(c*exp(2))*d^8*exp(1)^7-18*c^5*exp(2)*sqrt(c*exp(2))*d^8*exp(1)^5+9*c^5*exp(2)^2
*sqrt(c*exp(2))*d^8*exp(1)^3)/6/exp(1)^6/(-(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^2*exp(1)
+2*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)*d-c*d^2*exp(1))^3+(5*c^3*d^3*exp(
1)^6-30*c^3*exp(2)*d^3*exp(1)^4+45*c^3*exp(2)^2*d^3*exp(1)^2-20*c^3*exp(2)^3*d^3)/2/exp(1)^6/d/sqrt(c*exp(1)^2
-c*exp(2))*atan((-d*sqrt(c*exp(2))+(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)*exp(1))/d/sqrt(c
*exp(1)^2-c*exp(2))))
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maple [A] time = 0.05, size = 40, normalized size = 1.03 \[ \frac {\left (e x +2 d \right ) \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}} x}{2 \left (e x +d \right )^{5}} \]
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)^(5/2)/(e*x+d)^4,x)
[Out]
1/2*x*(e*x+2*d)*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)/(e*x+d)^5
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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)^(5/2)/(e*x+d)^4,x, algorithm="maxima")
[Out]
Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(5/2)/(d + e*x)^4,x)
[Out]
int((c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(5/2)/(d + e*x)^4, x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \left (d + e x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{4}}\, dx \]
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)**(5/2)/(e*x+d)**4,x)
[Out]
Integral((c*(d + e*x)**2)**(5/2)/(d + e*x)**4, x)
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