\(\int \frac {(d+e x)^{7/2}}{(a d e+(c d^2+a e^2) x+c d e x^2)^2} \, dx\) [2012]

Optimal result

Integrand size = 37, antiderivative size = 112 \[ \int \frac {(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {3 e \sqrt {d+e x}}{c^2 d^2}-\frac {(d+e x)^{3/2}}{c d (a e+c d x)}-\frac {3 e \sqrt {c d^2-a e^2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{5/2} d^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {640, 43, 52, 65, 214} \[ \int \frac {(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {3 e \sqrt {c d^2-a e^2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{5/2} d^{5/2}}-\frac {(d+e x)^{3/2}}{c d (a e+c d x)}+\frac {3 e \sqrt {d+e x}}{c^2 d^2} \]

[In]

Int[(d + e*x)^(7/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]

[Out]

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

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*} \text {integral}& = \int \frac {(d+e x)^{3/2}}{(a e+c d x)^2} \, dx \\ & = -\frac {(d+e x)^{3/2}}{c d (a e+c d x)}+\frac {(3 e) \int \frac {\sqrt {d+e x}}{a e+c d x} \, dx}{2 c d} \\ & = \frac {3 e \sqrt {d+e x}}{c^2 d^2}-\frac {(d+e x)^{3/2}}{c d (a e+c d x)}+\frac {\left (3 e \left (c d^2-a e^2\right )\right ) \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{2 c^2 d^2} \\ & = \frac {3 e \sqrt {d+e x}}{c^2 d^2}-\frac {(d+e x)^{3/2}}{c d (a e+c d x)}+\frac {\left (3 \left (c d^2-a e^2\right )\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^2 d^2} \\ & = \frac {3 e \sqrt {d+e x}}{c^2 d^2}-\frac {(d+e x)^{3/2}}{c d (a e+c d x)}-\frac {3 e \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 )}{c^{5/2} d^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.98 \[ \int \frac {(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {\sqrt {d+e x} \left (3 a e^2-c d (d-2 e x)\right )}{c^2 d^2 (a e+c d x)}-\frac {3 e \sqrt {-c d^2+a e^2} \arctan \left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{c^{5/2} d^{5/2}} \]

[In]

Integrate[(d + e*x)^(7/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]

[Out]

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

Maple [A] (verified)

Time = 3.98 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.15

method	result	size
pseudoelliptic	\(\frac {3 \left (-\frac {d \left (-2 e x +d \right ) c}{3}+e^{2} a \right ) \sqrt {e x +d}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}-3 e \left (e^{2} a -c \,d^{2}\right ) \left (c d x +a e \right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}\, c^{2} d^{2} \left (c d x +a e \right )}\)	\(129\)
derivativedivides	\(2 e \left (\frac {\sqrt {e x +d}}{c^{2} d^{2}}-\frac {\frac {\left (-\frac {e^{2} a}{2}+\frac {c \,d^{2}}{2}\right ) \sqrt {e x +d}}{c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {3 \left (e^{2} a -c \,d^{2}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}}{c^{2} d^{2}}\right )\)	\(130\)
default	\(2 e \left (\frac {\sqrt {e x +d}}{c^{2} d^{2}}-\frac {\frac {\left (-\frac {e^{2} a}{2}+\frac {c \,d^{2}}{2}\right ) \sqrt {e x +d}}{c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {3 \left (e^{2} a -c \,d^{2}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}}{c^{2} d^{2}}\right )\)	\(130\)
risch	\(\text {Expression too large to display}\)	\(3810\)

[In]

int((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.52 \[ \int \frac {(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\left [\frac {3 \, {\left (c d e x + a e^{2}\right )} \sqrt {\frac {c d^{2} - a e^{2}}{c d}} \log \left (\frac {c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt {e x + d} c d \sqrt {\frac {c d^{2} - a e^{2}}{c d}}}{c d x + a e}\right ) + 2 \, {\left (2 \, c d e x - c d^{2} + 3 \, a e^{2}\right )} \sqrt {e x + d}}{2 \, {\left (c^{3} d^{3} x + a c^{2} d^{2} e\right )}}, -\frac {3 \, {\left (c d e x + a e^{2}\right )} \sqrt {-\frac {c d^{2} - a e^{2}}{c d}} \arctan \left (-\frac {\sqrt {e x + d} c d \sqrt {-\frac {c d^{2} - a e^{2}}{c d}}}{c d^{2} - a e^{2}}\right ) - {\left (2 \, c d e x - c d^{2} + 3 \, a e^{2}\right )} \sqrt {e x + d}}{c^{3} d^{3} x + a c^{2} d^{2} e}\right ] \]

[In]

integrate((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="fricas")

[Out]

[1/2*(3*(c*d*e*x + a*e^2)*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*sqr
t((c*d^2 - a*e^2)/(c*d)))/(c*d*x + a*e)) + 2*(2*c*d*e*x - c*d^2 + 3*a*e^2)*sqrt(e*x + d))/(c^3*d^3*x + a*c^2*d
^2*e), -(3*(c*d*e*x + a*e^2)*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)) - (2*c*d*e*x - c*d^2 + 3*a*e^2)*sqrt(e*x + d))/(c^3*d^3*x + a*c^2*d^2*e)]

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\text {Timed out} \]

[In]

integrate((e*x+d)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((e*x+d)^(7/2)/(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(a*e^2-c*d^2>0)', see `assume?`
for more de

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.27 \[ \int \frac {(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {2 \, \sqrt {e x + d} e}{c^{2} d^{2}} + \frac {3 \, {\left (c d^{2} e - a e^{3}\right )} \arctan \left (\frac {\sqrt {e x + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{\sqrt {-c^{2} d^{3} + a c d e^{2}} c^{2} d^{2}} - \frac {\sqrt {e x + d} c d^{2} e - \sqrt {e x + d} a e^{3}}{{\left ({\left (e x + d\right )} c d - c d^{2} + a e^{2}\right )} c^{2} d^{2}} \]

[In]

integrate((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.25 \[ \int \frac {(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {\left (a\,e^3-c\,d^2\,e\right )\,\sqrt {d+e\,x}}{c^3\,d^3\,\left (d+e\,x\right )-c^3\,d^4+a\,c^2\,d^2\,e^2}+\frac {2\,e\,\sqrt {d+e\,x}}{c^2\,d^2}-\frac {3\,e\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,e\,\sqrt {a\,e^2-c\,d^2}\,\sqrt {d+e\,x}}{a\,e^3-c\,d^2\,e}\right )\,\sqrt {a\,e^2-c\,d^2}}{c^{5/2}\,d^{5/2}} \]

[In]

int((d + e*x)^(7/2)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^2,x)

[Out]

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