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

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

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Rubi [A]  time = 0.16, 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 = {626, 51, 63, 208} \begin {gather*} \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^{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 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 51

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 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 {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 ) \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 )}{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 [C]  time = 0.01, size = 57, normalized size = 0.31 \begin {gather*} \frac {2 \, _2F_1\left (-\frac {7}{2},1;-\frac {5}{2};\frac {c d (d+e x)}{c d^2-a e^2}\right )}{7 (d+e x)^{7/2} \left (c d^2-a e^2\right )} \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]

(2*Hypergeometric2F1[-7/2, 1, -5/2, (c*d*(d + e*x))/(c*d^2 - a*e^2)])/(7*(c*d^2 - a*e^2)*(d + e*x)^(7/2))

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IntegrateAlgebraic [A]  time = 0.29, size = 242, normalized size = 1.30 \begin {gather*} \frac {2 \left (-15 a^3 e^6+45 a^2 c d^2 e^4+21 a^2 c d e^4 (d+e x)-45 a c^2 d^4 e^2-42 a c^2 d^3 e^2 (d+e x)-35 a c^2 d^2 e^2 (d+e x)^2+15 c^3 d^6+21 c^3 d^5 (d+e x)+35 c^3 d^4 (d+e x)^2+105 c^3 d^3 (d+e x)^3\right )}{105 (d+e x)^{7/2} \left (c d^2-a e^2\right )^4}-\frac {2 c^{7/2} d^{7/2} \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 )}{\left (a e^2-c d^2\right )^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(15*c^3*d^6 - 45*a*c^2*d^4*e^2 + 45*a^2*c*d^2*e^4 - 15*a^3*e^6 + 21*c^3*d^5*(d + e*x) - 42*a*c^2*d^3*e^2*(d
+ e*x) + 21*a^2*c*d*e^4*(d + e*x) + 35*c^3*d^4*(d + e*x)^2 - 35*a*c^2*d^2*e^2*(d + e*x)^2 + 105*c^3*d^3*(d +
e*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[-(c*d^2) +
a*e^2]*Sqrt[d + e*x])/(c*d^2 - a*e^2)])/(-(c*d^2) + a*e^2)^(9/2)

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fricas [B]  time = 0.43, size = 1157, normalized size = 6.22 \begin {gather*} \left [\frac {105 \, {\left (c^{3} d^{3} e^{4} x^{4} + 4 \, c^{3} d^{4} e^{3} x^{3} + 6 \, c^{3} d^{5} e^{2} x^{2} + 4 \, c^{3} d^{6} e x + c^{3} d^{7}\right )} \sqrt {\frac {c d}{c d^{2} - a e^{2}}} \log \left (\frac {c d e x + 2 \, c d^{2} - a e^{2} - 2 \, {\left (c d^{2} - a e^{2}\right )} \sqrt {e x + d} \sqrt {\frac {c d}{c d^{2} - a e^{2}}}}{c d x + a e}\right ) + 2 \, {\left (105 \, c^{3} d^{3} e^{3} x^{3} + 176 \, c^{3} d^{6} - 122 \, a c^{2} d^{4} e^{2} + 66 \, a^{2} c d^{2} e^{4} - 15 \, a^{3} e^{6} + 35 \, {\left (10 \, c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 7 \, {\left (58 \, c^{3} d^{5} e - 16 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {e x + d}}{105 \, {\left (c^{4} d^{12} - 4 \, a c^{3} d^{10} e^{2} + 6 \, a^{2} c^{2} d^{8} e^{4} - 4 \, a^{3} c d^{6} e^{6} + a^{4} d^{4} e^{8} + {\left (c^{4} d^{8} e^{4} - 4 \, a c^{3} d^{6} e^{6} + 6 \, a^{2} c^{2} d^{4} e^{8} - 4 \, a^{3} c d^{2} e^{10} + a^{4} e^{12}\right )} x^{4} + 4 \, {\left (c^{4} d^{9} e^{3} - 4 \, a c^{3} d^{7} e^{5} + 6 \, a^{2} c^{2} d^{5} e^{7} - 4 \, a^{3} c d^{3} e^{9} + a^{4} d e^{11}\right )} x^{3} + 6 \, {\left (c^{4} d^{10} e^{2} - 4 \, a c^{3} d^{8} e^{4} + 6 \, a^{2} c^{2} d^{6} e^{6} - 4 \, a^{3} c d^{4} e^{8} + a^{4} d^{2} e^{10}\right )} x^{2} + 4 \, {\left (c^{4} d^{11} e - 4 \, a c^{3} d^{9} e^{3} + 6 \, a^{2} c^{2} d^{7} e^{5} - 4 \, a^{3} c d^{5} e^{7} + a^{4} d^{3} e^{9}\right )} x\right )}}, -\frac {2 \, {\left (105 \, {\left (c^{3} d^{3} e^{4} x^{4} + 4 \, c^{3} d^{4} e^{3} x^{3} + 6 \, c^{3} d^{5} e^{2} x^{2} + 4 \, c^{3} d^{6} e x + 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 {e x + d} \sqrt {-\frac {c d}{c d^{2} - a e^{2}}}}{c d e x + c d^{2}}\right ) - {\left (105 \, c^{3} d^{3} e^{3} x^{3} + 176 \, c^{3} d^{6} - 122 \, a c^{2} d^{4} e^{2} + 66 \, a^{2} c d^{2} e^{4} - 15 \, a^{3} e^{6} + 35 \, {\left (10 \, c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 7 \, {\left (58 \, c^{3} d^{5} e - 16 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {e x + d}\right )}}{105 \, {\left (c^{4} d^{12} - 4 \, a c^{3} d^{10} e^{2} + 6 \, a^{2} c^{2} d^{8} e^{4} - 4 \, a^{3} c d^{6} e^{6} + a^{4} d^{4} e^{8} + {\left (c^{4} d^{8} e^{4} - 4 \, a c^{3} d^{6} e^{6} + 6 \, a^{2} c^{2} d^{4} e^{8} - 4 \, a^{3} c d^{2} e^{10} + a^{4} e^{12}\right )} x^{4} + 4 \, {\left (c^{4} d^{9} e^{3} - 4 \, a c^{3} d^{7} e^{5} + 6 \, a^{2} c^{2} d^{5} e^{7} - 4 \, a^{3} c d^{3} e^{9} + a^{4} d e^{11}\right )} x^{3} + 6 \, {\left (c^{4} d^{10} e^{2} - 4 \, a c^{3} d^{8} e^{4} + 6 \, a^{2} c^{2} d^{6} e^{6} - 4 \, a^{3} c d^{4} e^{8} + a^{4} d^{2} e^{10}\right )} x^{2} + 4 \, {\left (c^{4} d^{11} e - 4 \, a c^{3} d^{9} e^{3} + 6 \, a^{2} c^{2} d^{7} e^{5} - 4 \, a^{3} c d^{5} e^{7} + a^{4} d^{3} e^{9}\right )} x\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*e^4*x^4 + 4*c^3*d^4*e^3*x^3 + 6*c^3*d^5*e^2*x^2 + 4*c^3*d^6*e*x + c^3*d^7)*sqrt(c*d/(c*d^
2 - a*e^2))*log((c*d*e*x + 2*c*d^2 - a*e^2 - 2*(c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt(c*d/(c*d^2 - a*e^2)))/(c*d*x
+ a*e)) + 2*(105*c^3*d^3*e^3*x^3 + 176*c^3*d^6 - 122*a*c^2*d^4*e^2 + 66*a^2*c*d^2*e^4 - 15*a^3*e^6 + 35*(10*c
^3*d^4*e^2 - a*c^2*d^2*e^4)*x^2 + 7*(58*c^3*d^5*e - 16*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*x)*sqrt(e*x + d))/(c^4*d
^12 - 4*a*c^3*d^10*e^2 + 6*a^2*c^2*d^8*e^4 - 4*a^3*c*d^6*e^6 + a^4*d^4*e^8 + (c^4*d^8*e^4 - 4*a*c^3*d^6*e^6 +
6*a^2*c^2*d^4*e^8 - 4*a^3*c*d^2*e^10 + a^4*e^12)*x^4 + 4*(c^4*d^9*e^3 - 4*a*c^3*d^7*e^5 + 6*a^2*c^2*d^5*e^7 -
4*a^3*c*d^3*e^9 + a^4*d*e^11)*x^3 + 6*(c^4*d^10*e^2 - 4*a*c^3*d^8*e^4 + 6*a^2*c^2*d^6*e^6 - 4*a^3*c*d^4*e^8 +
a^4*d^2*e^10)*x^2 + 4*(c^4*d^11*e - 4*a*c^3*d^9*e^3 + 6*a^2*c^2*d^7*e^5 - 4*a^3*c*d^5*e^7 + a^4*d^3*e^9)*x), -
2/105*(105*(c^3*d^3*e^4*x^4 + 4*c^3*d^4*e^3*x^3 + 6*c^3*d^5*e^2*x^2 + 4*c^3*d^6*e*x + c^3*d^7)*sqrt(-c*d/(c*d^
2 - a*e^2))*arctan(-(c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt(-c*d/(c*d^2 - a*e^2))/(c*d*e*x + c*d^2)) - (105*c^3*d^3
*e^3*x^3 + 176*c^3*d^6 - 122*a*c^2*d^4*e^2 + 66*a^2*c*d^2*e^4 - 15*a^3*e^6 + 35*(10*c^3*d^4*e^2 - a*c^2*d^2*e^
4)*x^2 + 7*(58*c^3*d^5*e - 16*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*x)*sqrt(e*x + d))/(c^4*d^12 - 4*a*c^3*d^10*e^2 +
6*a^2*c^2*d^8*e^4 - 4*a^3*c*d^6*e^6 + a^4*d^4*e^8 + (c^4*d^8*e^4 - 4*a*c^3*d^6*e^6 + 6*a^2*c^2*d^4*e^8 - 4*a^3
*c*d^2*e^10 + a^4*e^12)*x^4 + 4*(c^4*d^9*e^3 - 4*a*c^3*d^7*e^5 + 6*a^2*c^2*d^5*e^7 - 4*a^3*c*d^3*e^9 + a^4*d*e
^11)*x^3 + 6*(c^4*d^10*e^2 - 4*a*c^3*d^8*e^4 + 6*a^2*c^2*d^6*e^6 - 4*a^3*c*d^4*e^8 + a^4*d^2*e^10)*x^2 + 4*(c^
4*d^11*e - 4*a*c^3*d^9*e^3 + 6*a^2*c^2*d^7*e^5 - 4*a^3*c*d^5*e^7 + a^4*d^3*e^9)*x)]

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giac [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(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]

Timed out

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maple [A]  time = 0.06, size = 175, normalized size = 0.94 \begin {gather*} \frac {2 c^{4} d^{4} \arctan \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) c d}}\right )}{\left (a \,e^{2}-c \,d^{2}\right )^{4} \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) c d}}+\frac {2 c^{3} d^{3}}{\left (a \,e^{2}-c \,d^{2}\right )^{4} \sqrt {e x +d}}-\frac {2 c^{2} d^{2}}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 c d}{5 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {5}{2}}}-\frac {2}{7 \left (a \,e^{2}-c \,d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*c^4*d^4/(a*e^2-c*d^2)^4/((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)-2/7/(a
*e^2-c*d^2)/(e*x+d)^(7/2)-2/3*c^2*d^2/(a*e^2-c*d^2)^3/(e*x+d)^(3/2)+2/5*c*d/(a*e^2-c*d^2)^2/(e*x+d)^(5/2)+2*c^
3*d^3/(a*e^2-c*d^2)^4/(e*x+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(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(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 = 0.74, 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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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(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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