3.21.10 \(\int \frac {(d+e x)^{11/2}}{(a d e+(c d^2+a e^2) x+c d e x^2)^2} \, dx\) [2010]

Optimal. Leaf size=178 \[ \frac {7 e \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}{c^4 d^4}+\frac {7 e \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{3 c^3 d^3}+\frac {7 e (d+e x)^{5/2}}{5 c^2 d^2}-\frac {(d+e x)^{7/2}}{c d (a e+c d x)}-\frac {7 e \left (c d^2-a e^2\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{9/2} d^{9/2}} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {640, 43, 52, 65, 214} \begin {gather*} -\frac {7 e \left (c d^2-a e^2\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{9/2} d^{9/2}}+\frac {7 e \sqrt {d+e x} \left (c d^2-a e^2\right )^2}{c^4 d^4}+\frac {7 e (d+e x)^{3/2} \left (c d^2-a e^2\right )}{3 c^3 d^3}-\frac {(d+e x)^{7/2}}{c d (a e+c d x)}+\frac {7 e (d+e x)^{5/2}}{5 c^2 d^2} \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)^2,x]

[Out]

(7*e*(c*d^2 - a*e^2)^2*Sqrt[d + e*x])/(c^4*d^4) + (7*e*(c*d^2 - a*e^2)*(d + e*x)^(3/2))/(3*c^3*d^3) + (7*e*(d
+ e*x)^(5/2))/(5*c^2*d^2) - (d + e*x)^(7/2)/(c*d*(a*e + c*d*x)) - (7*e*(c*d^2 - a*e^2)^(5/2)*ArcTanh[(Sqrt[c]*
Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(c^(9/2)*d^(9/2))

Rule 43

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, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 52

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 65

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 214

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

Rule 640

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c/e)*x)^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 )^2} \, dx &=\int \frac {(d+e x)^{7/2}}{(a e+c d x)^2} \, dx\\ &=-\frac {(d+e x)^{7/2}}{c d (a e+c d x)}+\frac {(7 e) \int \frac {(d+e x)^{5/2}}{a e+c d x} \, dx}{2 c d}\\ &=\frac {7 e (d+e x)^{5/2}}{5 c^2 d^2}-\frac {(d+e x)^{7/2}}{c d (a e+c d x)}+\frac {\left (7 e \left (c d^2-a e^2\right )\right ) \int \frac {(d+e x)^{3/2}}{a e+c d x} \, dx}{2 c^2 d^2}\\ &=\frac {7 e \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{3 c^3 d^3}+\frac {7 e (d+e x)^{5/2}}{5 c^2 d^2}-\frac {(d+e x)^{7/2}}{c d (a e+c d x)}+\frac {\left (7 e \left (c d^2-a e^2\right )^2\right ) \int \frac {\sqrt {d+e x}}{a e+c d x} \, dx}{2 c^3 d^3}\\ &=\frac {7 e \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}{c^4 d^4}+\frac {7 e \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{3 c^3 d^3}+\frac {7 e (d+e x)^{5/2}}{5 c^2 d^2}-\frac {(d+e x)^{7/2}}{c d (a e+c d x)}+\frac {\left (7 e \left (c d^2-a e^2\right )^3\right ) \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{2 c^4 d^4}\\ &=\frac {7 e \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}{c^4 d^4}+\frac {7 e \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{3 c^3 d^3}+\frac {7 e (d+e x)^{5/2}}{5 c^2 d^2}-\frac {(d+e x)^{7/2}}{c d (a e+c d x)}+\frac {\left (7 \left (c d^2-a e^2\right )^3\right ) \text {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 )}{c^4 d^4}\\ &=\frac {7 e \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}{c^4 d^4}+\frac {7 e \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{3 c^3 d^3}+\frac {7 e (d+e x)^{5/2}}{5 c^2 d^2}-\frac {(d+e x)^{7/2}}{c d (a e+c d x)}-\frac {7 e \left (c d^2-a e^2\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{9/2} d^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.35, size = 191, normalized size = 1.07 \begin {gather*} \frac {\sqrt {d+e x} \left (105 a^3 e^6-35 a^2 c d e^4 (7 d-2 e x)+7 a c^2 d^2 e^2 \left (23 d^2-24 d e x-2 e^2 x^2\right )+c^3 d^3 \left (-15 d^3+116 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )\right )}{15 c^4 d^4 (a e+c d x)}-\frac {7 e \left (-c d^2+a e^2\right )^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{c^{9/2} d^{9/2}} \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)^2,x]

[Out]

(Sqrt[d + e*x]*(105*a^3*e^6 - 35*a^2*c*d*e^4*(7*d - 2*e*x) + 7*a*c^2*d^2*e^2*(23*d^2 - 24*d*e*x - 2*e^2*x^2) +
c^3*d^3*(-15*d^3 + 116*d^2*e*x + 32*d*e^2*x^2 + 6*e^3*x^3)))/(15*c^4*d^4*(a*e + c*d*x)) - (7*e*(-(c*d^2) + a*
e^2)^(5/2)*ArcTan[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[-(c*d^2) + a*e^2]])/(c^(9/2)*d^(9/2))

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Maple [A]
time = 0.77, size = 272, normalized size = 1.53

method	result	size
derivativedivides	\(2 e \left (\frac {\frac {c^{2} d^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 a c d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 c^{2} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+3 a^{2} e^{4} \sqrt {e x +d}-6 a c \,d^{2} e^{2} \sqrt {e x +d}+3 c^{2} d^{4} \sqrt {e x +d}}{c^{4} d^{4}}-\frac {\frac {\left (-\frac {1}{2} e^{6} a^{3}+\frac {3}{2} e^{4} d^{2} a^{2} c -\frac {3}{2} d^{4} e^{2} c^{2} a +\frac {1}{2} d^{6} c^{3}\right ) \sqrt {e x +d}}{c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {7 \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}}{c^{4} d^{4}}\right )\)	\(272\)
default	\(2 e \left (\frac {\frac {c^{2} d^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 a c d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 c^{2} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+3 a^{2} e^{4} \sqrt {e x +d}-6 a c \,d^{2} e^{2} \sqrt {e x +d}+3 c^{2} d^{4} \sqrt {e x +d}}{c^{4} d^{4}}-\frac {\frac {\left (-\frac {1}{2} e^{6} a^{3}+\frac {3}{2} e^{4} d^{2} a^{2} c -\frac {3}{2} d^{4} e^{2} c^{2} a +\frac {1}{2} d^{6} c^{3}\right ) \sqrt {e x +d}}{c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {7 \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}}{c^{4} d^{4}}\right )\)	\(272\)
risch	\(\frac {2 e \left (3 e^{2} x^{2} c^{2} d^{2}-10 a c d \,e^{3} x +16 c^{2} d^{3} e x +45 a^{2} e^{4}-100 a c \,d^{2} e^{2}+58 c^{2} d^{4}\right ) \sqrt {e x +d}}{15 c^{4} d^{4}}+\frac {e^{7} \sqrt {e x +d}\, a^{3}}{d^{4} c^{4} \left (c d e x +e^{2} a \right )}-\frac {3 e^{5} \sqrt {e x +d}\, a^{2}}{d^{2} c^{3} \left (c d e x +e^{2} a \right )}+\frac {3 e^{3} \sqrt {e x +d}\, a}{c^{2} \left (c d e x +e^{2} a \right )}-\frac {d^{2} e \sqrt {e x +d}}{c \left (c d e x +e^{2} a \right )}-\frac {7 e^{7} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right ) a^{3}}{d^{4} c^{4} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}+\frac {21 e^{5} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right ) a^{2}}{d^{2} c^{3} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}-\frac {21 e^{3} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right ) a}{c^{2} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}+\frac {7 d^{2} e \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{c \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\)	\(429\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(11/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x,method=_RETURNVERBOSE)

[Out]

2*e*(1/c^4/d^4*(1/5*c^2*d^2*(e*x+d)^(5/2)-2/3*a*c*d*e^2*(e*x+d)^(3/2)+2/3*c^2*d^3*(e*x+d)^(3/2)+3*a^2*e^4*(e*x
+d)^(1/2)-6*a*c*d^2*e^2*(e*x+d)^(1/2)+3*c^2*d^4*(e*x+d)^(1/2))-1/c^4/d^4*((-1/2*e^6*a^3+3/2*e^4*d^2*a^2*c-3/2*
d^4*e^2*c^2*a+1/2*d^6*c^3)*(e*x+d)^(1/2)/(c*d*(e*x+d)+e^2*a-c*d^2)+7/2*(a^3*e^6-3*a^2*c*d^2*e^4+3*a*c^2*d^4*e^
2-c^3*d^6)/((a*e^2-c*d^2)*c*d)^(1/2)*arctan(c*d*(e*x+d)^(1/2)/((a*e^2-c*d^2)*c*d)^(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((e*x+d)^(11/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^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(c*d^2-%e^2*a>0)', see `assume?
` for more d

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Fricas [A]
time = 4.05, size = 566, normalized size = 3.18 \begin {gather*} \left [\frac {105 \, {\left (c^{3} d^{5} x e - 2 \, a c^{2} d^{3} x e^{3} + a c^{2} d^{4} e^{2} + a^{2} c d x e^{5} - 2 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} \sqrt {\frac {c d^{2} - a e^{2}}{c d}} \log \left (\frac {c d x e + 2 \, c d^{2} - 2 \, \sqrt {x e + d} c d \sqrt {\frac {c d^{2} - a e^{2}}{c d}} - a e^{2}}{c d x + a e}\right ) + 2 \, {\left (116 \, c^{3} d^{5} x e - 15 \, c^{3} d^{6} + 70 \, a^{2} c d x e^{5} + 105 \, a^{3} e^{6} - 7 \, {\left (2 \, a c^{2} d^{2} x^{2} + 35 \, a^{2} c d^{2}\right )} e^{4} + 6 \, {\left (c^{3} d^{3} x^{3} - 28 \, a c^{2} d^{3} x\right )} e^{3} + {\left (32 \, c^{3} d^{4} x^{2} + 161 \, a c^{2} d^{4}\right )} e^{2}\right )} \sqrt {x e + d}}{30 \, {\left (c^{5} d^{5} x + a c^{4} d^{4} e\right )}}, -\frac {105 \, {\left (c^{3} d^{5} x e - 2 \, a c^{2} d^{3} x e^{3} + a c^{2} d^{4} e^{2} + a^{2} c d x e^{5} - 2 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} \sqrt {-\frac {c d^{2} - a e^{2}}{c d}} \arctan \left (-\frac {\sqrt {x e + d} c d \sqrt {-\frac {c d^{2} - a e^{2}}{c d}}}{c d^{2} - a e^{2}}\right ) - {\left (116 \, c^{3} d^{5} x e - 15 \, c^{3} d^{6} + 70 \, a^{2} c d x e^{5} + 105 \, a^{3} e^{6} - 7 \, {\left (2 \, a c^{2} d^{2} x^{2} + 35 \, a^{2} c d^{2}\right )} e^{4} + 6 \, {\left (c^{3} d^{3} x^{3} - 28 \, a c^{2} d^{3} x\right )} e^{3} + {\left (32 \, c^{3} d^{4} x^{2} + 161 \, a c^{2} d^{4}\right )} e^{2}\right )} \sqrt {x e + d}}{15 \, {\left (c^{5} d^{5} x + a c^{4} d^{4} e\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)^2,x, algorithm="fricas")

[Out]

[1/30*(105*(c^3*d^5*x*e - 2*a*c^2*d^3*x*e^3 + a*c^2*d^4*e^2 + a^2*c*d*x*e^5 - 2*a^2*c*d^2*e^4 + a^3*e^6)*sqrt(
(c*d^2 - a*e^2)/(c*d))*log((c*d*x*e + 2*c*d^2 - 2*sqrt(x*e + d)*c*d*sqrt((c*d^2 - a*e^2)/(c*d)) - a*e^2)/(c*d*
x + a*e)) + 2*(116*c^3*d^5*x*e - 15*c^3*d^6 + 70*a^2*c*d*x*e^5 + 105*a^3*e^6 - 7*(2*a*c^2*d^2*x^2 + 35*a^2*c*d
^2)*e^4 + 6*(c^3*d^3*x^3 - 28*a*c^2*d^3*x)*e^3 + (32*c^3*d^4*x^2 + 161*a*c^2*d^4)*e^2)*sqrt(x*e + d))/(c^5*d^5
*x + a*c^4*d^4*e), -1/15*(105*(c^3*d^5*x*e - 2*a*c^2*d^3*x*e^3 + a*c^2*d^4*e^2 + a^2*c*d*x*e^5 - 2*a^2*c*d^2*e
^4 + a^3*e^6)*sqrt(-(c*d^2 - a*e^2)/(c*d))*arctan(-sqrt(x*e + d)*c*d*sqrt(-(c*d^2 - a*e^2)/(c*d))/(c*d^2 - a*e
^2)) - (116*c^3*d^5*x*e - 15*c^3*d^6 + 70*a^2*c*d*x*e^5 + 105*a^3*e^6 - 7*(2*a*c^2*d^2*x^2 + 35*a^2*c*d^2)*e^4
+ 6*(c^3*d^3*x^3 - 28*a*c^2*d^3*x)*e^3 + (32*c^3*d^4*x^2 + 161*a*c^2*d^4)*e^2)*sqrt(x*e + d))/(c^5*d^5*x + a*
c^4*d^4*e)]

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Sympy [F(-1)] Timed out
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)**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (158) = 316\).
time = 2.96, size = 317, normalized size = 1.78 \begin {gather*} \frac {7 \, {\left (c^{3} d^{6} e - 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} - a^{3} e^{7}\right )} \arctan \left (\frac {\sqrt {x e + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{\sqrt {-c^{2} d^{3} + a c d e^{2}} c^{4} d^{4}} - \frac {\sqrt {x e + d} c^{3} d^{6} e - 3 \, \sqrt {x e + d} a c^{2} d^{4} e^{3} + 3 \, \sqrt {x e + d} a^{2} c d^{2} e^{5} - \sqrt {x e + d} a^{3} e^{7}}{{\left ({\left (x e + d\right )} c d - c d^{2} + a e^{2}\right )} c^{4} d^{4}} + \frac {2 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{8} d^{8} e + 10 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{8} d^{9} e + 45 \, \sqrt {x e + d} c^{8} d^{10} e - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{7} d^{7} e^{3} - 90 \, \sqrt {x e + d} a c^{7} d^{8} e^{3} + 45 \, \sqrt {x e + d} a^{2} c^{6} d^{6} e^{5}\right )}}{15 \, c^{10} d^{10}} \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)^2,x, algorithm="giac")

[Out]

7*(c^3*d^6*e - 3*a*c^2*d^4*e^3 + 3*a^2*c*d^2*e^5 - a^3*e^7)*arctan(sqrt(x*e + d)*c*d/sqrt(-c^2*d^3 + a*c*d*e^2
))/(sqrt(-c^2*d^3 + a*c*d*e^2)*c^4*d^4) - (sqrt(x*e + d)*c^3*d^6*e - 3*sqrt(x*e + d)*a*c^2*d^4*e^3 + 3*sqrt(x*
e + d)*a^2*c*d^2*e^5 - sqrt(x*e + d)*a^3*e^7)/(((x*e + d)*c*d - c*d^2 + a*e^2)*c^4*d^4) + 2/15*(3*(x*e + d)^(5
/2)*c^8*d^8*e + 10*(x*e + d)^(3/2)*c^8*d^9*e + 45*sqrt(x*e + d)*c^8*d^10*e - 10*(x*e + d)^(3/2)*a*c^7*d^7*e^3
- 90*sqrt(x*e + d)*a*c^7*d^8*e^3 + 45*sqrt(x*e + d)*a^2*c^6*d^6*e^5)/(c^10*d^10)

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Mupad [B]
time = 0.68, size = 290, normalized size = 1.63 \begin {gather*} \frac {\sqrt {d+e\,x}\,\left (a^3\,e^7-3\,a^2\,c\,d^2\,e^5+3\,a\,c^2\,d^4\,e^3-c^3\,d^6\,e\right )}{c^5\,d^5\,\left (d+e\,x\right )-c^5\,d^6+a\,c^4\,d^4\,e^2}-\left (\frac {2\,e\,{\left (a\,e^2-c\,d^2\right )}^2}{c^4\,d^4}-\frac {2\,e\,{\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )}^2}{c^6\,d^6}\right )\,\sqrt {d+e\,x}+\frac {2\,e\,{\left (d+e\,x\right )}^{5/2}}{5\,c^2\,d^2}+\frac {2\,e\,\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,c^4\,d^4}-\frac {7\,e\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,e\,{\left (a\,e^2-c\,d^2\right )}^{5/2}\,\sqrt {d+e\,x}}{a^3\,e^7-3\,a^2\,c\,d^2\,e^5+3\,a\,c^2\,d^4\,e^3-c^3\,d^6\,e}\right )\,{\left (a\,e^2-c\,d^2\right )}^{5/2}}{c^{9/2}\,d^{9/2}} \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)^2,x)

[Out]

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

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