3.2064 \(\int \frac {1}{(d+e x)^{7/2} \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\)

Optimal. Leaf size=269 \[ \frac {5 c^3 d^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{8 \sqrt {e} \left (c d^2-a e^2\right )^{7/2}}+\frac {5 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3}+\frac {5 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 (d+e x)^{7/2} \left (c d^2-a e^2\right )} \]

[Out]

5/8*c^3*d^3*arctan(e^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e^2+c*d^2)^(1/2)/(e*x+d)^(1/2))/(-a*e^2
+c*d^2)^(7/2)/e^(1/2)+1/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e^2+c*d^2)/(e*x+d)^(7/2)+5/12*c*d*(a*d*e
+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e^2+c*d^2)^2/(e*x+d)^(5/2)+5/8*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)
^(1/2)/(-a*e^2+c*d^2)^3/(e*x+d)^(3/2)

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Rubi [A]  time = 0.17, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {672, 660, 205} \[ \frac {5 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3}+\frac {5 c^3 d^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{8 \sqrt {e} \left (c d^2-a e^2\right )^{7/2}}+\frac {5 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 (d+e x)^{7/2} \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(3*(c*d^2 - a*e^2)*(d + e*x)^(7/2)) + (5*c*d*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2])/(12*(c*d^2 - a*e^2)^2*(d + e*x)^(5/2)) + (5*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c
*d*e*x^2])/(8*(c*d^2 - a*e^2)^3*(d + e*x)^(3/2)) + (5*c^3*d^3*ArcTan[(Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x +
c*d*e*x^2])/(Sqrt[c*d^2 - a*e^2]*Sqrt[d + e*x])])/(8*Sqrt[e]*(c*d^2 - a*e^2)^(7/2))

Rule 205

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

Rule 660

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e, Subst[Int[1/(
2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^
2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 672

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((m + p + 1)*(2*c*d - b*e)), x] + Dist[(c*(m + 2*p + 2))/((m + p + 1)*(2*c*d - b*e)), I
nt[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ
[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, 0] && NeQ[m + p + 1, 0] && IntegerQ[2*p]

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}+\frac {(5 c d) \int \frac {1}{(d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{6 \left (c d^2-a e^2\right )}\\ &=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}+\frac {5 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}+\frac {\left (5 c^2 d^2\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 \left (c d^2-a e^2\right )^2}\\ &=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}+\frac {5 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}+\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}+\frac {\left (5 c^3 d^3\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{16 \left (c d^2-a e^2\right )^3}\\ &=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}+\frac {5 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}+\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}+\frac {\left (5 c^3 d^3 e\right ) \operatorname {Subst}\left (\int \frac {1}{2 c d^2 e-e \left (c d^2+a e^2\right )+e^2 x^2} \, dx,x,\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}}\right )}{8 \left (c d^2-a e^2\right )^3}\\ &=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}+\frac {5 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}+\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}+\frac {5 c^3 d^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{8 \sqrt {e} \left (c d^2-a e^2\right )^{7/2}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 81, normalized size = 0.30 \[ \frac {2 c^3 d^3 \sqrt {(d+e x) (a e+c d x)} \, _2F_1\left (\frac {1}{2},4;\frac {3}{2};\frac {e (a e+c d x)}{a e^2-c d^2}\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*c^3*d^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*Hypergeometric2F1[1/2, 4, 3/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])
/((c*d^2 - a*e^2)^4*Sqrt[d + e*x])

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fricas [B]  time = 1.08, size = 1242, normalized size = 4.62 \[ \left [-\frac {15 \, {\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 {-c d^{2} e + a e^{3}} \log \left (-\frac {c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2} - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {-c d^{2} e + a e^{3}} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, {\left (33 \, c^{3} d^{6} e - 59 \, a c^{2} d^{4} e^{3} + 34 \, a^{2} c d^{2} e^{5} - 8 \, a^{3} e^{7} + 15 \, {\left (c^{3} d^{4} e^{3} - a c^{2} d^{2} e^{5}\right )} x^{2} + 10 \, {\left (4 \, c^{3} d^{5} e^{2} - 5 \, a c^{2} d^{3} e^{4} + a^{2} c d e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{48 \, {\left (c^{4} d^{12} e - 4 \, a c^{3} d^{10} e^{3} + 6 \, a^{2} c^{2} d^{8} e^{5} - 4 \, a^{3} c d^{6} e^{7} + a^{4} d^{4} e^{9} + {\left (c^{4} d^{8} e^{5} - 4 \, a c^{3} d^{6} e^{7} + 6 \, a^{2} c^{2} d^{4} e^{9} - 4 \, a^{3} c d^{2} e^{11} + a^{4} e^{13}\right )} x^{4} + 4 \, {\left (c^{4} d^{9} e^{4} - 4 \, a c^{3} d^{7} e^{6} + 6 \, a^{2} c^{2} d^{5} e^{8} - 4 \, a^{3} c d^{3} e^{10} + a^{4} d e^{12}\right )} x^{3} + 6 \, {\left (c^{4} d^{10} e^{3} - 4 \, a c^{3} d^{8} e^{5} + 6 \, a^{2} c^{2} d^{6} e^{7} - 4 \, a^{3} c d^{4} e^{9} + a^{4} d^{2} e^{11}\right )} x^{2} + 4 \, {\left (c^{4} d^{11} e^{2} - 4 \, a c^{3} d^{9} e^{4} + 6 \, a^{2} c^{2} d^{7} e^{6} - 4 \, a^{3} c d^{5} e^{8} + a^{4} d^{3} e^{10}\right )} x\right )}}, -\frac {15 \, {\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 {c d^{2} e - a e^{3}} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {c d^{2} e - a e^{3}} \sqrt {e x + d}}{c d e^{2} x^{2} + a d e^{2} + {\left (c d^{2} e + a e^{3}\right )} x}\right ) - {\left (33 \, c^{3} d^{6} e - 59 \, a c^{2} d^{4} e^{3} + 34 \, a^{2} c d^{2} e^{5} - 8 \, a^{3} e^{7} + 15 \, {\left (c^{3} d^{4} e^{3} - a c^{2} d^{2} e^{5}\right )} x^{2} + 10 \, {\left (4 \, c^{3} d^{5} e^{2} - 5 \, a c^{2} d^{3} e^{4} + a^{2} c d e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{24 \, {\left (c^{4} d^{12} e - 4 \, a c^{3} d^{10} e^{3} + 6 \, a^{2} c^{2} d^{8} e^{5} - 4 \, a^{3} c d^{6} e^{7} + a^{4} d^{4} e^{9} + {\left (c^{4} d^{8} e^{5} - 4 \, a c^{3} d^{6} e^{7} + 6 \, a^{2} c^{2} d^{4} e^{9} - 4 \, a^{3} c d^{2} e^{11} + a^{4} e^{13}\right )} x^{4} + 4 \, {\left (c^{4} d^{9} e^{4} - 4 \, a c^{3} d^{7} e^{6} + 6 \, a^{2} c^{2} d^{5} e^{8} - 4 \, a^{3} c d^{3} e^{10} + a^{4} d e^{12}\right )} x^{3} + 6 \, {\left (c^{4} d^{10} e^{3} - 4 \, a c^{3} d^{8} e^{5} + 6 \, a^{2} c^{2} d^{6} e^{7} - 4 \, a^{3} c d^{4} e^{9} + a^{4} d^{2} e^{11}\right )} x^{2} + 4 \, {\left (c^{4} d^{11} e^{2} - 4 \, a c^{3} d^{9} e^{4} + 6 \, a^{2} c^{2} d^{7} e^{6} - 4 \, a^{3} c d^{5} e^{8} + a^{4} d^{3} e^{10}\right )} x\right )}}\right ] \]

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)^(1/2),x, algorithm="fricas")

[Out]

[-1/48*(15*(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^2*e +
a*e^3)*log(-(c*d*e^2*x^2 + 2*a*e^3*x - c*d^3 + 2*a*d*e^2 - 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt
(-c*d^2*e + a*e^3)*sqrt(e*x + d))/(e^2*x^2 + 2*d*e*x + d^2)) - 2*(33*c^3*d^6*e - 59*a*c^2*d^4*e^3 + 34*a^2*c*d
^2*e^5 - 8*a^3*e^7 + 15*(c^3*d^4*e^3 - a*c^2*d^2*e^5)*x^2 + 10*(4*c^3*d^5*e^2 - 5*a*c^2*d^3*e^4 + a^2*c*d*e^6)
*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^4*d^12*e - 4*a*c^3*d^10*e^3 + 6*a^2*c^2*d^8*
e^5 - 4*a^3*c*d^6*e^7 + a^4*d^4*e^9 + (c^4*d^8*e^5 - 4*a*c^3*d^6*e^7 + 6*a^2*c^2*d^4*e^9 - 4*a^3*c*d^2*e^11 +
a^4*e^13)*x^4 + 4*(c^4*d^9*e^4 - 4*a*c^3*d^7*e^6 + 6*a^2*c^2*d^5*e^8 - 4*a^3*c*d^3*e^10 + a^4*d*e^12)*x^3 + 6*
(c^4*d^10*e^3 - 4*a*c^3*d^8*e^5 + 6*a^2*c^2*d^6*e^7 - 4*a^3*c*d^4*e^9 + a^4*d^2*e^11)*x^2 + 4*(c^4*d^11*e^2 -
4*a*c^3*d^9*e^4 + 6*a^2*c^2*d^7*e^6 - 4*a^3*c*d^5*e^8 + a^4*d^3*e^10)*x), -1/24*(15*(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^2*e - a*e^3)*arctan(sqrt(c*d*e*x^2 + a*d*e
+ (c*d^2 + a*e^2)*x)*sqrt(c*d^2*e - a*e^3)*sqrt(e*x + d)/(c*d*e^2*x^2 + a*d*e^2 + (c*d^2*e + a*e^3)*x)) - (33*
c^3*d^6*e - 59*a*c^2*d^4*e^3 + 34*a^2*c*d^2*e^5 - 8*a^3*e^7 + 15*(c^3*d^4*e^3 - a*c^2*d^2*e^5)*x^2 + 10*(4*c^3
*d^5*e^2 - 5*a*c^2*d^3*e^4 + a^2*c*d*e^6)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^4*d
^12*e - 4*a*c^3*d^10*e^3 + 6*a^2*c^2*d^8*e^5 - 4*a^3*c*d^6*e^7 + a^4*d^4*e^9 + (c^4*d^8*e^5 - 4*a*c^3*d^6*e^7
+ 6*a^2*c^2*d^4*e^9 - 4*a^3*c*d^2*e^11 + a^4*e^13)*x^4 + 4*(c^4*d^9*e^4 - 4*a*c^3*d^7*e^6 + 6*a^2*c^2*d^5*e^8
- 4*a^3*c*d^3*e^10 + a^4*d*e^12)*x^3 + 6*(c^4*d^10*e^3 - 4*a*c^3*d^8*e^5 + 6*a^2*c^2*d^6*e^7 - 4*a^3*c*d^4*e^9
+ a^4*d^2*e^11)*x^2 + 4*(c^4*d^11*e^2 - 4*a*c^3*d^9*e^4 + 6*a^2*c^2*d^7*e^6 - 4*a^3*c*d^5*e^8 + a^4*d^3*e^10)
*x)]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

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)^(1/2),x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 0.07, size = 454, normalized size = 1.69 \[ \frac {\sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, \left (15 c^{3} d^{3} e^{3} x^{3} \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )+45 c^{3} d^{4} e^{2} x^{2} \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )+45 c^{3} d^{5} e x \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )+15 c^{3} d^{6} \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )-15 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c^{2} d^{2} e^{2} x^{2}+10 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a c d \,e^{3} x -40 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c^{2} d^{3} e x -8 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a^{2} e^{4}+26 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a c \,d^{2} e^{2}-33 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, c^{2} d^{4}\right )}{24 \left (e x +d \right )^{\frac {7}{2}} \sqrt {c d x +a e}\, \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}} \]

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)^(1/2),x)

[Out]

1/24*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(15*arctanh((c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2)*e)*x^3*c^3*
d^3*e^3+45*arctanh((c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2)*e)*x^2*c^3*d^4*e^2+45*arctanh((c*d*x+a*e)^(1/2)/(
(a*e^2-c*d^2)*e)^(1/2)*e)*x*c^3*d^5*e+15*arctanh((c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2)*e)*c^3*d^6-15*(c*d*
x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)*c^2*d^2*e^2*x^2+10*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)*a*c*d*e^3*x-
40*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)*c^2*d^3*e*x-8*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a^2*e^4+2
6*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a*c*d^2*e^2-33*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*c^2*d^4)/
(e*x+d)^(7/2)/(c*d*x+a*e)^(1/2)/(a*e^2-c*d^2)^3/((a*e^2-c*d^2)*e)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (e x + d\right )}^{\frac {7}{2}}}\,{d x} \]

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)^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (d+e\,x\right )}^{7/2}\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \]

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)^(1/2)),x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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)**(1/2),x)

[Out]

Timed out

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