3.2023 \(\int \frac {(d+e x)^{5/2}}{(a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx\)

Optimal. Leaf size=146 \[ \frac {3 e \sqrt {d+e x}}{4 \left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac {\sqrt {d+e x}}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2}-\frac {3 e^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{4 \sqrt {c} \sqrt {d} \left (c d^2-a e^2\right )^{5/2}} \]

[Out]

-3/4*e^2*arctanh(c^(1/2)*d^(1/2)*(e*x+d)^(1/2)/(-a*e^2+c*d^2)^(1/2))/(-a*e^2+c*d^2)^(5/2)/c^(1/2)/d^(1/2)-1/2*
(e*x+d)^(1/2)/(-a*e^2+c*d^2)/(c*d*x+a*e)^2+3/4*e*(e*x+d)^(1/2)/(-a*e^2+c*d^2)^2/(c*d*x+a*e)

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Rubi [A]  time = 0.09, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {626, 51, 63, 208} \[ \frac {3 e \sqrt {d+e x}}{4 \left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac {\sqrt {d+e x}}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2}-\frac {3 e^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{4 \sqrt {c} \sqrt {d} \left (c d^2-a e^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(5/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

-Sqrt[d + e*x]/(2*(c*d^2 - a*e^2)*(a*e + c*d*x)^2) + (3*e*Sqrt[d + e*x])/(4*(c*d^2 - a*e^2)^2*(a*e + c*d*x)) -
(3*e^2*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(4*Sqrt[c]*Sqrt[d]*(c*d^2 - a*e^2)^(5/2)
)

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx &=\int \frac {1}{(a e+c d x)^3 \sqrt {d+e x}} \, dx\\ &=-\frac {\sqrt {d+e x}}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2}-\frac {(3 e) \int \frac {1}{(a e+c d x)^2 \sqrt {d+e x}} \, dx}{4 \left (c d^2-a e^2\right )}\\ &=-\frac {\sqrt {d+e x}}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2}+\frac {3 e \sqrt {d+e x}}{4 \left (c d^2-a e^2\right )^2 (a e+c d x)}+\frac {\left (3 e^2\right ) \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{8 \left (c d^2-a e^2\right )^2}\\ &=-\frac {\sqrt {d+e x}}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2}+\frac {3 e \sqrt {d+e x}}{4 \left (c d^2-a e^2\right )^2 (a e+c d x)}+\frac {(3 e) \operatorname {Subst}\left (\int \frac {1}{-\frac {c d^2}{e}+a e+\frac {c d x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{4 \left (c d^2-a e^2\right )^2}\\ &=-\frac {\sqrt {d+e x}}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2}+\frac {3 e \sqrt {d+e x}}{4 \left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac {3 e^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{4 \sqrt {c} \sqrt {d} \left (c d^2-a e^2\right )^{5/2}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 59, normalized size = 0.40 \[ \frac {2 e^2 \sqrt {d+e x} \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};-\frac {c d (d+e x)}{a e^2-c d^2}\right )}{\left (a e^2-c d^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(5/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

(2*e^2*Sqrt[d + e*x]*Hypergeometric2F1[1/2, 3, 3/2, -((c*d*(d + e*x))/(-(c*d^2) + a*e^2))])/(-(c*d^2) + a*e^2)
^3

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fricas [B]  time = 0.85, size = 654, normalized size = 4.48 \[ \left [\frac {3 \, {\left (c^{2} d^{2} e^{2} x^{2} + 2 \, a c d e^{3} x + a^{2} e^{4}\right )} \sqrt {c^{2} d^{3} - a c d e^{2}} \log \left (\frac {c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt {c^{2} d^{3} - a c d e^{2}} \sqrt {e x + d}}{c d x + a e}\right ) - 2 \, {\left (2 \, c^{3} d^{5} - 7 \, a c^{2} d^{3} e^{2} + 5 \, a^{2} c d e^{4} - 3 \, {\left (c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{8 \, {\left (a^{2} c^{4} d^{7} e^{2} - 3 \, a^{3} c^{3} d^{5} e^{4} + 3 \, a^{4} c^{2} d^{3} e^{6} - a^{5} c d e^{8} + {\left (c^{6} d^{9} - 3 \, a c^{5} d^{7} e^{2} + 3 \, a^{2} c^{4} d^{5} e^{4} - a^{3} c^{3} d^{3} e^{6}\right )} x^{2} + 2 \, {\left (a c^{5} d^{8} e - 3 \, a^{2} c^{4} d^{6} e^{3} + 3 \, a^{3} c^{3} d^{4} e^{5} - a^{4} c^{2} d^{2} e^{7}\right )} x\right )}}, \frac {3 \, {\left (c^{2} d^{2} e^{2} x^{2} + 2 \, a c d e^{3} x + a^{2} e^{4}\right )} \sqrt {-c^{2} d^{3} + a c d e^{2}} \arctan \left (\frac {\sqrt {-c^{2} d^{3} + a c d e^{2}} \sqrt {e x + d}}{c d e x + c d^{2}}\right ) - {\left (2 \, c^{3} d^{5} - 7 \, a c^{2} d^{3} e^{2} + 5 \, a^{2} c d e^{4} - 3 \, {\left (c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{4 \, {\left (a^{2} c^{4} d^{7} e^{2} - 3 \, a^{3} c^{3} d^{5} e^{4} + 3 \, a^{4} c^{2} d^{3} e^{6} - a^{5} c d e^{8} + {\left (c^{6} d^{9} - 3 \, a c^{5} d^{7} e^{2} + 3 \, a^{2} c^{4} d^{5} e^{4} - a^{3} c^{3} d^{3} e^{6}\right )} x^{2} + 2 \, {\left (a c^{5} d^{8} e - 3 \, a^{2} c^{4} d^{6} e^{3} + 3 \, a^{3} c^{3} d^{4} e^{5} - a^{4} c^{2} d^{2} e^{7}\right )} x\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="fricas")

[Out]

[1/8*(3*(c^2*d^2*e^2*x^2 + 2*a*c*d*e^3*x + a^2*e^4)*sqrt(c^2*d^3 - a*c*d*e^2)*log((c*d*e*x + 2*c*d^2 - a*e^2 -
2*sqrt(c^2*d^3 - a*c*d*e^2)*sqrt(e*x + d))/(c*d*x + a*e)) - 2*(2*c^3*d^5 - 7*a*c^2*d^3*e^2 + 5*a^2*c*d*e^4 -
3*(c^3*d^4*e - a*c^2*d^2*e^3)*x)*sqrt(e*x + d))/(a^2*c^4*d^7*e^2 - 3*a^3*c^3*d^5*e^4 + 3*a^4*c^2*d^3*e^6 - a^5
*c*d*e^8 + (c^6*d^9 - 3*a*c^5*d^7*e^2 + 3*a^2*c^4*d^5*e^4 - a^3*c^3*d^3*e^6)*x^2 + 2*(a*c^5*d^8*e - 3*a^2*c^4*
d^6*e^3 + 3*a^3*c^3*d^4*e^5 - a^4*c^2*d^2*e^7)*x), 1/4*(3*(c^2*d^2*e^2*x^2 + 2*a*c*d*e^3*x + a^2*e^4)*sqrt(-c^
2*d^3 + a*c*d*e^2)*arctan(sqrt(-c^2*d^3 + a*c*d*e^2)*sqrt(e*x + d)/(c*d*e*x + c*d^2)) - (2*c^3*d^5 - 7*a*c^2*d
^3*e^2 + 5*a^2*c*d*e^4 - 3*(c^3*d^4*e - a*c^2*d^2*e^3)*x)*sqrt(e*x + d))/(a^2*c^4*d^7*e^2 - 3*a^3*c^3*d^5*e^4
+ 3*a^4*c^2*d^3*e^6 - a^5*c*d*e^8 + (c^6*d^9 - 3*a*c^5*d^7*e^2 + 3*a^2*c^4*d^5*e^4 - a^3*c^3*d^3*e^6)*x^2 + 2*
(a*c^5*d^8*e - 3*a^2*c^4*d^6*e^3 + 3*a^3*c^3*d^4*e^5 - a^4*c^2*d^2*e^7)*x)]

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giac [B]  time = 82.08, size = 5749, normalized size = 39.38 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="giac")

[Out]

3/8*(4*c^6*d^11*e^2 + 2*sqrt(2)*sqrt(-c^2*d^3 + a*c*d*e^2 - sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*c^5*d
^10*e^2 + 6*c^6*d^10*e^2 - 20*a*c^5*d^9*e^4 + 4*sqrt(2)*sqrt(-c^2*d^3 + a*c*d*e^2 - sqrt(c^2*d^4 - 2*a*c*d^2*e
^2 + a^2*e^4)*c*d)*c^5*d^9*e^2 - 10*sqrt(2)*sqrt(-c^2*d^3 + a*c*d*e^2 - sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4
)*c*d)*a*c^4*d^8*e^4 - 24*a*c^5*d^8*e^4 + 2*sqrt(2)*sqrt(-c^2*d^3 + a*c*d*e^2 - sqrt(c^2*d^4 - 2*a*c*d^2*e^2 +
a^2*e^4)*c*d)*c^5*d^8*e^2 - 3*sqrt(2)*sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(-c^2*d^3 + a*c*d*e^2 - sqr
t(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*c^4*d^8*e^2 + 40*a^2*c^4*d^7*e^6 - 16*sqrt(2)*sqrt(-c^2*d^3 + a*c*d*
e^2 - sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*a*c^4*d^7*e^4 - 6*sqrt(2)*sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^
2*e^4)*sqrt(-c^2*d^3 + a*c*d*e^2 - sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*c^4*d^7*e^2 + 20*sqrt(2)*sqrt(
-c^2*d^3 + a*c*d*e^2 - sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*a^2*c^3*d^6*e^6 + 36*a^2*c^4*d^6*e^6 - 6*s
qrt(2)*sqrt(-c^2*d^3 + a*c*d*e^2 - sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*a*c^4*d^6*e^4 - 4*(c^2*d^4 - 2
*a*c*d^2*e^2 + a^2*e^4)*c^4*d^7*e^2 + 12*sqrt(2)*sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(-c^2*d^3 + a*c*d
*e^2 - sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*a*c^3*d^6*e^4 - 3*sqrt(2)*sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a
^2*e^4)*sqrt(-c^2*d^3 + a*c*d*e^2 - sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*c^4*d^6*e^2 - 40*a^3*c^3*d^5*
e^8 + 24*sqrt(2)*sqrt(-c^2*d^3 + a*c*d*e^2 - sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*a^2*c^3*d^5*e^6 - 6*
(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c^4*d^6*e^2 + 18*sqrt(2)*sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(-c^2
*d^3 + a*c*d*e^2 - sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*a*c^3*d^5*e^4 - 20*sqrt(2)*sqrt(-c^2*d^3 + a*c
*d*e^2 - sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*a^3*c^2*d^4*e^8 - 24*a^3*c^3*d^4*e^8 + 6*sqrt(2)*sqrt(-c
^2*d^3 + a*c*d*e^2 - sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*a^2*c^3*d^4*e^6 + 12*(c^2*d^4 - 2*a*c*d^2*e^
2 + a^2*e^4)*a*c^3*d^5*e^4 - 18*sqrt(2)*sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(-c^2*d^3 + a*c*d*e^2 - sq
rt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*a^2*c^2*d^4*e^6 + 6*sqrt(2)*sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)
*sqrt(-c^2*d^3 + a*c*d*e^2 - sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*a*c^3*d^4*e^4 + 20*a^4*c^2*d^3*e^10
- 16*sqrt(2)*sqrt(-c^2*d^3 + a*c*d*e^2 - sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*a^3*c^2*d^3*e^8 + 12*(c^
2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*a*c^3*d^4*e^4 - 18*sqrt(2)*sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(-c^2*
d^3 + a*c*d*e^2 - sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*a^2*c^2*d^3*e^6 + 10*sqrt(2)*sqrt(-c^2*d^3 + a*
c*d*e^2 - sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*a^4*c*d^2*e^10 + 6*a^4*c^2*d^2*e^10 - 2*sqrt(2)*sqrt(-c
^2*d^3 + a*c*d*e^2 - sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*a^3*c^2*d^2*e^8 - 12*(c^2*d^4 - 2*a*c*d^2*e^
2 + a^2*e^4)*a^2*c^2*d^3*e^6 + 12*sqrt(2)*sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(-c^2*d^3 + a*c*d*e^2 -
sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*a^3*c*d^2*e^8 - 3*sqrt(2)*sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)
*sqrt(-c^2*d^3 + a*c*d*e^2 - sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*a^2*c^2*d^2*e^6 - 4*a^5*c*d*e^12 + 4
*sqrt(2)*sqrt(-c^2*d^3 + a*c*d*e^2 - sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*a^4*c*d*e^10 - 6*(c^2*d^4 -
2*a*c*d^2*e^2 + a^2*e^4)*a^2*c^2*d^2*e^6 + 6*sqrt(2)*sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(-c^2*d^3 + a
*c*d*e^2 - sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*a^3*c*d*e^8 - 2*sqrt(2)*sqrt(-c^2*d^3 + a*c*d*e^2 - sq
rt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*a^5*e^12 + 4*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*a^3*c*d*e^8 - 3*sq
rt(2)*sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(-c^2*d^3 + a*c*d*e^2 - sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e
^4)*c*d)*a^4*e^10)*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 1
0*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10 + sqrt((c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^
3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10)^2))/(c^5*d^9 - 4*a*c^4*d^7*e^2 + 6*a^2*c^3*d^5*e^4 - 4*a^3*c^2*d^3
*e^6 + a^4*c*d*e^8)))/((c^8*d^16 + 2*c^8*d^15 - 8*a*c^7*d^14*e^2 + c^8*d^14 - 14*a*c^7*d^13*e^2 + 28*a^2*c^6*d
^12*e^4 - 6*a*c^7*d^12*e^2 + 42*a^2*c^6*d^11*e^4 - 56*a^3*c^5*d^10*e^6 + 15*a^2*c^6*d^10*e^4 - 70*a^3*c^5*d^9*
e^6 + 70*a^4*c^4*d^8*e^8 - 20*a^3*c^5*d^8*e^6 + 70*a^4*c^4*d^7*e^8 - 56*a^5*c^3*d^6*e^10 + 15*a^4*c^4*d^6*e^8
- 42*a^5*c^3*d^5*e^10 + 28*a^6*c^2*d^4*e^12 - 6*a^5*c^3*d^4*e^10 + 14*a^6*c^2*d^3*e^12 - 8*a^7*c*d^2*e^14 + a^
6*c^2*d^2*e^12 - 2*a^7*c*d*e^14 + a^8*e^16)*abs(c)*abs(d)) - 3/8*(4*c^6*d^11*e^2 - 2*sqrt(2)*sqrt(-c^2*d^3 + a
*c*d*e^2 + sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*c^5*d^10*e^2 + 6*c^6*d^10*e^2 - 20*a*c^5*d^9*e^4 - 4*s
qrt(2)*sqrt(-c^2*d^3 + a*c*d*e^2 + sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*c^5*d^9*e^2 + 10*sqrt(2)*sqrt(
-c^2*d^3 + a*c*d*e^2 + sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*a*c^4*d^8*e^4 - 24*a*c^5*d^8*e^4 - 2*sqrt(
2)*sqrt(-c^2*d^3 + a*c*d*e^2 + sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*c^5*d^8*e^2 - 3*sqrt(2)*sqrt(c^2*d
^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(-c^2*d^3 + a*c*d*e^2 + sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*c^4*d^8
*e^2 + 40*a^2*c^4*d^7*e^6 + 16*sqrt(2)*sqrt(-c^2*d^3 + a*c*d*e^2 + sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d
)*a*c^4*d^7*e^4 - 6*sqrt(2)*sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(-c^2*d^3 + a*c*d*e^2 + sqrt(c^2*d^4 -
2*a*c*d^2*e^2 + a^2*e^4)*c*d)*c^4*d^7*e^2 - 20*sqrt(2)*sqrt(-c^2*d^3 + a*c*d*e^2 + sqrt(c^2*d^4 - 2*a*c*d^2*e
^2 + a^2*e^4)*c*d)*a^2*c^3*d^6*e^6 + 36*a^2*c^4*d^6*e^6 + 6*sqrt(2)*sqrt(-c^2*d^3 + a*c*d*e^2 + sqrt(c^2*d^4 -
2*a*c*d^2*e^2 + a^2*e^4)*c*d)*a*c^4*d^6*e^4 - 4*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c^4*d^7*e^2 + 12*sqrt(2)*
sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(-c^2*d^3 + a*c*d*e^2 + sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*
d)*a*c^3*d^6*e^4 - 3*sqrt(2)*sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(-c^2*d^3 + a*c*d*e^2 + sqrt(c^2*d^4
- 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*c^4*d^6*e^2 - 40*a^3*c^3*d^5*e^8 - 24*sqrt(2)*sqrt(-c^2*d^3 + a*c*d*e^2 + sqrt
(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*a^2*c^3*d^5*e^6 - 6*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c^4*d^6*e^2 +
18*sqrt(2)*sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(-c^2*d^3 + a*c*d*e^2 + sqrt(c^2*d^4 - 2*a*c*d^2*e^2 +
a^2*e^4)*c*d)*a*c^3*d^5*e^4 + 20*sqrt(2)*sqrt(-c^2*d^3 + a*c*d*e^2 + sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*
c*d)*a^3*c^2*d^4*e^8 - 24*a^3*c^3*d^4*e^8 - 6*sqrt(2)*sqrt(-c^2*d^3 + a*c*d*e^2 + sqrt(c^2*d^4 - 2*a*c*d^2*e^2
+ a^2*e^4)*c*d)*a^2*c^3*d^4*e^6 + 12*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*a*c^3*d^5*e^4 - 18*sqrt(2)*sqrt(c^2*
d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(-c^2*d^3 + a*c*d*e^2 + sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*a^2*c^
2*d^4*e^6 + 6*sqrt(2)*sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(-c^2*d^3 + a*c*d*e^2 + sqrt(c^2*d^4 - 2*a*c
*d^2*e^2 + a^2*e^4)*c*d)*a*c^3*d^4*e^4 + 20*a^4*c^2*d^3*e^10 + 16*sqrt(2)*sqrt(-c^2*d^3 + a*c*d*e^2 + sqrt(c^2
*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*a^3*c^2*d^3*e^8 + 12*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*a*c^3*d^4*e^4 -
18*sqrt(2)*sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(-c^2*d^3 + a*c*d*e^2 + sqrt(c^2*d^4 - 2*a*c*d^2*e^2 +
a^2*e^4)*c*d)*a^2*c^2*d^3*e^6 - 10*sqrt(2)*sqrt(-c^2*d^3 + a*c*d*e^2 + sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)
*c*d)*a^4*c*d^2*e^10 + 6*a^4*c^2*d^2*e^10 + 2*sqrt(2)*sqrt(-c^2*d^3 + a*c*d*e^2 + sqrt(c^2*d^4 - 2*a*c*d^2*e^2
+ a^2*e^4)*c*d)*a^3*c^2*d^2*e^8 - 12*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*a^2*c^2*d^3*e^6 + 12*sqrt(2)*sqrt(c^
2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(-c^2*d^3 + a*c*d*e^2 + sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*a^3*
c*d^2*e^8 - 3*sqrt(2)*sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(-c^2*d^3 + a*c*d*e^2 + sqrt(c^2*d^4 - 2*a*c
*d^2*e^2 + a^2*e^4)*c*d)*a^2*c^2*d^2*e^6 - 4*a^5*c*d*e^12 - 4*sqrt(2)*sqrt(-c^2*d^3 + a*c*d*e^2 + sqrt(c^2*d^4
- 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*a^4*c*d*e^10 - 6*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*a^2*c^2*d^2*e^6 + 6*sqrt
(2)*sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(-c^2*d^3 + a*c*d*e^2 + sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4
)*c*d)*a^3*c*d*e^8 + 2*sqrt(2)*sqrt(-c^2*d^3 + a*c*d*e^2 + sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*a^5*e^
12 + 4*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*a^3*c*d*e^8 - 3*sqrt(2)*sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqr
t(-c^2*d^3 + a*c*d*e^2 + sqrt(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*c*d)*a^4*e^10)*arctan(2*sqrt(1/2)*sqrt(x*e +
d)/sqrt(-(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10 -
sqrt((c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10)^2))/(
c^5*d^9 - 4*a*c^4*d^7*e^2 + 6*a^2*c^3*d^5*e^4 - 4*a^3*c^2*d^3*e^6 + a^4*c*d*e^8)))/((c^8*d^16 + 2*c^8*d^15 - 8
*a*c^7*d^14*e^2 + c^8*d^14 - 14*a*c^7*d^13*e^2 + 28*a^2*c^6*d^12*e^4 - 6*a*c^7*d^12*e^2 + 42*a^2*c^6*d^11*e^4
- 56*a^3*c^5*d^10*e^6 + 15*a^2*c^6*d^10*e^4 - 70*a^3*c^5*d^9*e^6 + 70*a^4*c^4*d^8*e^8 - 20*a^3*c^5*d^8*e^6 + 7
0*a^4*c^4*d^7*e^8 - 56*a^5*c^3*d^6*e^10 + 15*a^4*c^4*d^6*e^8 - 42*a^5*c^3*d^5*e^10 + 28*a^6*c^2*d^4*e^12 - 6*a
^5*c^3*d^4*e^10 + 14*a^6*c^2*d^3*e^12 - 8*a^7*c*d^2*e^14 + a^6*c^2*d^2*e^12 - 2*a^7*c*d*e^14 + a^8*e^16)*abs(c
)*abs(d)) + 1/4*(3*(x*e + d)^(7/2)*c^3*d^5*e^2 - 5*(x*e + d)^(5/2)*c^3*d^6*e^2 - 6*(x*e + d)^(7/2)*a*c^2*d^3*e
^4 + 15*(x*e + d)^(5/2)*a*c^2*d^4*e^4 + 3*(x*e + d)^(7/2)*a^2*c*d*e^6 - 15*(x*e + d)^(5/2)*a^2*c*d^2*e^6 + 5*(
x*e + d)^(5/2)*a^3*e^8)/((c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8)*((x*e + d
)^2*c*d - (x*e + d)*c*d^2 + (x*e + d)*a*e^2)^2)

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maple [A]  time = 0.07, size = 144, normalized size = 0.99 \[ \frac {3 e^{2} \arctan \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) c d}}\right )}{4 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) c d}}+\frac {\sqrt {e x +d}\, e^{2}}{2 \left (a \,e^{2}-c \,d^{2}\right ) \left (c d e x +a \,e^{2}\right )^{2}}+\frac {3 \sqrt {e x +d}\, e^{2}}{4 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (c d e x +a \,e^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(5/2)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^3,x)

[Out]

1/2*e^2*(e*x+d)^(1/2)/(a*e^2-c*d^2)/(c*d*e*x+a*e^2)^2+3/4*e^2/(a*e^2-c*d^2)^2*(e*x+d)^(1/2)/(c*d*e*x+a*e^2)+3/
4*e^2/(a*e^2-c*d^2)^2/((a*e^2-c*d^2)*c*d)^(1/2)*arctan((e*x+d)^(1/2)/((a*e^2-c*d^2)*c*d)^(1/2)*c*d)

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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((e*x+d)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,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 positive or negative?

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mupad [B]  time = 0.72, size = 177, normalized size = 1.21 \[ \frac {\frac {5\,e^2\,\sqrt {d+e\,x}}{4\,\left (a\,e^2-c\,d^2\right )}+\frac {3\,c\,d\,e^2\,{\left (d+e\,x\right )}^{3/2}}{4\,{\left (a\,e^2-c\,d^2\right )}^2}}{a^2\,e^4+c^2\,d^4-\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )\,\left (d+e\,x\right )+c^2\,d^2\,{\left (d+e\,x\right )}^2-2\,a\,c\,d^2\,e^2}+\frac {3\,e^2\,\mathrm {atan}\left (\frac {c\,d\,\sqrt {d+e\,x}}{\sqrt {c\,d}\,\sqrt {a\,e^2-c\,d^2}}\right )}{4\,\sqrt {c\,d}\,{\left (a\,e^2-c\,d^2\right )}^{5/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(5/2)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3,x)

[Out]

((5*e^2*(d + e*x)^(1/2))/(4*(a*e^2 - c*d^2)) + (3*c*d*e^2*(d + e*x)^(3/2))/(4*(a*e^2 - c*d^2)^2))/(a^2*e^4 + c
^2*d^4 - (2*c^2*d^3 - 2*a*c*d*e^2)*(d + e*x) + c^2*d^2*(d + e*x)^2 - 2*a*c*d^2*e^2) + (3*e^2*atan((c*d*(d + e*
x)^(1/2))/((c*d)^(1/2)*(a*e^2 - c*d^2)^(1/2))))/(4*(c*d)^(1/2)*(a*e^2 - c*d^2)^(5/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(5/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

Timed out

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