3.21.17 \(\int \frac {1}{(d+e x)^{3/2} (a d e+(c d^2+a e^2) x+c d e x^2)^2} \, dx\) [2017]

Optimal. Leaf size=192 \[ -\frac {7 e}{5 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}-\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x) (d+e x)^{5/2}}-\frac {7 c d e}{3 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}-\frac {7 c^2 d^2 e}{\left (c d^2-a e^2\right )^4 \sqrt {d+e x}}+\frac {7 c^{5/2} d^{5/2} e \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{9/2}} \]

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 192, 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, 44, 53, 65, 214} \begin {gather*} \frac {7 c^{5/2} d^{5/2} e \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{9/2}}-\frac {7 c^2 d^2 e}{\sqrt {d+e x} \left (c d^2-a e^2\right )^4}-\frac {7 c d e}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3}-\frac {1}{(d+e x)^{5/2} \left (c d^2-a e^2\right ) (a e+c d x)}-\frac {7 e}{5 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*e)/(5*(c*d^2 - a*e^2)^2*(d + e*x)^(5/2)) - 1/((c*d^2 - a*e^2)*(a*e + c*d*x)*(d + e*x)^(5/2)) - (7*c*d*e)/(
3*(c*d^2 - a*e^2)^3*(d + e*x)^(3/2)) - (7*c^2*d^2*e)/((c*d^2 - a*e^2)^4*Sqrt[d + e*x]) + (7*c^(5/2)*d^(5/2)*e*
ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(c*d^2 - a*e^2)^(9/2)

Rule 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 53

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 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 {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=\int \frac {1}{(a e+c d x)^2 (d+e x)^{7/2}} \, dx\\ &=-\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x) (d+e x)^{5/2}}-\frac {(7 e) \int \frac {1}{(a e+c d x) (d+e x)^{7/2}} \, dx}{2 \left (c d^2-a e^2\right )}\\ &=-\frac {7 e}{5 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}-\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x) (d+e x)^{5/2}}-\frac {(7 c d e) \int \frac {1}{(a e+c d x) (d+e x)^{5/2}} \, dx}{2 \left (c d^2-a e^2\right )^2}\\ &=-\frac {7 e}{5 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}-\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x) (d+e x)^{5/2}}-\frac {7 c d e}{3 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}-\frac {\left (7 c^2 d^2 e\right ) \int \frac {1}{(a e+c d x) (d+e x)^{3/2}} \, dx}{2 \left (c d^2-a e^2\right )^3}\\ &=-\frac {7 e}{5 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}-\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x) (d+e x)^{5/2}}-\frac {7 c d e}{3 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}-\frac {7 c^2 d^2 e}{\left (c d^2-a e^2\right )^4 \sqrt {d+e x}}-\frac {\left (7 c^3 d^3 e\right ) \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{2 \left (c d^2-a e^2\right )^4}\\ &=-\frac {7 e}{5 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}-\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x) (d+e x)^{5/2}}-\frac {7 c d e}{3 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}-\frac {7 c^2 d^2 e}{\left (c d^2-a e^2\right )^4 \sqrt {d+e x}}-\frac {\left (7 c^3 d^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 )}{\left (c d^2-a e^2\right )^4}\\ &=-\frac {7 e}{5 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}-\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x) (d+e x)^{5/2}}-\frac {7 c d e}{3 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}-\frac {7 c^2 d^2 e}{\left (c d^2-a e^2\right )^4 \sqrt {d+e x}}+\frac {7 c^{5/2} d^{5/2} e \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.42, size = 200, normalized size = 1.04 \begin {gather*} \frac {-6 a^3 e^6+2 a^2 c d e^4 (16 d+7 e x)-2 a c^2 d^2 e^2 \left (58 d^2+84 d e x+35 e^2 x^2\right )-c^3 d^3 \left (15 d^3+161 d^2 e x+245 d e^2 x^2+105 e^3 x^3\right )}{15 \left (c d^2-a e^2\right )^4 (a e+c d x) (d+e x)^{5/2}}-\frac {7 c^{5/2} d^{5/2} e \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{\left (-c d^2+a e^2\right )^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*a^3*e^6 + 2*a^2*c*d*e^4*(16*d + 7*e*x) - 2*a*c^2*d^2*e^2*(58*d^2 + 84*d*e*x + 35*e^2*x^2) - c^3*d^3*(15*d^
3 + 161*d^2*e*x + 245*d*e^2*x^2 + 105*e^3*x^3))/(15*(c*d^2 - a*e^2)^4*(a*e + c*d*x)*(d + e*x)^(5/2)) - (7*c^(5
/2)*d^(5/2)*e*ArcTan[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[-(c*d^2) + a*e^2]])/(-(c*d^2) + a*e^2)^(9/2)

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Maple [A]
time = 0.76, size = 183, normalized size = 0.95

method	result	size
derivativedivides	\(2 e \left (-\frac {c^{3} d^{3} \left (\frac {\sqrt {e x +d}}{2 c d \left (e x +d \right )+2 e^{2} a -2 c \,d^{2}}+\frac {7 \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{\left (e^{2} a -c \,d^{2}\right )^{4}}-\frac {1}{5 \left (e^{2} a -c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {5}{2}}}-\frac {3 c^{2} d^{2}}{\left (e^{2} a -c \,d^{2}\right )^{4} \sqrt {e x +d}}+\frac {2 c d}{3 \left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {3}{2}}}\right )\)	\(183\)
default	\(2 e \left (-\frac {c^{3} d^{3} \left (\frac {\sqrt {e x +d}}{2 c d \left (e x +d \right )+2 e^{2} a -2 c \,d^{2}}+\frac {7 \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{\left (e^{2} a -c \,d^{2}\right )^{4}}-\frac {1}{5 \left (e^{2} a -c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {5}{2}}}-\frac {3 c^{2} d^{2}}{\left (e^{2} a -c \,d^{2}\right )^{4} \sqrt {e x +d}}+\frac {2 c d}{3 \left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {3}{2}}}\right )\)	\(183\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x,method=_RETURNVERBOSE)

[Out]

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

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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)^(3/2)/(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 [B] Leaf count of result is larger than twice the leaf count of optimal. 661 vs. \(2 (170) = 340\).
time = 2.65, size = 1337, normalized size = 6.96 \begin {gather*} \left [\frac {105 \, {\left (c^{3} d^{6} x e + a c^{2} d^{2} x^{3} e^{5} + {\left (c^{3} d^{3} x^{4} + 3 \, a c^{2} d^{3} x^{2}\right )} e^{4} + 3 \, {\left (c^{3} d^{4} x^{3} + a c^{2} d^{4} x\right )} e^{3} + {\left (3 \, c^{3} d^{5} x^{2} + a c^{2} d^{5}\right )} e^{2}\right )} \sqrt {\frac {c d}{c d^{2} - a e^{2}}} \log \left (\frac {c d x e + 2 \, c d^{2} + 2 \, {\left (c d^{2} - a e^{2}\right )} \sqrt {x e + d} \sqrt {\frac {c d}{c d^{2} - a e^{2}}} - a e^{2}}{c d x + a e}\right ) - 2 \, {\left (161 \, c^{3} d^{5} x e + 15 \, c^{3} d^{6} - 14 \, a^{2} c d x e^{5} + 6 \, a^{3} e^{6} + 2 \, {\left (35 \, a c^{2} d^{2} x^{2} - 16 \, a^{2} c d^{2}\right )} e^{4} + 21 \, {\left (5 \, c^{3} d^{3} x^{3} + 8 \, a c^{2} d^{3} x\right )} e^{3} + {\left (245 \, c^{3} d^{4} x^{2} + 116 \, a c^{2} d^{4}\right )} e^{2}\right )} \sqrt {x e + d}}{30 \, {\left (c^{5} d^{12} x + a^{5} x^{3} e^{12} + {\left (a^{4} c d x^{4} + 3 \, a^{5} d x^{2}\right )} e^{11} - {\left (a^{4} c d^{2} x^{3} - 3 \, a^{5} d^{2} x\right )} e^{10} - {\left (4 \, a^{3} c^{2} d^{3} x^{4} + 9 \, a^{4} c d^{3} x^{2} - a^{5} d^{3}\right )} e^{9} - {\left (6 \, a^{3} c^{2} d^{4} x^{3} + 11 \, a^{4} c d^{4} x\right )} e^{8} + 2 \, {\left (3 \, a^{2} c^{3} d^{5} x^{4} + 3 \, a^{3} c^{2} d^{5} x^{2} - 2 \, a^{4} c d^{5}\right )} e^{7} + 14 \, {\left (a^{2} c^{3} d^{6} x^{3} + a^{3} c^{2} d^{6} x\right )} e^{6} - 2 \, {\left (2 \, a c^{4} d^{7} x^{4} - 3 \, a^{2} c^{3} d^{7} x^{2} - 3 \, a^{3} c^{2} d^{7}\right )} e^{5} - {\left (11 \, a c^{4} d^{8} x^{3} + 6 \, a^{2} c^{3} d^{8} x\right )} e^{4} + {\left (c^{5} d^{9} x^{4} - 9 \, a c^{4} d^{9} x^{2} - 4 \, a^{2} c^{3} d^{9}\right )} e^{3} + {\left (3 \, c^{5} d^{10} x^{3} - a c^{4} d^{10} x\right )} e^{2} + {\left (3 \, c^{5} d^{11} x^{2} + a c^{4} d^{11}\right )} e\right )}}, \frac {105 \, {\left (c^{3} d^{6} x e + a c^{2} d^{2} x^{3} e^{5} + {\left (c^{3} d^{3} x^{4} + 3 \, a c^{2} d^{3} x^{2}\right )} e^{4} + 3 \, {\left (c^{3} d^{4} x^{3} + a c^{2} d^{4} x\right )} e^{3} + {\left (3 \, c^{3} d^{5} x^{2} + a c^{2} d^{5}\right )} e^{2}\right )} \sqrt {-\frac {c d}{c d^{2} - a e^{2}}} \arctan \left (-\frac {{\left (c d^{2} - a e^{2}\right )} \sqrt {x e + d} \sqrt {-\frac {c d}{c d^{2} - a e^{2}}}}{c d x e + c d^{2}}\right ) - {\left (161 \, c^{3} d^{5} x e + 15 \, c^{3} d^{6} - 14 \, a^{2} c d x e^{5} + 6 \, a^{3} e^{6} + 2 \, {\left (35 \, a c^{2} d^{2} x^{2} - 16 \, a^{2} c d^{2}\right )} e^{4} + 21 \, {\left (5 \, c^{3} d^{3} x^{3} + 8 \, a c^{2} d^{3} x\right )} e^{3} + {\left (245 \, c^{3} d^{4} x^{2} + 116 \, a c^{2} d^{4}\right )} e^{2}\right )} \sqrt {x e + d}}{15 \, {\left (c^{5} d^{12} x + a^{5} x^{3} e^{12} + {\left (a^{4} c d x^{4} + 3 \, a^{5} d x^{2}\right )} e^{11} - {\left (a^{4} c d^{2} x^{3} - 3 \, a^{5} d^{2} x\right )} e^{10} - {\left (4 \, a^{3} c^{2} d^{3} x^{4} + 9 \, a^{4} c d^{3} x^{2} - a^{5} d^{3}\right )} e^{9} - {\left (6 \, a^{3} c^{2} d^{4} x^{3} + 11 \, a^{4} c d^{4} x\right )} e^{8} + 2 \, {\left (3 \, a^{2} c^{3} d^{5} x^{4} + 3 \, a^{3} c^{2} d^{5} x^{2} - 2 \, a^{4} c d^{5}\right )} e^{7} + 14 \, {\left (a^{2} c^{3} d^{6} x^{3} + a^{3} c^{2} d^{6} x\right )} e^{6} - 2 \, {\left (2 \, a c^{4} d^{7} x^{4} - 3 \, a^{2} c^{3} d^{7} x^{2} - 3 \, a^{3} c^{2} d^{7}\right )} e^{5} - {\left (11 \, a c^{4} d^{8} x^{3} + 6 \, a^{2} c^{3} d^{8} x\right )} e^{4} + {\left (c^{5} d^{9} x^{4} - 9 \, a c^{4} d^{9} x^{2} - 4 \, a^{2} c^{3} d^{9}\right )} e^{3} + {\left (3 \, c^{5} d^{10} x^{3} - a c^{4} d^{10} x\right )} e^{2} + {\left (3 \, c^{5} d^{11} x^{2} + a c^{4} d^{11}\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(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^6*x*e + a*c^2*d^2*x^3*e^5 + (c^3*d^3*x^4 + 3*a*c^2*d^3*x^2)*e^4 + 3*(c^3*d^4*x^3 + a*c^2*d^4
*x)*e^3 + (3*c^3*d^5*x^2 + a*c^2*d^5)*e^2)*sqrt(c*d/(c*d^2 - a*e^2))*log((c*d*x*e + 2*c*d^2 + 2*(c*d^2 - a*e^2
)*sqrt(x*e + d)*sqrt(c*d/(c*d^2 - a*e^2)) - a*e^2)/(c*d*x + a*e)) - 2*(161*c^3*d^5*x*e + 15*c^3*d^6 - 14*a^2*c
*d*x*e^5 + 6*a^3*e^6 + 2*(35*a*c^2*d^2*x^2 - 16*a^2*c*d^2)*e^4 + 21*(5*c^3*d^3*x^3 + 8*a*c^2*d^3*x)*e^3 + (245
*c^3*d^4*x^2 + 116*a*c^2*d^4)*e^2)*sqrt(x*e + d))/(c^5*d^12*x + a^5*x^3*e^12 + (a^4*c*d*x^4 + 3*a^5*d*x^2)*e^1
1 - (a^4*c*d^2*x^3 - 3*a^5*d^2*x)*e^10 - (4*a^3*c^2*d^3*x^4 + 9*a^4*c*d^3*x^2 - a^5*d^3)*e^9 - (6*a^3*c^2*d^4*
x^3 + 11*a^4*c*d^4*x)*e^8 + 2*(3*a^2*c^3*d^5*x^4 + 3*a^3*c^2*d^5*x^2 - 2*a^4*c*d^5)*e^7 + 14*(a^2*c^3*d^6*x^3
+ a^3*c^2*d^6*x)*e^6 - 2*(2*a*c^4*d^7*x^4 - 3*a^2*c^3*d^7*x^2 - 3*a^3*c^2*d^7)*e^5 - (11*a*c^4*d^8*x^3 + 6*a^2
*c^3*d^8*x)*e^4 + (c^5*d^9*x^4 - 9*a*c^4*d^9*x^2 - 4*a^2*c^3*d^9)*e^3 + (3*c^5*d^10*x^3 - a*c^4*d^10*x)*e^2 +
(3*c^5*d^11*x^2 + a*c^4*d^11)*e), 1/15*(105*(c^3*d^6*x*e + a*c^2*d^2*x^3*e^5 + (c^3*d^3*x^4 + 3*a*c^2*d^3*x^2)
*e^4 + 3*(c^3*d^4*x^3 + a*c^2*d^4*x)*e^3 + (3*c^3*d^5*x^2 + a*c^2*d^5)*e^2)*sqrt(-c*d/(c*d^2 - a*e^2))*arctan(
-(c*d^2 - a*e^2)*sqrt(x*e + d)*sqrt(-c*d/(c*d^2 - a*e^2))/(c*d*x*e + c*d^2)) - (161*c^3*d^5*x*e + 15*c^3*d^6 -
14*a^2*c*d*x*e^5 + 6*a^3*e^6 + 2*(35*a*c^2*d^2*x^2 - 16*a^2*c*d^2)*e^4 + 21*(5*c^3*d^3*x^3 + 8*a*c^2*d^3*x)*e
^3 + (245*c^3*d^4*x^2 + 116*a*c^2*d^4)*e^2)*sqrt(x*e + d))/(c^5*d^12*x + a^5*x^3*e^12 + (a^4*c*d*x^4 + 3*a^5*d
*x^2)*e^11 - (a^4*c*d^2*x^3 - 3*a^5*d^2*x)*e^10 - (4*a^3*c^2*d^3*x^4 + 9*a^4*c*d^3*x^2 - a^5*d^3)*e^9 - (6*a^3
*c^2*d^4*x^3 + 11*a^4*c*d^4*x)*e^8 + 2*(3*a^2*c^3*d^5*x^4 + 3*a^3*c^2*d^5*x^2 - 2*a^4*c*d^5)*e^7 + 14*(a^2*c^3
*d^6*x^3 + a^3*c^2*d^6*x)*e^6 - 2*(2*a*c^4*d^7*x^4 - 3*a^2*c^3*d^7*x^2 - 3*a^3*c^2*d^7)*e^5 - (11*a*c^4*d^8*x^
3 + 6*a^2*c^3*d^8*x)*e^4 + (c^5*d^9*x^4 - 9*a*c^4*d^9*x^2 - 4*a^2*c^3*d^9)*e^3 + (3*c^5*d^10*x^3 - a*c^4*d^10*
x)*e^2 + (3*c^5*d^11*x^2 + a*c^4*d^11)*e)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d + e x\right )^{\frac {7}{2}} \left (a e + c d x\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(3/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)

[Out]

Integral(1/((d + e*x)**(7/2)*(a*e + c*d*x)**2), x)

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Giac [A]
time = 2.97, size = 334, normalized size = 1.74 \begin {gather*} -\frac {7 \, c^{3} d^{3} \arctan \left (\frac {\sqrt {x e + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right ) e}{{\left (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}\right )} \sqrt {-c^{2} d^{3} + a c d e^{2}}} - \frac {\sqrt {x e + d} c^{3} d^{3} e}{{\left (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}\right )} {\left ({\left (x e + d\right )} c d - c d^{2} + a e^{2}\right )}} - \frac {2 \, {\left (45 \, {\left (x e + d\right )}^{2} c^{2} d^{2} e + 10 \, {\left (x e + d\right )} c^{2} d^{3} e + 3 \, c^{2} d^{4} e - 10 \, {\left (x e + d\right )} a c d e^{3} - 6 \, a c d^{2} e^{3} + 3 \, a^{2} e^{5}\right )}}{15 \, {\left (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}\right )} {\left (x e + d\right )}^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.83, size = 244, normalized size = 1.27 \begin {gather*} -\frac {\frac {2\,e}{5\,\left (a\,e^2-c\,d^2\right )}-\frac {14\,c\,d\,e\,\left (d+e\,x\right )}{15\,{\left (a\,e^2-c\,d^2\right )}^2}+\frac {14\,c^2\,d^2\,e\,{\left (d+e\,x\right )}^2}{3\,{\left (a\,e^2-c\,d^2\right )}^3}+\frac {7\,c^3\,d^3\,e\,{\left (d+e\,x\right )}^3}{{\left (a\,e^2-c\,d^2\right )}^4}}{\left (a\,e^2-c\,d^2\right )\,{\left (d+e\,x\right )}^{5/2}+c\,d\,{\left (d+e\,x\right )}^{7/2}}-\frac {7\,c^{5/2}\,d^{5/2}\,e\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,\sqrt {d+e\,x}\,\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 )}{{\left (a\,e^2-c\,d^2\right )}^{9/2}}\right )}{{\left (a\,e^2-c\,d^2\right )}^{9/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d + e*x)^(3/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^2),x)

[Out]

- ((2*e)/(5*(a*e^2 - c*d^2)) - (14*c*d*e*(d + e*x))/(15*(a*e^2 - c*d^2)^2) + (14*c^2*d^2*e*(d + e*x)^2)/(3*(a*
e^2 - c*d^2)^3) + (7*c^3*d^3*e*(d + e*x)^3)/(a*e^2 - c*d^2)^4)/((a*e^2 - c*d^2)*(d + e*x)^(5/2) + c*d*(d + e*x
)^(7/2)) - (7*c^(5/2)*d^(5/2)*e*atan((c^(1/2)*d^(1/2)*(d + e*x)^(1/2)*(a^4*e^8 + c^4*d^8 - 4*a*c^3*d^6*e^2 - 4
*a^3*c*d^2*e^6 + 6*a^2*c^2*d^4*e^4))/(a*e^2 - c*d^2)^(9/2)))/(a*e^2 - c*d^2)^(9/2)

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