3.9.71 \(\int \frac {(c d^2+2 c d e x+c e^2 x^2)^{5/2}}{(d+e x)^6} \, dx\)
Optimal. Leaf size=42 \[ \frac {c^3 (d+e x) \log (d+e x)}{e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used =
{642, 608, 31} \begin {gather*} \frac {c^3 (d+e x) \log (d+e x)}{e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)/(d + e*x)^6,x]
[Out]
(c^3*(d + e*x)*Log[d + e*x])/(e*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2])
Rule 31
Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]
Rule 608
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(b/2 + c*x)/Sqrt[a + b*x + c*x^2], Int[1/(b/2
+ c*x), x], x] /; FreeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0]
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)^6} \, dx &=c^3 \int \frac {1}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx\\ &=\frac {\left (c^3 \left (c d e+c e^2 x\right )\right ) \int \frac {1}{c d e+c e^2 x} \, dx}{\sqrt {c d^2+2 c d e x+c e^2 x^2}}\\ &=\frac {c^3 (d+e x) \log (d+e x)}{e \sqrt {c d^2+2 c d e x+c e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 31, normalized size = 0.74 \begin {gather*} \frac {c^3 (d+e x) \log (d+e x)}{e \sqrt {c (d+e x)^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)/(d + e*x)^6,x]
[Out]
(c^3*(d + e*x)*Log[d + e*x])/(e*Sqrt[c*(d + e*x)^2])
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IntegrateAlgebraic [B] time = 0.40, size = 136, normalized size = 3.24 \begin {gather*} -\frac {c^{5/2} \tanh ^{-1}\left (\frac {x \sqrt {c e^2}}{\sqrt {c} d}-\frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{\sqrt {c} d}\right )}{e}-\frac {c^2 \sqrt {c e^2} \log \left (x \left (c d e+c e^2 x\right )-x \sqrt {c e^2} \sqrt {c d^2+2 c d e x+c e^2 x^2}\right )}{2 e^2} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)/(d + e*x)^6,x]
[Out]
-((c^(5/2)*ArcTanh[(Sqrt[c*e^2]*x)/(Sqrt[c]*d) - Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]/(Sqrt[c]*d)])/e) - (c^2*S
qrt[c*e^2]*Log[x*(c*d*e + c*e^2*x) - Sqrt[c*e^2]*x*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]])/(2*e^2)
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fricas [A] time = 0.41, size = 43, normalized size = 1.02 \begin {gather*} \frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c^{2} \log \left (e x + d\right )}{e^{2} x + d e} \end {gather*}
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)^6,x, algorithm="fricas")
[Out]
sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*c^2*log(e*x + d)/(e^2*x + d*e)
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
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)^6,x, algorithm="giac")
[Out]
Exception raised: NotImplementedError >> Unable to parse Giac output: 2*(-c^2*sqrt(c*exp(2))/2/exp(2)*ln(abs(e
xp(2)*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)-sqrt(c*exp(2))*exp(1)*d))+(3*c^3*exp(1)^6*d+c
^3*exp(2)*exp(1)^4*d+4*c^3*exp(2)^2*exp(1)^2*d-8*c^3*exp(2)^3*d)/4/exp(1)^6/2/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)))+(45*c^3*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^9*exp(1)^10*d+15*c^3*exp(2)*(sqrt(c*d
^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^9*exp(1)^8*d-660*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)^9*exp(1)^6*d+600*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)^9*exp(1)^4*d-405*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)^8*exp(1
)^9*d^2+1305*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)^8*exp(1)^7*d
^2+2700*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)^8*exp(1)^5*d^2-
3600*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)^8*exp(1)^3*d^2+210
*c^4*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^7*exp(1)^10*d^3-760*c^4*exp(2)*(sqrt(c*d^2+2*c
*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^7*exp(1)^8*d^3-5250*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)^7*exp(1)^6*d^3-3000*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)^7*exp(1)^4*d^3+8800*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)^7*exp(1)^2
*d^3+450*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)^6*exp(1)^9*d^4+5200*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)^6*exp(1)^7*d^4+9750*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)^6*exp(1)^5*d^4-5400*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)^6*exp(1)^3*d^4-10000*c^4*exp(2)^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)^6*exp(1)*d^4-384*c^5*(sqrt(c*d^2+2*c
*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^5*exp(1)^10*d^5-3590*c^5*exp(2)*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*e
xp(2))-sqrt(c*exp(2))*x)^5*exp(1)^8*d^5-9670*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)^5*exp(1)^6*d^5-6180*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)^5*exp(1)^4*
d^5+15440*c^5*exp(2)^4*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^5*exp(1)^2*d^5+4384*c^5*exp(
2)^5*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^5*d^5+1920*c^5*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*exp(1)^9*d^6+6250*c^5*exp(2)*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*e
xp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^4*exp(1)^7*d^6+8910*c^5*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*exp(1)^5*d^6-7080*c^5*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*exp(1)^3*d^6-10000*c^5*exp(2)^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)^4*exp(1)*d^6-210*c^6*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^
3*exp(1)^10*d^7-3280*c^6*exp(2)*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^3*exp(1)^8*d^7-4950
*c^6*exp(2)^2*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^3*exp(1)^6*d^7-840*c^6*exp(2)^3*(sqrt
(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^3*exp(1)^4*d^7+9280*c^6*exp(2)^4*(sqrt(c*d^2+2*c*d*x*exp
(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^3*exp(1)^2*d^7+630*c^6*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*exp(1)^9*d^8+2160*c^6*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*exp(1)^7*d^8+1530*c^6*exp(2)^2*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqr
t(c*exp(2))*x)^2*exp(1)^5*d^8-4320*c^6*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*exp(1)^3*d^8-45*c^7*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)*exp(1)^10*d^9-465
*c^7*exp(2)*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)*exp(1)^8*d^9-510*c^7*exp(2)^2*(sqrt(c*d
^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)*exp(1)^6*d^9+1020*c^7*exp(2)^3*(sqrt(c*d^2+2*c*d*x*exp(1)+c*
x^2*exp(2))-sqrt(c*exp(2))*x)*exp(1)^4*d^9+45*c^7*sqrt(c*exp(2))*exp(1)^9*d^10+45*c^7*exp(2)*sqrt(c*exp(2))*ex
p(1)^7*d^10-90*c^7*exp(2)^2*sqrt(c*exp(2))*exp(1)^5*d^10)/120/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*
exp(1)*d^2)^5)
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maple [A] time = 0.05, size = 40, normalized size = 0.95 \begin {gather*} \frac {\left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}} \ln \left (e x +d \right )}{\left (e x +d \right )^{5} e} \end {gather*}
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)^6,x)
[Out]
(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 [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
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)^6,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 = -0.02 \begin {gather*} \int \frac {{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^6} \,d x \end {gather*}
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)^6,x)
[Out]
int((c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(5/2)/(d + e*x)^6, x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c \left (d + e x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{6}}\, dx \end {gather*}
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)**6,x)
[Out]
Integral((c*(d + e*x)**2)**(5/2)/(d + e*x)**6, x)
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