3.21.8 \(\int \frac {1}{(d+e x)^{7/2} (a d e+(c d^2+a e^2) x+c d e x^2)} \, dx\) [2008]

Optimal. Leaf size=186 \[ \frac {2}{7 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}+\frac {2 c d}{5 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}+\frac {2 c^2 d^2}{3 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}+\frac {2 c^3 d^3}{\left (c d^2-a e^2\right )^4 \sqrt {d+e x}}-\frac {2 c^{7/2} d^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{9/2}} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {640, 53, 65, 214} \begin {gather*} -\frac {2 c^{7/2} d^{7/2} \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 {2 c^3 d^3}{\sqrt {d+e x} \left (c d^2-a e^2\right )^4}+\frac {2 c^2 d^2}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3}+\frac {2 c d}{5 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2}+\frac {2}{7 (d+e x)^{7/2} \left (c d^2-a e^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(7/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]

[Out]

2/(7*(c*d^2 - a*e^2)*(d + e*x)^(7/2)) + (2*c*d)/(5*(c*d^2 - a*e^2)^2*(d + e*x)^(5/2)) + (2*c^2*d^2)/(3*(c*d^2
- a*e^2)^3*(d + e*x)^(3/2)) + (2*c^3*d^3)/((c*d^2 - a*e^2)^4*Sqrt[d + e*x]) - (2*c^(7/2)*d^(7/2)*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 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)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx &=\int \frac {1}{(a e+c d x) (d+e x)^{9/2}} \, dx\\ &=\frac {2}{7 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}+\frac {(c d) \int \frac {1}{(a e+c d x) (d+e x)^{7/2}} \, dx}{c d^2-a e^2}\\ &=\frac {2}{7 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}+\frac {2 c d}{5 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}+\frac {\left (c^2 d^2\right ) \int \frac {1}{(a e+c d x) (d+e x)^{5/2}} \, dx}{\left (c d^2-a e^2\right )^2}\\ &=\frac {2}{7 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}+\frac {2 c d}{5 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}+\frac {2 c^2 d^2}{3 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}+\frac {\left (c^3 d^3\right ) \int \frac {1}{(a e+c d x) (d+e x)^{3/2}} \, dx}{\left (c d^2-a e^2\right )^3}\\ &=\frac {2}{7 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}+\frac {2 c d}{5 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}+\frac {2 c^2 d^2}{3 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}+\frac {2 c^3 d^3}{\left (c d^2-a e^2\right )^4 \sqrt {d+e x}}+\frac {\left (c^4 d^4\right ) \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{\left (c d^2-a e^2\right )^4}\\ &=\frac {2}{7 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}+\frac {2 c d}{5 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}+\frac {2 c^2 d^2}{3 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}+\frac {2 c^3 d^3}{\left (c d^2-a e^2\right )^4 \sqrt {d+e x}}+\frac {\left (2 c^4 d^4\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 )}{e \left (c d^2-a e^2\right )^4}\\ &=\frac {2}{7 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}+\frac {2 c d}{5 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}+\frac {2 c^2 d^2}{3 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}+\frac {2 c^3 d^3}{\left (c d^2-a e^2\right )^4 \sqrt {d+e x}}-\frac {2 c^{7/2} d^{7/2} \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 = 0.25, size = 189, normalized size = 1.02 \begin {gather*} \frac {-30 a^3 e^6+6 a^2 c d e^4 (22 d+7 e x)-2 a c^2 d^2 e^2 \left (122 d^2+112 d e x+35 e^2 x^2\right )+2 c^3 d^3 \left (176 d^3+406 d^2 e x+350 d e^2 x^2+105 e^3 x^3\right )}{105 \left (c d^2-a e^2\right )^4 (d+e x)^{7/2}}+\frac {2 c^{7/2} d^{7/2} \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)^(7/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]

[Out]

(-30*a^3*e^6 + 6*a^2*c*d*e^4*(22*d + 7*e*x) - 2*a*c^2*d^2*e^2*(122*d^2 + 112*d*e*x + 35*e^2*x^2) + 2*c^3*d^3*(
176*d^3 + 406*d^2*e*x + 350*d*e^2*x^2 + 105*e^3*x^3))/(105*(c*d^2 - a*e^2)^4*(d + e*x)^(7/2)) + (2*c^(7/2)*d^(
7/2)*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 = 175, normalized size = 0.94

method	result	size
derivativedivides	\(\frac {2 c^{4} d^{4} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{\left (e^{2} a -c \,d^{2}\right )^{4} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}-\frac {2}{7 \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}-\frac {2 c^{2} d^{2}}{3 \left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 c^{3} d^{3}}{\left (e^{2} a -c \,d^{2}\right )^{4} \sqrt {e x +d}}+\frac {2 c d}{5 \left (e^{2} a -c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {5}{2}}}\)	\(175\)
default	\(\frac {2 c^{4} d^{4} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{\left (e^{2} a -c \,d^{2}\right )^{4} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}-\frac {2}{7 \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}-\frac {2 c^{2} d^{2}}{3 \left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 c^{3} d^{3}}{\left (e^{2} a -c \,d^{2}\right )^{4} \sqrt {e x +d}}+\frac {2 c d}{5 \left (e^{2} a -c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {5}{2}}}\)	\(175\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x,method=_RETURNVERBOSE)

[Out]

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

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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)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^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. 575 vs. \(2 (157) = 314\).
time = 4.34, size = 1165, normalized size = 6.26 \begin {gather*} \left [\frac {105 \, {\left (c^{3} d^{3} x^{4} e^{4} + 4 \, c^{3} d^{4} x^{3} e^{3} + 6 \, c^{3} d^{5} x^{2} e^{2} + 4 \, c^{3} d^{6} x e + c^{3} d^{7}\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 (406 \, c^{3} d^{5} x e + 176 \, c^{3} d^{6} + 21 \, a^{2} c d x e^{5} - 15 \, a^{3} e^{6} - {\left (35 \, a c^{2} d^{2} x^{2} - 66 \, a^{2} c d^{2}\right )} e^{4} + 7 \, {\left (15 \, c^{3} d^{3} x^{3} - 16 \, a c^{2} d^{3} x\right )} e^{3} + 2 \, {\left (175 \, c^{3} d^{4} x^{2} - 61 \, a c^{2} d^{4}\right )} e^{2}\right )} \sqrt {x e + d}}{105 \, {\left (4 \, c^{4} d^{11} x e + c^{4} d^{12} + a^{4} x^{4} e^{12} + 4 \, a^{4} d x^{3} e^{11} - 2 \, {\left (2 \, a^{3} c d^{2} x^{4} - 3 \, a^{4} d^{2} x^{2}\right )} e^{10} - 4 \, {\left (4 \, a^{3} c d^{3} x^{3} - a^{4} d^{3} x\right )} e^{9} + {\left (6 \, a^{2} c^{2} d^{4} x^{4} - 24 \, a^{3} c d^{4} x^{2} + a^{4} d^{4}\right )} e^{8} + 8 \, {\left (3 \, a^{2} c^{2} d^{5} x^{3} - 2 \, a^{3} c d^{5} x\right )} e^{7} - 4 \, {\left (a c^{3} d^{6} x^{4} - 9 \, a^{2} c^{2} d^{6} x^{2} + a^{3} c d^{6}\right )} e^{6} - 8 \, {\left (2 \, a c^{3} d^{7} x^{3} - 3 \, a^{2} c^{2} d^{7} x\right )} e^{5} + {\left (c^{4} d^{8} x^{4} - 24 \, a c^{3} d^{8} x^{2} + 6 \, a^{2} c^{2} d^{8}\right )} e^{4} + 4 \, {\left (c^{4} d^{9} x^{3} - 4 \, a c^{3} d^{9} x\right )} e^{3} + 2 \, {\left (3 \, c^{4} d^{10} x^{2} - 2 \, a c^{3} d^{10}\right )} e^{2}\right )}}, -\frac {2 \, {\left (105 \, {\left (c^{3} d^{3} x^{4} e^{4} + 4 \, c^{3} d^{4} x^{3} e^{3} + 6 \, c^{3} d^{5} x^{2} e^{2} + 4 \, c^{3} d^{6} x e + c^{3} d^{7}\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 (406 \, c^{3} d^{5} x e + 176 \, c^{3} d^{6} + 21 \, a^{2} c d x e^{5} - 15 \, a^{3} e^{6} - {\left (35 \, a c^{2} d^{2} x^{2} - 66 \, a^{2} c d^{2}\right )} e^{4} + 7 \, {\left (15 \, c^{3} d^{3} x^{3} - 16 \, a c^{2} d^{3} x\right )} e^{3} + 2 \, {\left (175 \, c^{3} d^{4} x^{2} - 61 \, a c^{2} d^{4}\right )} e^{2}\right )} \sqrt {x e + d}\right )}}{105 \, {\left (4 \, c^{4} d^{11} x e + c^{4} d^{12} + a^{4} x^{4} e^{12} + 4 \, a^{4} d x^{3} e^{11} - 2 \, {\left (2 \, a^{3} c d^{2} x^{4} - 3 \, a^{4} d^{2} x^{2}\right )} e^{10} - 4 \, {\left (4 \, a^{3} c d^{3} x^{3} - a^{4} d^{3} x\right )} e^{9} + {\left (6 \, a^{2} c^{2} d^{4} x^{4} - 24 \, a^{3} c d^{4} x^{2} + a^{4} d^{4}\right )} e^{8} + 8 \, {\left (3 \, a^{2} c^{2} d^{5} x^{3} - 2 \, a^{3} c d^{5} x\right )} e^{7} - 4 \, {\left (a c^{3} d^{6} x^{4} - 9 \, a^{2} c^{2} d^{6} x^{2} + a^{3} c d^{6}\right )} e^{6} - 8 \, {\left (2 \, a c^{3} d^{7} x^{3} - 3 \, a^{2} c^{2} d^{7} x\right )} e^{5} + {\left (c^{4} d^{8} x^{4} - 24 \, a c^{3} d^{8} x^{2} + 6 \, a^{2} c^{2} d^{8}\right )} e^{4} + 4 \, {\left (c^{4} d^{9} x^{3} - 4 \, a c^{3} d^{9} x\right )} e^{3} + 2 \, {\left (3 \, c^{4} d^{10} x^{2} - 2 \, a c^{3} d^{10}\right )} e^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="fricas")

[Out]

[1/105*(105*(c^3*d^3*x^4*e^4 + 4*c^3*d^4*x^3*e^3 + 6*c^3*d^5*x^2*e^2 + 4*c^3*d^6*x*e + c^3*d^7)*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*(406*c^3*d^5*x*e + 176*c^3*d^6 + 21*a^2*c*d*x*e^5 - 15*a^3*e^6 - (35*a*c^2*d^2*x^2 - 66*a^2*c*d^2
)*e^4 + 7*(15*c^3*d^3*x^3 - 16*a*c^2*d^3*x)*e^3 + 2*(175*c^3*d^4*x^2 - 61*a*c^2*d^4)*e^2)*sqrt(x*e + d))/(4*c^
4*d^11*x*e + c^4*d^12 + a^4*x^4*e^12 + 4*a^4*d*x^3*e^11 - 2*(2*a^3*c*d^2*x^4 - 3*a^4*d^2*x^2)*e^10 - 4*(4*a^3*
c*d^3*x^3 - a^4*d^3*x)*e^9 + (6*a^2*c^2*d^4*x^4 - 24*a^3*c*d^4*x^2 + a^4*d^4)*e^8 + 8*(3*a^2*c^2*d^5*x^3 - 2*a
^3*c*d^5*x)*e^7 - 4*(a*c^3*d^6*x^4 - 9*a^2*c^2*d^6*x^2 + a^3*c*d^6)*e^6 - 8*(2*a*c^3*d^7*x^3 - 3*a^2*c^2*d^7*x
)*e^5 + (c^4*d^8*x^4 - 24*a*c^3*d^8*x^2 + 6*a^2*c^2*d^8)*e^4 + 4*(c^4*d^9*x^3 - 4*a*c^3*d^9*x)*e^3 + 2*(3*c^4*
d^10*x^2 - 2*a*c^3*d^10)*e^2), -2/105*(105*(c^3*d^3*x^4*e^4 + 4*c^3*d^4*x^3*e^3 + 6*c^3*d^5*x^2*e^2 + 4*c^3*d^
6*x*e + c^3*d^7)*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)) - (406*c^3*d^5*x*e + 176*c^3*d^6 + 21*a^2*c*d*x*e^5 - 15*a^3*e^6 - (35*a*c^2*d^2*x^2 - 66*a^
2*c*d^2)*e^4 + 7*(15*c^3*d^3*x^3 - 16*a*c^2*d^3*x)*e^3 + 2*(175*c^3*d^4*x^2 - 61*a*c^2*d^4)*e^2)*sqrt(x*e + d)
)/(4*c^4*d^11*x*e + c^4*d^12 + a^4*x^4*e^12 + 4*a^4*d*x^3*e^11 - 2*(2*a^3*c*d^2*x^4 - 3*a^4*d^2*x^2)*e^10 - 4*
(4*a^3*c*d^3*x^3 - a^4*d^3*x)*e^9 + (6*a^2*c^2*d^4*x^4 - 24*a^3*c*d^4*x^2 + a^4*d^4)*e^8 + 8*(3*a^2*c^2*d^5*x^
3 - 2*a^3*c*d^5*x)*e^7 - 4*(a*c^3*d^6*x^4 - 9*a^2*c^2*d^6*x^2 + a^3*c*d^6)*e^6 - 8*(2*a*c^3*d^7*x^3 - 3*a^2*c^
2*d^7*x)*e^5 + (c^4*d^8*x^4 - 24*a*c^3*d^8*x^2 + 6*a^2*c^2*d^8)*e^4 + 4*(c^4*d^9*x^3 - 4*a*c^3*d^9*x)*e^3 + 2*
(3*c^4*d^10*x^2 - 2*a*c^3*d^10)*e^2)]

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

[Out]

Timed out

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Giac [A]
time = 1.00, size = 302, normalized size = 1.62 \begin {gather*} \frac {2 \, c^{4} d^{4} \arctan \left (\frac {\sqrt {x e + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{{\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 {2 \, {\left (105 \, {\left (x e + d\right )}^{3} c^{3} d^{3} + 35 \, {\left (x e + d\right )}^{2} c^{3} d^{4} + 21 \, {\left (x e + d\right )} c^{3} d^{5} + 15 \, c^{3} d^{6} - 35 \, {\left (x e + d\right )}^{2} a c^{2} d^{2} e^{2} - 42 \, {\left (x e + d\right )} a c^{2} d^{3} e^{2} - 45 \, a c^{2} d^{4} e^{2} + 21 \, {\left (x e + d\right )} a^{2} c d e^{4} + 45 \, a^{2} c d^{2} e^{4} - 15 \, a^{3} e^{6}\right )}}{105 \, {\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 {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="giac")

[Out]

2*c^4*d^4*arctan(sqrt(x*e + d)*c*d/sqrt(-c^2*d^3 + a*c*d*e^2))/((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)) + 2/105*(105*(x*e + d)^3*c^3*d^3 + 35*(x*e + d)^2*c^
3*d^4 + 21*(x*e + d)*c^3*d^5 + 15*c^3*d^6 - 35*(x*e + d)^2*a*c^2*d^2*e^2 - 42*(x*e + d)*a*c^2*d^3*e^2 - 45*a*c
^2*d^4*e^2 + 21*(x*e + d)*a^2*c*d*e^4 + 45*a^2*c*d^2*e^4 - 15*a^3*e^6)/((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)^(7/2))

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Mupad [B]
time = 0.74, size = 213, normalized size = 1.15 \begin {gather*} \frac {2\,c^{7/2}\,d^{7/2}\,\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}}-\frac {\frac {2}{7\,\left (a\,e^2-c\,d^2\right )}+\frac {2\,c^2\,d^2\,{\left (d+e\,x\right )}^2}{3\,{\left (a\,e^2-c\,d^2\right )}^3}-\frac {2\,c^3\,d^3\,{\left (d+e\,x\right )}^3}{{\left (a\,e^2-c\,d^2\right )}^4}-\frac {2\,c\,d\,\left (d+e\,x\right )}{5\,{\left (a\,e^2-c\,d^2\right )}^2}}{{\left (d+e\,x\right )}^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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