3.18.4 \(\int \frac {(d+e x)^{11/2}}{(a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx\)

Optimal. Leaf size=152 \[ -\frac {15 e^2 \sqrt {c d^2-a e^2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{4 c^{7/2} d^{7/2}}-\frac {5 e (d+e x)^{3/2}}{4 c^2 d^2 (a e+c d x)}-\frac {(d+e x)^{5/2}}{2 c d (a e+c d x)^2}+\frac {15 e^2 \sqrt {d+e x}}{4 c^3 d^3} \]

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Rubi [A]  time = 0.10, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {626, 47, 50, 63, 208} \begin {gather*} -\frac {15 e^2 \sqrt {c d^2-a e^2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{4 c^{7/2} d^{7/2}}-\frac {5 e (d+e x)^{3/2}}{4 c^2 d^2 (a e+c d x)}-\frac {(d+e x)^{5/2}}{2 c d (a e+c d x)^2}+\frac {15 e^2 \sqrt {d+e x}}{4 c^3 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(11/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

(15*e^2*Sqrt[d + e*x])/(4*c^3*d^3) - (5*e*(d + e*x)^(3/2))/(4*c^2*d^2*(a*e + c*d*x)) - (d + e*x)^(5/2)/(2*c*d*
(a*e + c*d*x)^2) - (15*e^2*Sqrt[c*d^2 - a*e^2]*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(
4*c^(7/2)*d^(7/2))

Rule 47

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

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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)^{11/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx &=\int \frac {(d+e x)^{5/2}}{(a e+c d x)^3} \, dx\\ &=-\frac {(d+e x)^{5/2}}{2 c d (a e+c d x)^2}+\frac {(5 e) \int \frac {(d+e x)^{3/2}}{(a e+c d x)^2} \, dx}{4 c d}\\ &=-\frac {5 e (d+e x)^{3/2}}{4 c^2 d^2 (a e+c d x)}-\frac {(d+e x)^{5/2}}{2 c d (a e+c d x)^2}+\frac {\left (15 e^2\right ) \int \frac {\sqrt {d+e x}}{a e+c d x} \, dx}{8 c^2 d^2}\\ &=\frac {15 e^2 \sqrt {d+e x}}{4 c^3 d^3}-\frac {5 e (d+e x)^{3/2}}{4 c^2 d^2 (a e+c d x)}-\frac {(d+e x)^{5/2}}{2 c d (a e+c d x)^2}+\frac {\left (15 e^2 \left (c d^2-a e^2\right )\right ) \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{8 c^3 d^3}\\ &=\frac {15 e^2 \sqrt {d+e x}}{4 c^3 d^3}-\frac {5 e (d+e x)^{3/2}}{4 c^2 d^2 (a e+c d x)}-\frac {(d+e x)^{5/2}}{2 c d (a e+c d x)^2}+\frac {\left (15 e \left (c d^2-a e^2\right )\right ) \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 c^3 d^3}\\ &=\frac {15 e^2 \sqrt {d+e x}}{4 c^3 d^3}-\frac {5 e (d+e x)^{3/2}}{4 c^2 d^2 (a e+c d x)}-\frac {(d+e x)^{5/2}}{2 c d (a e+c d x)^2}-\frac {15 e^2 \sqrt {c d^2-a e^2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{4 c^{7/2} d^{7/2}}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 61, normalized size = 0.40 \begin {gather*} \frac {2 e^2 (d+e x)^{7/2} \, _2F_1\left (3,\frac {7}{2};\frac {9}{2};-\frac {c d (d+e x)}{a e^2-c d^2}\right )}{7 \left (a e^2-c d^2\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(11/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

(2*e^2*(d + e*x)^(7/2)*Hypergeometric2F1[3, 7/2, 9/2, -((c*d*(d + e*x))/(-(c*d^2) + a*e^2))])/(7*(-(c*d^2) + a
*e^2)^3)

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IntegrateAlgebraic [A]  time = 0.69, size = 204, normalized size = 1.34 \begin {gather*} \frac {\sqrt {d+e x} \left (15 a^2 e^6-30 a c d^2 e^4+25 a c d e^4 (d+e x)+15 c^2 d^4 e^2-25 c^2 d^3 e^2 (d+e x)+8 c^2 d^2 e^2 (d+e x)^2\right )}{4 c^3 d^3 \left (-a e^2+c d^2-c d (d+e x)\right )^2}+\frac {15 e^2 \sqrt {a e^2-c d^2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x} \sqrt {a e^2-c d^2}}{c d^2-a e^2}\right )}{4 c^{7/2} d^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)^(11/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

(Sqrt[d + e*x]*(15*c^2*d^4*e^2 - 30*a*c*d^2*e^4 + 15*a^2*e^6 - 25*c^2*d^3*e^2*(d + e*x) + 25*a*c*d*e^4*(d + e*
x) + 8*c^2*d^2*e^2*(d + e*x)^2))/(4*c^3*d^3*(c*d^2 - a*e^2 - c*d*(d + e*x))^2) + (15*e^2*Sqrt[-(c*d^2) + a*e^2
]*ArcTan[(Sqrt[c]*Sqrt[d]*Sqrt[-(c*d^2) + a*e^2]*Sqrt[d + e*x])/(c*d^2 - a*e^2)])/(4*c^(7/2)*d^(7/2))

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fricas [A]  time = 0.43, size = 440, normalized size = 2.89 \begin {gather*} \left [\frac {15 \, {\left (c^{2} d^{2} e^{2} x^{2} + 2 \, a c d e^{3} x + a^{2} e^{4}\right )} \sqrt {\frac {c d^{2} - a e^{2}}{c d}} \log \left (\frac {c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt {e x + d} c d \sqrt {\frac {c d^{2} - a e^{2}}{c d}}}{c d x + a e}\right ) + 2 \, {\left (8 \, c^{2} d^{2} e^{2} x^{2} - 2 \, c^{2} d^{4} - 5 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4} - {\left (9 \, c^{2} d^{3} e - 25 \, a c d e^{3}\right )} x\right )} \sqrt {e x + d}}{8 \, {\left (c^{5} d^{5} x^{2} + 2 \, a c^{4} d^{4} e x + a^{2} c^{3} d^{3} e^{2}\right )}}, -\frac {15 \, {\left (c^{2} d^{2} e^{2} x^{2} + 2 \, a c d e^{3} x + a^{2} e^{4}\right )} \sqrt {-\frac {c d^{2} - a e^{2}}{c d}} \arctan \left (-\frac {\sqrt {e x + d} c d \sqrt {-\frac {c d^{2} - a e^{2}}{c d}}}{c d^{2} - a e^{2}}\right ) - {\left (8 \, c^{2} d^{2} e^{2} x^{2} - 2 \, c^{2} d^{4} - 5 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4} - {\left (9 \, c^{2} d^{3} e - 25 \, a c d e^{3}\right )} x\right )} \sqrt {e x + d}}{4 \, {\left (c^{5} d^{5} x^{2} + 2 \, a c^{4} d^{4} e x + a^{2} c^{3} d^{3} e^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(11/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="fricas")

[Out]

[1/8*(15*(c^2*d^2*e^2*x^2 + 2*a*c*d*e^3*x + a^2*e^4)*sqrt((c*d^2 - a*e^2)/(c*d))*log((c*d*e*x + 2*c*d^2 - a*e^
2 - 2*sqrt(e*x + d)*c*d*sqrt((c*d^2 - a*e^2)/(c*d)))/(c*d*x + a*e)) + 2*(8*c^2*d^2*e^2*x^2 - 2*c^2*d^4 - 5*a*c
*d^2*e^2 + 15*a^2*e^4 - (9*c^2*d^3*e - 25*a*c*d*e^3)*x)*sqrt(e*x + d))/(c^5*d^5*x^2 + 2*a*c^4*d^4*e*x + a^2*c^
3*d^3*e^2), -1/4*(15*(c^2*d^2*e^2*x^2 + 2*a*c*d*e^3*x + a^2*e^4)*sqrt(-(c*d^2 - a*e^2)/(c*d))*arctan(-sqrt(e*x
+ d)*c*d*sqrt(-(c*d^2 - a*e^2)/(c*d))/(c*d^2 - a*e^2)) - (8*c^2*d^2*e^2*x^2 - 2*c^2*d^4 - 5*a*c*d^2*e^2 + 15*
a^2*e^4 - (9*c^2*d^3*e - 25*a*c*d*e^3)*x)*sqrt(e*x + d))/(c^5*d^5*x^2 + 2*a*c^4*d^4*e*x + a^2*c^3*d^3*e^2)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(11/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming [a,c,d,exp(1),exp(2)]=[-83,-8,-38,29,40]Precision problem choosing root in common_EXT, current p
recision 14Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.
The choice was done assuming [a,c,d,exp(1),exp(2)]=[69,-62,15,83,-10]Warning, need to choose a branch for the
root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,c,d,exp(1),exp(2)]=[
61,93,-2,-73,84]Unable to divide, perhaps due to rounding error%%%{-7680,[6,6,12,6,0]%%%}+%%%{16384,[5,7,14,5,
0]%%%}+%%%{3840,[5,7,13,5,0]%%%}+%%%{1920,[5,7,12,5,0]%%%}+%%%{13312,[5,6,12,6,1]%%%}+%%%{-3840,[5,6,11,6,1]%%
%}+%%%{16384,[5,5,10,7,2]%%%}+%%%{-12736,[4,8,16,4,0]%%%}+%%%{-7232,[4,8,15,4,0]%%%}+%%%{-3616,[4,8,14,4,0]%%%
}+%%%{-30976,[4,7,14,5,1]%%%}+%%%{2496,[4,7,13,5,1]%%%}+%%%{-2368,[4,7,12,5,1]%%%}+%%%{-27776,[4,6,12,6,2]%%%}
+%%%{-2496,[4,6,11,6,2]%%%}+%%%{-3616,[4,6,10,6,2]%%%}+%%%{-30976,[4,5,10,7,3]%%%}+%%%{7232,[4,5,9,7,3]%%%}+%%
%{-12736,[4,4,8,8,4]%%%}+%%%{4928,[3,9,18,3,0]%%%}+%%%{4560,[3,9,17,3,0]%%%}+%%%{2280,[3,9,16,3,0]%%%}+%%%{213
76,[3,8,16,4,1]%%%}+%%%{6128,[3,8,15,4,1]%%%}+%%%{5344,[3,8,14,4,1]%%%}+%%%{33984,[3,7,14,5,2]%%%}+%%%{-2784,[
3,7,13,5,2]%%%}+%%%{3952,[3,7,12,5,2]%%%}+%%%{33024,[3,6,12,6,3]%%%}+%%%{2784,[3,6,11,6,3]%%%}+%%%{5344,[3,6,1
0,6,3]%%%}+%%%{33984,[3,5,10,7,4]%%%}+%%%{-6128,[3,5,9,7,4]%%%}+%%%{2280,[3,5,8,7,4]%%%}+%%%{21376,[3,4,8,8,5]
%%%}+%%%{-4560,[3,4,7,8,5]%%%}+%%%{4928,[3,3,6,9,6]%%%}+%%%{-1030,[2,10,20,2,0]%%%}+%%%{-1324,[2,10,19,2,0]%%%
}+%%%{-662,[2,10,18,2,0]%%%}+%%%{-6544,[2,9,18,3,1]%%%}+%%%{-4412,[2,9,17,3,1]%%%}+%%%{-2868,[2,9,16,3,1]%%%}+
%%%{-16552,[2,8,16,4,2]%%%}+%%%{-2796,[2,8,15,4,2]%%%}+%%%{-4266,[2,8,14,4,2]%%%}+%%%{-22256,[2,7,14,5,3]%%%}+
%%%{1316,[2,7,13,5,3]%%%}+%%%{-3608,[2,7,12,5,3]%%%}+%%%{-22436,[2,6,12,6,4]%%%}+%%%{-1316,[2,6,11,6,4]%%%}+%%
%{-4266,[2,6,10,6,4]%%%}+%%%{-22256,[2,5,10,7,5]%%%}+%%%{2796,[2,5,9,7,5]%%%}+%%%{-2868,[2,5,8,7,5]%%%}+%%%{-1
6552,[2,4,8,8,6]%%%}+%%%{4412,[2,4,7,8,6]%%%}+%%%{-662,[2,4,6,8,6]%%%}+%%%{-6544,[2,3,6,9,7]%%%}+%%%{1324,[2,3
,5,9,7]%%%}+%%%{-1030,[2,2,4,10,8]%%%}+%%%{112,[1,11,22,1,0]%%%}+%%%{184,[1,11,21,1,0]%%%}+%%%{92,[1,11,20,1,0
]%%%}+%%%{940,[1,10,20,2,1]%%%}+%%%{992,[1,10,19,2,1]%%%}+%%%{588,[1,10,18,2,1]%%%}+%%%{3344,[1,9,18,3,2]%%%}+
%%%{1768,[1,9,17,3,2]%%%}+%%%{1472,[1,9,16,3,2]%%%}+%%%{6480,[1,8,16,4,3]%%%}+%%%{680,[1,8,15,4,3]%%%}+%%%{181
2,[1,8,14,4,3]%%%}+%%%{8064,[1,7,14,5,4]%%%}+%%%{-280,[1,7,13,5,4]%%%}+%%%{1672,[1,7,12,5,4]%%%}+%%%{8200,[1,6
,12,6,5]%%%}+%%%{280,[1,6,11,6,5]%%%}+%%%{1812,[1,6,10,6,5]%%%}+%%%{8064,[1,5,10,7,6]%%%}+%%%{-680,[1,5,9,7,6]
%%%}+%%%{1472,[1,5,8,7,6]%%%}+%%%{6480,[1,4,8,8,7]%%%}+%%%{-1768,[1,4,7,8,7]%%%}+%%%{588,[1,4,6,8,7]%%%}+%%%{3
344,[1,3,6,9,8]%%%}+%%%{-992,[1,3,5,9,8]%%%}+%%%{92,[1,3,4,9,8]%%%}+%%%{940,[1,2,4,10,9]%%%}+%%%{-184,[1,2,3,1
0,9]%%%}+%%%{112,[1,1,2,11,10]%%%}+%%%{-5,[0,12,24,0,0]%%%}+%%%{-10,[0,12,23,0,0]%%%}+%%%{-5,[0,12,22,0,0]%%%}
+%%%{-52,[0,11,22,1,1]%%%}+%%%{-74,[0,11,21,1,1]%%%}+%%%{-42,[0,11,20,1,1]%%%}+%%%{-240,[0,10,20,2,2]%%%}+%%%{
-218,[0,10,19,2,2]%%%}+%%%{-151,[0,10,18,2,2]%%%}+%%%{-628,[0,9,18,3,3]%%%}+%%%{-266,[0,9,17,3,3]%%%}+%%%{-284
,[0,9,16,3,3]%%%}+%%%{-1043,[0,8,16,4,4]%%%}+%%%{-80,[0,8,15,4,4]%%%}+%%%{-324,[0,8,14,4,4]%%%}+%%%{-1240,[0,7
,14,5,5]%%%}+%%%{32,[0,7,13,5,5]%%%}+%%%{-308,[0,7,12,5,5]%%%}+%%%{-1264,[0,6,12,6,6]%%%}+%%%{-32,[0,6,11,6,6]
%%%}+%%%{-324,[0,6,10,6,6]%%%}+%%%{-1240,[0,5,10,7,7]%%%}+%%%{80,[0,5,9,7,7]%%%}+%%%{-284,[0,5,8,7,7]%%%}+%%%{
-1043,[0,4,8,8,8]%%%}+%%%{266,[0,4,7,8,8]%%%}+%%%{-151,[0,4,6,8,8]%%%}+%%%{-628,[0,3,6,9,9]%%%}+%%%{218,[0,3,5
,9,9]%%%}+%%%{-42,[0,3,4,9,9]%%%}+%%%{-240,[0,2,4,10,10]%%%}+%%%{74,[0,2,3,10,10]%%%}+%%%{-5,[0,2,2,10,10]%%%}
+%%%{-52,[0,1,2,11,11]%%%}+%%%{10,[0,1,1,11,11]%%%}+%%%{-5,[0,0,0,12,12]%%%} / %%%{16,[2,2,4,2,0]%%%} Error: B
ad Argument ValueWarning, need to choose a branch for the root of a polynomial with parameters. This might be
wrong.The choice was done assuming [a,c,d,exp(1),exp(2)]=[-92,88,73,-29,8]Unable to divide, perhaps due to rou
nding error%%%{-7680,[6,6,12,6,0]%%%}+%%%{16384,[5,7,14,5,0]%%%}+%%%{3840,[5,7,13,5,0]%%%}+%%%{1920,[5,7,12,5,
0]%%%}+%%%{13312,[5,6,12,6,1]%%%}+%%%{-3840,[5,6,11,6,1]%%%}+%%%{16384,[5,5,10,7,2]%%%}+%%%{-12736,[4,8,16,4,0
]%%%}+%%%{-7232,[4,8,15,4,0]%%%}+%%%{-3616,[4,8,14,4,0]%%%}+%%%{-30976,[4,7,14,5,1]%%%}+%%%{2496,[4,7,13,5,1]%
%%}+%%%{-2368,[4,7,12,5,1]%%%}+%%%{-27776,[4,6,12,6,2]%%%}+%%%{-2496,[4,6,11,6,2]%%%}+%%%{-3616,[4,6,10,6,2]%%
%}+%%%{-30976,[4,5,10,7,3]%%%}+%%%{7232,[4,5,9,7,3]%%%}+%%%{-12736,[4,4,8,8,4]%%%}+%%%{4928,[3,9,18,3,0]%%%}+%
%%{4560,[3,9,17,3,0]%%%}+%%%{2280,[3,9,16,3,0]%%%}+%%%{21376,[3,8,16,4,1]%%%}+%%%{6128,[3,8,15,4,1]%%%}+%%%{53
44,[3,8,14,4,1]%%%}+%%%{33984,[3,7,14,5,2]%%%}+%%%{-2784,[3,7,13,5,2]%%%}+%%%{3952,[3,7,12,5,2]%%%}+%%%{33024,
[3,6,12,6,3]%%%}+%%%{2784,[3,6,11,6,3]%%%}+%%%{5344,[3,6,10,6,3]%%%}+%%%{33984,[3,5,10,7,4]%%%}+%%%{-6128,[3,5
,9,7,4]%%%}+%%%{2280,[3,5,8,7,4]%%%}+%%%{21376,[3,4,8,8,5]%%%}+%%%{-4560,[3,4,7,8,5]%%%}+%%%{4928,[3,3,6,9,6]%
%%}+%%%{-1030,[2,10,20,2,0]%%%}+%%%{-1324,[2,10,19,2,0]%%%}+%%%{-662,[2,10,18,2,0]%%%}+%%%{-6544,[2,9,18,3,1]%
%%}+%%%{-4412,[2,9,17,3,1]%%%}+%%%{-2868,[2,9,16,3,1]%%%}+%%%{-16552,[2,8,16,4,2]%%%}+%%%{-2796,[2,8,15,4,2]%%
%}+%%%{-4266,[2,8,14,4,2]%%%}+%%%{-22256,[2,7,14,5,3]%%%}+%%%{1316,[2,7,13,5,3]%%%}+%%%{-3608,[2,7,12,5,3]%%%}
+%%%{-22436,[2,6,12,6,4]%%%}+%%%{-1316,[2,6,11,6,4]%%%}+%%%{-4266,[2,6,10,6,4]%%%}+%%%{-22256,[2,5,10,7,5]%%%}
+%%%{2796,[2,5,9,7,5]%%%}+%%%{-2868,[2,5,8,7,5]%%%}+%%%{-16552,[2,4,8,8,6]%%%}+%%%{4412,[2,4,7,8,6]%%%}+%%%{-6
62,[2,4,6,8,6]%%%}+%%%{-6544,[2,3,6,9,7]%%%}+%%%{1324,[2,3,5,9,7]%%%}+%%%{-1030,[2,2,4,10,8]%%%}+%%%{112,[1,11
,22,1,0]%%%}+%%%{184,[1,11,21,1,0]%%%}+%%%{92,[1,11,20,1,0]%%%}+%%%{940,[1,10,20,2,1]%%%}+%%%{992,[1,10,19,2,1
]%%%}+%%%{588,[1,10,18,2,1]%%%}+%%%{3344,[1,9,18,3,2]%%%}+%%%{1768,[1,9,17,3,2]%%%}+%%%{1472,[1,9,16,3,2]%%%}+
%%%{6480,[1,8,16,4,3]%%%}+%%%{680,[1,8,15,4,3]%%%}+%%%{1812,[1,8,14,4,3]%%%}+%%%{8064,[1,7,14,5,4]%%%}+%%%{-28
0,[1,7,13,5,4]%%%}+%%%{1672,[1,7,12,5,4]%%%}+%%%{8200,[1,6,12,6,5]%%%}+%%%{280,[1,6,11,6,5]%%%}+%%%{1812,[1,6,
10,6,5]%%%}+%%%{8064,[1,5,10,7,6]%%%}+%%%{-680,[1,5,9,7,6]%%%}+%%%{1472,[1,5,8,7,6]%%%}+%%%{6480,[1,4,8,8,7]%%
%}+%%%{-1768,[1,4,7,8,7]%%%}+%%%{588,[1,4,6,8,7]%%%}+%%%{3344,[1,3,6,9,8]%%%}+%%%{-992,[1,3,5,9,8]%%%}+%%%{92,
[1,3,4,9,8]%%%}+%%%{940,[1,2,4,10,9]%%%}+%%%{-184,[1,2,3,10,9]%%%}+%%%{112,[1,1,2,11,10]%%%}+%%%{-5,[0,12,24,0
,0]%%%}+%%%{-10,[0,12,23,0,0]%%%}+%%%{-5,[0,12,22,0,0]%%%}+%%%{-52,[0,11,22,1,1]%%%}+%%%{-74,[0,11,21,1,1]%%%}
+%%%{-42,[0,11,20,1,1]%%%}+%%%{-240,[0,10,20,2,2]%%%}+%%%{-218,[0,10,19,2,2]%%%}+%%%{-151,[0,10,18,2,2]%%%}+%%
%{-628,[0,9,18,3,3]%%%}+%%%{-266,[0,9,17,3,3]%%%}+%%%{-284,[0,9,16,3,3]%%%}+%%%{-1043,[0,8,16,4,4]%%%}+%%%{-80
,[0,8,15,4,4]%%%}+%%%{-324,[0,8,14,4,4]%%%}+%%%{-1240,[0,7,14,5,5]%%%}+%%%{32,[0,7,13,5,5]%%%}+%%%{-308,[0,7,1
2,5,5]%%%}+%%%{-1264,[0,6,12,6,6]%%%}+%%%{-32,[0,6,11,6,6]%%%}+%%%{-324,[0,6,10,6,6]%%%}+%%%{-1240,[0,5,10,7,7
]%%%}+%%%{80,[0,5,9,7,7]%%%}+%%%{-284,[0,5,8,7,7]%%%}+%%%{-1043,[0,4,8,8,8]%%%}+%%%{266,[0,4,7,8,8]%%%}+%%%{-1
51,[0,4,6,8,8]%%%}+%%%{-628,[0,3,6,9,9]%%%}+%%%{218,[0,3,5,9,9]%%%}+%%%{-42,[0,3,4,9,9]%%%}+%%%{-240,[0,2,4,10
,10]%%%}+%%%{74,[0,2,3,10,10]%%%}+%%%{-5,[0,2,2,10,10]%%%}+%%%{-52,[0,1,2,11,11]%%%}+%%%{10,[0,1,1,11,11]%%%}+
%%%{-5,[0,0,0,12,12]%%%} / %%%{16,[2,2,4,2,0]%%%} Error: Bad Argument ValueEvaluation time: 251.21Done

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maple [B]  time = 0.07, size = 288, normalized size = 1.89 \begin {gather*} \frac {7 \sqrt {e x +d}\, a^{2} e^{6}}{4 \left (c d e x +a \,e^{2}\right )^{2} c^{3} d^{3}}-\frac {7 \sqrt {e x +d}\, a \,e^{4}}{2 \left (c d e x +a \,e^{2}\right )^{2} c^{2} d}+\frac {7 \sqrt {e x +d}\, d \,e^{2}}{4 \left (c d e x +a \,e^{2}\right )^{2} c}+\frac {9 \left (e x +d \right )^{\frac {3}{2}} a \,e^{4}}{4 \left (c d e x +a \,e^{2}\right )^{2} c^{2} d^{2}}-\frac {9 \left (e x +d \right )^{\frac {3}{2}} e^{2}}{4 \left (c d e x +a \,e^{2}\right )^{2} c}-\frac {15 a \,e^{4} \arctan \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) c d}}\right )}{4 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) c d}\, c^{3} d^{3}}+\frac {15 e^{2} \arctan \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) c d}}\right )}{4 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) c d}\, c^{2} d}+\frac {2 \sqrt {e x +d}\, e^{2}}{c^{3} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(11/2)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^3,x)

[Out]

2*e^2*(e*x+d)^(1/2)/c^3/d^3+9/4*e^4/c^2/d^2/(c*d*e*x+a*e^2)^2*(e*x+d)^(3/2)*a-9/4*e^2/c/(c*d*e*x+a*e^2)^2*(e*x
+d)^(3/2)+7/4*e^6/c^3/d^3/(c*d*e*x+a*e^2)^2*(e*x+d)^(1/2)*a^2-7/2*e^4/c^2/d/(c*d*e*x+a*e^2)^2*(e*x+d)^(1/2)*a+
7/4*e^2/c*d/(c*d*e*x+a*e^2)^2*(e*x+d)^(1/2)-15/4*e^4/c^3/d^3/((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)*a+15/4*e^2/c^2/d/((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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(11/2)/(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 = 0.17, size = 240, normalized size = 1.58 \begin {gather*} \frac {2\,e^2\,\sqrt {d+e\,x}}{c^3\,d^3}-\frac {\left (\frac {9\,c^2\,d^3\,e^2}{4}-\frac {9\,a\,c\,d\,e^4}{4}\right )\,{\left (d+e\,x\right )}^{3/2}-\sqrt {d+e\,x}\,\left (\frac {7\,a^2\,e^6}{4}-\frac {7\,a\,c\,d^2\,e^4}{2}+\frac {7\,c^2\,d^4\,e^2}{4}\right )}{c^5\,d^7-\left (2\,c^5\,d^6-2\,a\,c^4\,d^4\,e^2\right )\,\left (d+e\,x\right )+c^5\,d^5\,{\left (d+e\,x\right )}^2-2\,a\,c^4\,d^5\,e^2+a^2\,c^3\,d^3\,e^4}-\frac {15\,e^2\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,e^2\,\sqrt {a\,e^2-c\,d^2}\,\sqrt {d+e\,x}}{a\,e^4-c\,d^2\,e^2}\right )\,\sqrt {a\,e^2-c\,d^2}}{4\,c^{7/2}\,d^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(11/2)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3,x)

[Out]

(2*e^2*(d + e*x)^(1/2))/(c^3*d^3) - (((9*c^2*d^3*e^2)/4 - (9*a*c*d*e^4)/4)*(d + e*x)^(3/2) - (d + e*x)^(1/2)*(
(7*a^2*e^6)/4 + (7*c^2*d^4*e^2)/4 - (7*a*c*d^2*e^4)/2))/(c^5*d^7 - (2*c^5*d^6 - 2*a*c^4*d^4*e^2)*(d + e*x) + c
^5*d^5*(d + e*x)^2 - 2*a*c^4*d^5*e^2 + a^2*c^3*d^3*e^4) - (15*e^2*atan((c^(1/2)*d^(1/2)*e^2*(a*e^2 - c*d^2)^(1
/2)*(d + e*x)^(1/2))/(a*e^4 - c*d^2*e^2))*(a*e^2 - c*d^2)^(1/2))/(4*c^(7/2)*d^(7/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(11/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

Timed out

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