3.1952 \(\int \frac {1}{(d+e x)^3 \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\)
Optimal. Leaf size=171 \[ \frac {16 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 (d+e x) \left (c d^2-a e^2\right )^3}+\frac {8 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 (d+e x)^2 \left (c d^2-a e^2\right )^2}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 (d+e x)^3 \left (c d^2-a e^2\right )} \]
[Out]
2/5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e^2+c*d^2)/(e*x+d)^3+8/15*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2
)^(1/2)/(-a*e^2+c*d^2)^2/(e*x+d)^2+16/15*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e^2+c*d^2)^3/(e*x
+d)
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Rubi [A] time = 0.08, antiderivative size = 171, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used =
{658, 650} \[ \frac {16 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 (d+e x) \left (c d^2-a e^2\right )^3}+\frac {8 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 (d+e x)^2 \left (c d^2-a e^2\right )^2}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 (d+e x)^3 \left (c d^2-a e^2\right )} \]
Antiderivative was successfully verified.
[In]
Int[1/((d + e*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]
[Out]
(2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*(c*d^2 - a*e^2)*(d + e*x)^3) + (8*c*d*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2])/(15*(c*d^2 - a*e^2)^2*(d + e*x)^2) + (16*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2])/(15*(c*d^2 - a*e^2)^3*(d + e*x))
Rule 650
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((p + 1)*(2*c*d - b*e)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] &&
EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && EqQ[m + 2*p + 2, 0]
Rule 658
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((m + p + 1)*(2*c*d - b*e)), x] + Dist[(c*Simplify[m + 2*p + 2])/((m + p + 1)*(2*c*d -
b*e)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c
, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2], 0]
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 \left (c d^2-a e^2\right ) (d+e x)^3}+\frac {(4 c d) \int \frac {1}{(d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{5 \left (c d^2-a e^2\right )}\\ &=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 \left (c d^2-a e^2\right ) (d+e x)^3}+\frac {8 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 \left (c d^2-a e^2\right )^2 (d+e x)^2}+\frac {\left (8 c^2 d^2\right ) \int \frac {1}{(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{15 \left (c d^2-a e^2\right )^2}\\ &=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 \left (c d^2-a e^2\right ) (d+e x)^3}+\frac {8 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 \left (c d^2-a e^2\right )^2 (d+e x)^2}+\frac {16 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 \left (c d^2-a e^2\right )^3 (d+e x)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 94, normalized size = 0.55 \[ \frac {2 \sqrt {(d+e x) (a e+c d x)} \left (3 a^2 e^4-2 a c d e^2 (5 d+2 e x)+c^2 d^2 \left (15 d^2+20 d e x+8 e^2 x^2\right )\right )}{15 (d+e x)^3 \left (c d^2-a e^2\right )^3} \]
Antiderivative was successfully verified.
[In]
Integrate[1/((d + e*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]
[Out]
(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(3*a^2*e^4 - 2*a*c*d*e^2*(5*d + 2*e*x) + c^2*d^2*(15*d^2 + 20*d*e*x + 8*e^2*x
^2)))/(15*(c*d^2 - a*e^2)^3*(d + e*x)^3)
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fricas [A] time = 3.75, size = 279, normalized size = 1.63 \[ \frac {2 \, {\left (8 \, c^{2} d^{2} e^{2} x^{2} + 15 \, c^{2} d^{4} - 10 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 4 \, {\left (5 \, c^{2} d^{3} e - a c d e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{15 \, {\left (c^{3} d^{9} - 3 \, a c^{2} d^{7} e^{2} + 3 \, a^{2} c d^{5} e^{4} - a^{3} d^{3} e^{6} + {\left (c^{3} d^{6} e^{3} - 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} - a^{3} e^{9}\right )} x^{3} + 3 \, {\left (c^{3} d^{7} e^{2} - 3 \, a c^{2} d^{5} e^{4} + 3 \, a^{2} c d^{3} e^{6} - a^{3} d e^{8}\right )} x^{2} + 3 \, {\left (c^{3} d^{8} e - 3 \, a c^{2} d^{6} e^{3} + 3 \, a^{2} c d^{4} e^{5} - a^{3} d^{2} e^{7}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="fricas")
[Out]
2/15*(8*c^2*d^2*e^2*x^2 + 15*c^2*d^4 - 10*a*c*d^2*e^2 + 3*a^2*e^4 + 4*(5*c^2*d^3*e - a*c*d*e^3)*x)*sqrt(c*d*e*
x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(c^3*d^9 - 3*a*c^2*d^7*e^2 + 3*a^2*c*d^5*e^4 - a^3*d^3*e^6 + (c^3*d^6*e^3 - 3
*a*c^2*d^4*e^5 + 3*a^2*c*d^2*e^7 - a^3*e^9)*x^3 + 3*(c^3*d^7*e^2 - 3*a*c^2*d^5*e^4 + 3*a^2*c*d^3*e^6 - a^3*d*e
^8)*x^2 + 3*(c^3*d^8*e - 3*a*c^2*d^6*e^3 + 3*a^2*c*d^4*e^5 - a^3*d^2*e^7)*x)
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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(1/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="giac")
[Out]
Exception raised: NotImplementedError >> Unable to parse Giac output: 2*((-3*exp(1)^2*(sqrt(c*d*exp(1)*x^2+a*d
*exp(1)+(c*d^2+a*exp(2))*x)-sqrt(c*d*exp(1))*x)^3*a^2*exp(2)^2+2*c*d^2*exp(1)^2*(sqrt(c*d*exp(1)*x^2+a*d*exp(1
)+(c*d^2+a*exp(2))*x)-sqrt(c*d*exp(1))*x)^3*a*exp(2)+4*c*d^2*exp(1)^4*(sqrt(c*d*exp(1)*x^2+a*d*exp(1)+(c*d^2+a
*exp(2))*x)-sqrt(c*d*exp(1))*x)^3*a-3*c^2*d^4*exp(1)^2*(sqrt(c*d*exp(1)*x^2+a*d*exp(1)+(c*d^2+a*exp(2))*x)-sqr
t(c*d*exp(1))*x)^3+9*d*exp(1)*sqrt(c*d*exp(1))*(sqrt(c*d*exp(1)*x^2+a*d*exp(1)+(c*d^2+a*exp(2))*x)-sqrt(c*d*ex
p(1))*x)^2*a^2*exp(2)^2-6*c*d^3*exp(1)*sqrt(c*d*exp(1))*(sqrt(c*d*exp(1)*x^2+a*d*exp(1)+(c*d^2+a*exp(2))*x)-sq
rt(c*d*exp(1))*x)^2*a*exp(2)-12*c*d^3*exp(1)^3*sqrt(c*d*exp(1))*(sqrt(c*d*exp(1)*x^2+a*d*exp(1)+(c*d^2+a*exp(2
))*x)-sqrt(c*d*exp(1))*x)^2*a+9*c^2*d^5*exp(1)*sqrt(c*d*exp(1))*(sqrt(c*d*exp(1)*x^2+a*d*exp(1)+(c*d^2+a*exp(2
))*x)-sqrt(c*d*exp(1))*x)^2-5*d*exp(1)*(sqrt(c*d*exp(1)*x^2+a*d*exp(1)+(c*d^2+a*exp(2))*x)-sqrt(c*d*exp(1))*x)
*a^3*exp(2)^3+5*d*exp(1)^3*(sqrt(c*d*exp(1)*x^2+a*d*exp(1)+(c*d^2+a*exp(2))*x)-sqrt(c*d*exp(1))*x)*a^3*exp(2)^
2+5*c*d^3*exp(1)*(sqrt(c*d*exp(1)*x^2+a*d*exp(1)+(c*d^2+a*exp(2))*x)-sqrt(c*d*exp(1))*x)*a^2*exp(2)^2-18*c*d^3
*exp(1)^3*(sqrt(c*d*exp(1)*x^2+a*d*exp(1)+(c*d^2+a*exp(2))*x)-sqrt(c*d*exp(1))*x)*a^2*exp(2)+4*c*d^3*exp(1)^5*
(sqrt(c*d*exp(1)*x^2+a*d*exp(1)+(c*d^2+a*exp(2))*x)-sqrt(c*d*exp(1))*x)*a^2+c^2*d^5*exp(1)*(sqrt(c*d*exp(1)*x^
2+a*d*exp(1)+(c*d^2+a*exp(2))*x)-sqrt(c*d*exp(1))*x)*a*exp(2)+17*c^2*d^5*exp(1)^3*(sqrt(c*d*exp(1)*x^2+a*d*exp
(1)+(c*d^2+a*exp(2))*x)-sqrt(c*d*exp(1))*x)*a-9*c^3*d^7*exp(1)*(sqrt(c*d*exp(1)*x^2+a*d*exp(1)+(c*d^2+a*exp(2)
)*x)-sqrt(c*d*exp(1))*x)-3*d^2*sqrt(c*d*exp(1))*a^3*exp(2)^3+11*d^2*exp(1)^2*sqrt(c*d*exp(1))*a^3*exp(2)^2-8*d
^2*exp(1)^4*sqrt(c*d*exp(1))*a^3*exp(2)-3*c*d^4*sqrt(c*d*exp(1))*a^2*exp(2)^2+2*c*d^4*exp(1)^2*sqrt(c*d*exp(1)
)*a^2*exp(2)+4*c*d^4*exp(1)^4*sqrt(c*d*exp(1))*a^2+3*c^2*d^6*sqrt(c*d*exp(1))*a*exp(2)-9*c^2*d^6*exp(1)^2*sqrt
(c*d*exp(1))*a+3*c^3*d^8*sqrt(c*d*exp(1)))/(-8*d^2*exp(1)*a^2*exp(2)^2+16*d^2*exp(1)^3*a^2*exp(2)-8*d^2*exp(1)
^5*a^2)/(-exp(1)*(sqrt(c*d*exp(1)*x^2+a*d*exp(1)+(c*d^2+a*exp(2))*x)-sqrt(c*d*exp(1))*x)^2+2*d*sqrt(c*d*exp(1)
)*(sqrt(c*d*exp(1)*x^2+a*d*exp(1)+(c*d^2+a*exp(2))*x)-sqrt(c*d*exp(1))*x)-d*a*exp(2)+d*exp(1)^2*a-c*d^3)^2+(3*
a^2*exp(2)^2-2*c*d^2*a*exp(2)-4*c*d^2*exp(1)^2*a+3*c^2*d^4)/2/(4*d^2*a^2*exp(2)^2-8*d^2*exp(1)^2*a^2*exp(2)+4*
d^2*exp(1)^4*a^2)/sqrt(-a*d*exp(1)^3+a*d*exp(1)*exp(2))*atan((-d*sqrt(c*d*exp(1))+(sqrt(c*d*exp(1)*x^2+a*d*exp
(1)+(c*d^2+a*exp(2))*x)-sqrt(c*d*exp(1))*x)*exp(1))/sqrt(-a*d*exp(1)^3+a*d*exp(1)*exp(2))))
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maple [A] time = 0.05, size = 146, normalized size = 0.85 \[ -\frac {2 \left (c d x +a e \right ) \left (8 c^{2} d^{2} e^{2} x^{2}-4 a c d \,e^{3} x +20 c^{2} d^{3} e x +3 a^{2} e^{4}-10 a c \,d^{2} e^{2}+15 c^{2} d^{4}\right )}{15 \left (e x +d \right )^{2} \left (a^{3} e^{6}-3 a^{2} c \,d^{2} e^{4}+3 a \,c^{2} d^{4} e^{2}-c^{3} d^{6}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(e*x+d)^3/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2),x)
[Out]
-2/15*(c*d*x+a*e)*(8*c^2*d^2*e^2*x^2-4*a*c*d*e^3*x+20*c^2*d^3*e*x+3*a^2*e^4-10*a*c*d^2*e^2+15*c^2*d^4)/(e*x+d)
^2/(a^3*e^6-3*a^2*c*d^2*e^4+3*a*c^2*d^4*e^2-c^3*d^6)/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e^2-c*d^2>0)', see `assume?`
for more details)Is a*e^2-c*d^2 zero or nonzero?
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mupad [B] time = 0.78, size = 110, normalized size = 0.64 \[ -\frac {2\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (3\,a^2\,e^4-10\,a\,c\,d^2\,e^2-4\,a\,c\,d\,e^3\,x+15\,c^2\,d^4+20\,c^2\,d^3\,e\,x+8\,c^2\,d^2\,e^2\,x^2\right )}{15\,{\left (a\,e^2-c\,d^2\right )}^3\,{\left (d+e\,x\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((d + e*x)^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)),x)
[Out]
-(2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*(3*a^2*e^4 + 15*c^2*d^4 + 8*c^2*d^2*e^2*x^2 - 10*a*c*d^2*e^2
+ 20*c^2*d^3*e*x - 4*a*c*d*e^3*x))/(15*(a*e^2 - c*d^2)^3*(d + e*x)^3)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)
[Out]
Integral(1/(sqrt((d + e*x)*(a*e + c*d*x))*(d + e*x)**3), x)
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