3.17.39 \(\int \frac {1}{(d+e x)^4 \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\)
Optimal. Leaf size=231 \[ \frac {32 c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 (d+e x) \left (c d^2-a e^2\right )^4}+\frac {16 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 (d+e x)^2 \left (c d^2-a e^2\right )^3}+\frac {12 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 (d+e x)^3 \left (c d^2-a e^2\right )^2}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 (d+e x)^4 \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.12, antiderivative size = 231, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used =
{658, 650} \begin {gather*} \frac {32 c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 (d+e x) \left (c d^2-a e^2\right )^4}+\frac {16 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 (d+e x)^2 \left (c d^2-a e^2\right )^3}+\frac {12 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 (d+e x)^3 \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}}{7 (d+e x)^4 \left (c d^2-a e^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/((d + e*x)^4*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])/(7*(c*d^2 - a*e^2)*(d + e*x)^4) + (12*c*d*Sqrt[a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2])/(35*(c*d^2 - a*e^2)^2*(d + e*x)^3) + (16*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d
*e*x^2])/(35*(c*d^2 - a*e^2)^3*(d + e*x)^2) + (32*c^3*d^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*(c*
d^2 - a*e^2)^4*(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)^4 \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}}{7 \left (c d^2-a e^2\right ) (d+e x)^4}+\frac {(6 c d) \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}{7 \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}}{7 \left (c d^2-a e^2\right ) (d+e x)^4}+\frac {12 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 \left (c d^2-a e^2\right )^2 (d+e x)^3}+\frac {\left (24 c^2 d^2\right ) \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}{35 \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}}{7 \left (c d^2-a e^2\right ) (d+e x)^4}+\frac {12 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 \left (c d^2-a e^2\right )^2 (d+e x)^3}+\frac {16 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 \left (c d^2-a e^2\right )^3 (d+e x)^2}+\frac {\left (16 c^3 d^3\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}{35 \left (c d^2-a e^2\right )^3}\\ &=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 \left (c d^2-a e^2\right ) (d+e x)^4}+\frac {12 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 \left (c d^2-a e^2\right )^2 (d+e x)^3}+\frac {16 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 \left (c d^2-a e^2\right )^3 (d+e x)^2}+\frac {32 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 \left (c d^2-a e^2\right )^4 (d+e x)}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 138, normalized size = 0.60 \begin {gather*} \frac {2 \sqrt {(d+e x) (a e+c d x)} \left (-5 a^3 e^6+3 a^2 c d e^4 (7 d+2 e x)-a c^2 d^2 e^2 \left (35 d^2+28 d e x+8 e^2 x^2\right )+c^3 d^3 \left (35 d^3+70 d^2 e x+56 d e^2 x^2+16 e^3 x^3\right )\right )}{35 (d+e x)^4 \left (c d^2-a e^2\right )^4} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/((d + e*x)^4*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)]*(-5*a^3*e^6 + 3*a^2*c*d*e^4*(7*d + 2*e*x) - a*c^2*d^2*e^2*(35*d^2 + 28*d*e*x
+ 8*e^2*x^2) + c^3*d^3*(35*d^3 + 70*d^2*e*x + 56*d*e^2*x^2 + 16*e^3*x^3)))/(35*(c*d^2 - a*e^2)^4*(d + e*x)^4)
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IntegrateAlgebraic [F] time = 180.01, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[1/((d + e*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]
[Out]
$Aborted
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fricas [B] time = 5.43, size = 452, normalized size = 1.96 \begin {gather*} \frac {2 \, {\left (16 \, c^{3} d^{3} e^{3} x^{3} + 35 \, c^{3} d^{6} - 35 \, a c^{2} d^{4} e^{2} + 21 \, a^{2} c d^{2} e^{4} - 5 \, a^{3} e^{6} + 8 \, {\left (7 \, c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (35 \, c^{3} d^{5} e - 14 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{35 \, {\left (c^{4} d^{12} - 4 \, a c^{3} d^{10} e^{2} + 6 \, a^{2} c^{2} d^{8} e^{4} - 4 \, a^{3} c d^{6} e^{6} + a^{4} d^{4} e^{8} + {\left (c^{4} d^{8} e^{4} - 4 \, a c^{3} d^{6} e^{6} + 6 \, a^{2} c^{2} d^{4} e^{8} - 4 \, a^{3} c d^{2} e^{10} + a^{4} e^{12}\right )} x^{4} + 4 \, {\left (c^{4} d^{9} e^{3} - 4 \, a c^{3} d^{7} e^{5} + 6 \, a^{2} c^{2} d^{5} e^{7} - 4 \, a^{3} c d^{3} e^{9} + a^{4} d e^{11}\right )} x^{3} + 6 \, {\left (c^{4} d^{10} e^{2} - 4 \, a c^{3} d^{8} e^{4} + 6 \, a^{2} c^{2} d^{6} e^{6} - 4 \, a^{3} c d^{4} e^{8} + a^{4} d^{2} e^{10}\right )} x^{2} + 4 \, {\left (c^{4} d^{11} e - 4 \, a c^{3} d^{9} e^{3} + 6 \, a^{2} c^{2} d^{7} e^{5} - 4 \, a^{3} c d^{5} e^{7} + a^{4} d^{3} e^{9}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="fricas")
[Out]
2/35*(16*c^3*d^3*e^3*x^3 + 35*c^3*d^6 - 35*a*c^2*d^4*e^2 + 21*a^2*c*d^2*e^4 - 5*a^3*e^6 + 8*(7*c^3*d^4*e^2 - a
*c^2*d^2*e^4)*x^2 + 2*(35*c^3*d^5*e - 14*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a
*e^2)*x)/(c^4*d^12 - 4*a*c^3*d^10*e^2 + 6*a^2*c^2*d^8*e^4 - 4*a^3*c*d^6*e^6 + a^4*d^4*e^8 + (c^4*d^8*e^4 - 4*a
*c^3*d^6*e^6 + 6*a^2*c^2*d^4*e^8 - 4*a^3*c*d^2*e^10 + a^4*e^12)*x^4 + 4*(c^4*d^9*e^3 - 4*a*c^3*d^7*e^5 + 6*a^2
*c^2*d^5*e^7 - 4*a^3*c*d^3*e^9 + a^4*d*e^11)*x^3 + 6*(c^4*d^10*e^2 - 4*a*c^3*d^8*e^4 + 6*a^2*c^2*d^6*e^6 - 4*a
^3*c*d^4*e^8 + a^4*d^2*e^10)*x^2 + 4*(c^4*d^11*e - 4*a*c^3*d^9*e^3 + 6*a^2*c^2*d^7*e^5 - 4*a^3*c*d^5*e^7 + a^4
*d^3*e^9)*x)
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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(1/(e*x+d)^4/(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*((-15*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)^5*a^3*exp(2)^3+9*c*d^2*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)^5*a^2*exp(2)^2+36*c*d^2*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)^5*a^2*exp(2)-9*c^2*d^4*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)^5*a*exp(2)-36*c^2*d^4*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)^5*a+15*c^3*d^6*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)^5+75*d*exp(1)^2*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)^4*a^3*exp(2)^3-45*c*d^3*exp(1)^2*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)^4*a^2*exp(2)^2-180*c*d^3*exp(1)^4*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)^4*a^2*exp(2)+45*c^2*d^5*exp(1)^2*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)^4*a*exp(2)+180*c^2*d^5*exp(1)^4*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)^4*a-75*c^3*d^7*exp(1)^2*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)^4-40*d*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^4*exp(2)^4+40*d*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^4*exp(2)^3-126*c*d^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^3*exp(2)^3+72*c*d^3*exp(1)^4*(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*a^3*exp(2)^2-96*c*d^3*exp(1)^6*(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*exp(2)+66*c^2*d^5*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+288*c^2*d^5*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^2*exp(2)+96*c^2*d^5*exp(1)^6*(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^2-50*c^3*d^7*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
00*c^3*d^7*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+150*c^4*d^9*ex
p(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+120*d^2*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*a^4*exp(2)^4-120*d^2*exp(1)^3*sqr
t(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^4*exp(2)^3-18*c*d^4*
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*a^3*exp(2)^3
+72*c*d^4*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^3*exp(2)^2+96*c*d^4*exp(1)^7*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*e
xp(1))*x)^2*a^3-18*c^2*d^6*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*a^2*exp(2)^2-144*c^2*d^6*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^2*exp(2)-288*c^2*d^6*exp(1)^5*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^2-30*c^3*d^8*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*a*exp(2)+480*c^3*d^8*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-150*c^4*d^10*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)^2-33*d^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)*a^5*exp(2)^5+66*d^2*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^5*exp(2)^4-33*d^2*exp(1)^5*(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)*a^5*exp(2)^3+15*c*d^4*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)*a^4*exp(2)^4-186*c*d^4*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^4*exp(2)^3+207*c*d^4*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^4
*exp(2)^2-36*c*d^4*exp(1)^7*(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^4*exp(2)
+54*c^2*d^6*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)^3-54*c^2
*d^6*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+81*c^2*d^6*
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^3*exp(2)-156*c^2*d^6*exp(1)
^7*(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+6*c^3*d^8*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)^2-102*c^3*d^8*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)+321*c^3*d^8*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+75*c^4*d^10*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)-300*c^4*d^10*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+75*c^5*d^12*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)-15*d^3*sqrt(c*d*exp(1))*a^5*exp(2)^5+78*d^3*exp(1)^2*sqrt(c*d*exp(1))*a^5*exp(2)^4-111*d^3*exp(1)^4*s
qrt(c*d*exp(1))*a^5*exp(2)^3+48*d^3*exp(1)^6*sqrt(c*d*exp(1))*a^5*exp(2)^2-31*c*d^5*sqrt(c*d*exp(1))*a^4*exp(2
)^4+106*c*d^5*exp(1)^2*sqrt(c*d*exp(1))*a^4*exp(2)^3-111*c*d^5*exp(1)^4*sqrt(c*d*exp(1))*a^4*exp(2)^2+68*c*d^5
*exp(1)^6*sqrt(c*d*exp(1))*a^4*exp(2)-32*c*d^5*exp(1)^8*sqrt(c*d*exp(1))*a^4-18*c^2*d^7*sqrt(c*d*exp(1))*a^3*e
xp(2)^3+54*c^2*d^7*exp(1)^2*sqrt(c*d*exp(1))*a^3*exp(2)^2-129*c^2*d^7*exp(1)^4*sqrt(c*d*exp(1))*a^3*exp(2)+108
*c^2*d^7*exp(1)^6*sqrt(c*d*exp(1))*a^3-18*c^3*d^9*sqrt(c*d*exp(1))*a^2*exp(2)^2+102*c^3*d^9*exp(1)^2*sqrt(c*d*
exp(1))*a^2*exp(2)-129*c^3*d^9*exp(1)^4*sqrt(c*d*exp(1))*a^2-31*c^4*d^11*sqrt(c*d*exp(1))*a*exp(2)+76*c^4*d^11
*exp(1)^2*sqrt(c*d*exp(1))*a-15*c^5*d^13*sqrt(c*d*exp(1)))/(48*d^3*exp(1)*a^3*exp(2)^3-144*d^3*exp(1)^3*a^3*ex
p(2)^2+144*d^3*exp(1)^5*a^3*exp(2)-48*d^3*exp(1)^7*a^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)^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)^3+(5*a^3*exp(2)^3-3*c*d^2*a^2*exp(2)^2-12*c*d^2*exp(1)^2*a^2*exp(2)
+3*c^2*d^4*a*exp(2)+12*c^2*d^4*exp(1)^2*a-5*c^3*d^6)/2/(8*d^3*a^3*exp(2)^3-24*d^3*exp(1)^2*a^3*exp(2)^2+24*d^3
*exp(1)^4*a^3*exp(2)-8*d^3*exp(1)^6*a^3)/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 = 0.06, size = 217, normalized size = 0.94 \begin {gather*} -\frac {2 \left (c d x +a e \right ) \left (-16 c^{3} d^{3} e^{3} x^{3}+8 a \,c^{2} d^{2} e^{4} x^{2}-56 c^{3} d^{4} e^{2} x^{2}-6 a^{2} c d \,e^{5} x +28 a \,c^{2} d^{3} e^{3} x -70 c^{3} d^{5} e x +5 a^{3} e^{6}-21 a^{2} c \,d^{2} e^{4}+35 a \,c^{2} d^{4} e^{2}-35 c^{3} d^{6}\right )}{35 \left (e x +d \right )^{3} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(e*x+d)^4/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2),x)
[Out]
-2/35*(c*d*x+a*e)*(-16*c^3*d^3*e^3*x^3+8*a*c^2*d^2*e^4*x^2-56*c^3*d^4*e^2*x^2-6*a^2*c*d*e^5*x+28*a*c^2*d^3*e^3
*x-70*c^3*d^5*e*x+5*a^3*e^6-21*a^2*c*d^2*e^4+35*a*c^2*d^4*e^2-35*c^3*d^6)/(e*x+d)^3/(a^4*e^8-4*a^3*c*d^2*e^6+6
*a^2*c^2*d^4*e^4-4*a*c^3*d^6*e^2+c^4*d^8)/(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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^4/(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 = 0.81, size = 252, normalized size = 1.09 \begin {gather*} \frac {32\,c^3\,d^3\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{35\,{\left (a\,e^2-c\,d^2\right )}^4\,\left (d+e\,x\right )}-\frac {2\,e\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{\left (7\,a\,e^3-7\,c\,d^2\,e\right )\,{\left (d+e\,x\right )}^4}-\frac {48\,c^2\,d^2\,e\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{35\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )\,{\left (d+e\,x\right )}^2}+\frac {12\,c\,d\,e\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{7\,\left (a\,e^2-c\,d^2\right )\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )\,{\left (d+e\,x\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((d + e*x)^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)),x)
[Out]
(32*c^3*d^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(35*(a*e^2 - c*d^2)^4*(d + e*x)) - (2*e*(x*(a*e^2 +
c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/((7*a*e^3 - 7*c*d^2*e)*(d + e*x)^4) - (48*c^2*d^2*e*(x*(a*e^2 + c*d^2) + a
*d*e + c*d*e*x^2)^(1/2))/(35*(a*e^2 - c*d^2)^2*(3*a*e^3 - 3*c*d^2*e)*(d + e*x)^2) + (12*c*d*e*(x*(a*e^2 + c*d^
2) + a*d*e + c*d*e*x^2)^(1/2))/(7*(a*e^2 - c*d^2)*(5*a*e^3 - 5*c*d^2*e)*(d + e*x)^3)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (d + e x\right )^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)**4/(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)**4), x)
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