3.18.39 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{15/2}} \, dx\)

Optimal. Leaf size=301 \[ \frac {5 c^4 d^4 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{64 e^{7/2} \left (c d^2-a e^2\right )^{3/2}}+\frac {5 c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 e^3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}-\frac {5 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{32 e^3 (d+e x)^{5/2}}-\frac {5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 e^2 (d+e x)^{9/2}}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4 e (d+e x)^{13/2}} \]

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Rubi [A]  time = 0.23, antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {662, 672, 660, 205} \begin {gather*} \frac {5 c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 e^3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}-\frac {5 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{32 e^3 (d+e x)^{5/2}}+\frac {5 c^4 d^4 \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 )}{64 e^{7/2} \left (c d^2-a e^2\right )^{3/2}}-\frac {5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 e^2 (d+e x)^{9/2}}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4 e (d+e x)^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(15/2),x]

[Out]

(-5*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(32*e^3*(d + e*x)^(5/2)) + (5*c^3*d^3*Sqrt[a*d*e + (c
*d^2 + a*e^2)*x + c*d*e*x^2])/(64*e^3*(c*d^2 - a*e^2)*(d + e*x)^(3/2)) - (5*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c
*d*e*x^2)^(3/2))/(24*e^2*(d + e*x)^(9/2)) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(4*e*(d + e*x)^(13/2
)) + (5*c^4*d^4*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
])])/(64*e^(7/2)*(c*d^2 - a*e^2)^(3/2))

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 662

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

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 {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4 e (d+e x)^{13/2}}+\frac {(5 c d) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{11/2}} \, dx}{8 e}\\ &=-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 e^2 (d+e x)^{9/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4 e (d+e x)^{13/2}}+\frac {\left (5 c^2 d^2\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{7/2}} \, dx}{16 e^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}}{32 e^3 (d+e x)^{5/2}}-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 e^2 (d+e x)^{9/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4 e (d+e x)^{13/2}}+\frac {\left (5 c^3 d^3\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}{64 e^3}\\ &=-\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 e^3 (d+e x)^{5/2}}+\frac {5 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 e^3 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 e^2 (d+e x)^{9/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4 e (d+e x)^{13/2}}+\frac {\left (5 c^4 d^4\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}{128 e^3 \left (c d^2-a e^2\right )}\\ &=-\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 e^3 (d+e x)^{5/2}}+\frac {5 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 e^3 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 e^2 (d+e x)^{9/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4 e (d+e x)^{13/2}}+\frac {\left (5 c^4 d^4\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 )}{64 e^2 \left (c d^2-a e^2\right )}\\ &=-\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 e^3 (d+e x)^{5/2}}+\frac {5 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 e^3 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 e^2 (d+e x)^{9/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4 e (d+e x)^{13/2}}+\frac {5 c^4 d^4 \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 )}{64 e^{7/2} \left (c d^2-a e^2\right )^{3/2}}\\ \end {align*}

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Mathematica [C]  time = 0.07, size = 83, normalized size = 0.28 \begin {gather*} \frac {2 c^4 d^4 ((d+e x) (a e+c d x))^{7/2} \, _2F_1\left (\frac {7}{2},5;\frac {9}{2};\frac {e (a e+c d x)}{a e^2-c d^2}\right )}{7 (d+e x)^{7/2} \left (c d^2-a e^2\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(15/2),x]

[Out]

(2*c^4*d^4*((a*e + c*d*x)*(d + e*x))^(7/2)*Hypergeometric2F1[7/2, 5, 9/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)
])/(7*(c*d^2 - a*e^2)^5*(d + e*x)^(7/2))

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IntegrateAlgebraic [F]  time = 180.01, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(15/2),x]

[Out]

$Aborted

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fricas [B]  time = 0.46, size = 1205, normalized size = 4.00 \begin {gather*} \left [-\frac {15 \, {\left (c^{4} d^{4} e^{5} x^{5} + 5 \, c^{4} d^{5} e^{4} x^{4} + 10 \, c^{4} d^{6} e^{3} x^{3} + 10 \, c^{4} d^{7} e^{2} x^{2} + 5 \, c^{4} d^{8} e x + c^{4} d^{9}\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 (15 \, c^{4} d^{8} e - 5 \, a c^{3} d^{6} e^{3} - 2 \, a^{2} c^{2} d^{4} e^{5} - 56 \, a^{3} c d^{2} e^{7} + 48 \, a^{4} e^{9} - 15 \, {\left (c^{4} d^{5} e^{4} - a c^{3} d^{3} e^{6}\right )} x^{3} + {\left (73 \, c^{4} d^{6} e^{3} - 191 \, a c^{3} d^{4} e^{5} + 118 \, a^{2} c^{2} d^{2} e^{7}\right )} x^{2} + {\left (55 \, c^{4} d^{7} e^{2} - 19 \, a c^{3} d^{5} e^{4} - 172 \, a^{2} c^{2} d^{3} e^{6} + 136 \, a^{3} c d e^{8}\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}}{384 \, {\left (c^{2} d^{9} e^{4} - 2 \, a c d^{7} e^{6} + a^{2} d^{5} e^{8} + {\left (c^{2} d^{4} e^{9} - 2 \, a c d^{2} e^{11} + a^{2} e^{13}\right )} x^{5} + 5 \, {\left (c^{2} d^{5} e^{8} - 2 \, a c d^{3} e^{10} + a^{2} d e^{12}\right )} x^{4} + 10 \, {\left (c^{2} d^{6} e^{7} - 2 \, a c d^{4} e^{9} + a^{2} d^{2} e^{11}\right )} x^{3} + 10 \, {\left (c^{2} d^{7} e^{6} - 2 \, a c d^{5} e^{8} + a^{2} d^{3} e^{10}\right )} x^{2} + 5 \, {\left (c^{2} d^{8} e^{5} - 2 \, a c d^{6} e^{7} + a^{2} d^{4} e^{9}\right )} x\right )}}, -\frac {15 \, {\left (c^{4} d^{4} e^{5} x^{5} + 5 \, c^{4} d^{5} e^{4} x^{4} + 10 \, c^{4} d^{6} e^{3} x^{3} + 10 \, c^{4} d^{7} e^{2} x^{2} + 5 \, c^{4} d^{8} e x + c^{4} d^{9}\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 (15 \, c^{4} d^{8} e - 5 \, a c^{3} d^{6} e^{3} - 2 \, a^{2} c^{2} d^{4} e^{5} - 56 \, a^{3} c d^{2} e^{7} + 48 \, a^{4} e^{9} - 15 \, {\left (c^{4} d^{5} e^{4} - a c^{3} d^{3} e^{6}\right )} x^{3} + {\left (73 \, c^{4} d^{6} e^{3} - 191 \, a c^{3} d^{4} e^{5} + 118 \, a^{2} c^{2} d^{2} e^{7}\right )} x^{2} + {\left (55 \, c^{4} d^{7} e^{2} - 19 \, a c^{3} d^{5} e^{4} - 172 \, a^{2} c^{2} d^{3} e^{6} + 136 \, a^{3} c d e^{8}\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}}{192 \, {\left (c^{2} d^{9} e^{4} - 2 \, a c d^{7} e^{6} + a^{2} d^{5} e^{8} + {\left (c^{2} d^{4} e^{9} - 2 \, a c d^{2} e^{11} + a^{2} e^{13}\right )} x^{5} + 5 \, {\left (c^{2} d^{5} e^{8} - 2 \, a c d^{3} e^{10} + a^{2} d e^{12}\right )} x^{4} + 10 \, {\left (c^{2} d^{6} e^{7} - 2 \, a c d^{4} e^{9} + a^{2} d^{2} e^{11}\right )} x^{3} + 10 \, {\left (c^{2} d^{7} e^{6} - 2 \, a c d^{5} e^{8} + a^{2} d^{3} e^{10}\right )} x^{2} + 5 \, {\left (c^{2} d^{8} e^{5} - 2 \, a c d^{6} e^{7} + a^{2} d^{4} e^{9}\right )} x\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(15/2),x, algorithm="fricas")

[Out]

[-1/384*(15*(c^4*d^4*e^5*x^5 + 5*c^4*d^5*e^4*x^4 + 10*c^4*d^6*e^3*x^3 + 10*c^4*d^7*e^2*x^2 + 5*c^4*d^8*e*x + c
^4*d^9)*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*(15*c^4*d^8*e - 5*a*c^
3*d^6*e^3 - 2*a^2*c^2*d^4*e^5 - 56*a^3*c*d^2*e^7 + 48*a^4*e^9 - 15*(c^4*d^5*e^4 - a*c^3*d^3*e^6)*x^3 + (73*c^4
*d^6*e^3 - 191*a*c^3*d^4*e^5 + 118*a^2*c^2*d^2*e^7)*x^2 + (55*c^4*d^7*e^2 - 19*a*c^3*d^5*e^4 - 172*a^2*c^2*d^3
*e^6 + 136*a^3*c*d*e^8)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^2*d^9*e^4 - 2*a*c*d^7
*e^6 + a^2*d^5*e^8 + (c^2*d^4*e^9 - 2*a*c*d^2*e^11 + a^2*e^13)*x^5 + 5*(c^2*d^5*e^8 - 2*a*c*d^3*e^10 + a^2*d*e
^12)*x^4 + 10*(c^2*d^6*e^7 - 2*a*c*d^4*e^9 + a^2*d^2*e^11)*x^3 + 10*(c^2*d^7*e^6 - 2*a*c*d^5*e^8 + a^2*d^3*e^1
0)*x^2 + 5*(c^2*d^8*e^5 - 2*a*c*d^6*e^7 + a^2*d^4*e^9)*x), -1/192*(15*(c^4*d^4*e^5*x^5 + 5*c^4*d^5*e^4*x^4 + 1
0*c^4*d^6*e^3*x^3 + 10*c^4*d^7*e^2*x^2 + 5*c^4*d^8*e*x + c^4*d^9)*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)
) + (15*c^4*d^8*e - 5*a*c^3*d^6*e^3 - 2*a^2*c^2*d^4*e^5 - 56*a^3*c*d^2*e^7 + 48*a^4*e^9 - 15*(c^4*d^5*e^4 - a*
c^3*d^3*e^6)*x^3 + (73*c^4*d^6*e^3 - 191*a*c^3*d^4*e^5 + 118*a^2*c^2*d^2*e^7)*x^2 + (55*c^4*d^7*e^2 - 19*a*c^3
*d^5*e^4 - 172*a^2*c^2*d^3*e^6 + 136*a^3*c*d*e^8)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)
)/(c^2*d^9*e^4 - 2*a*c*d^7*e^6 + a^2*d^5*e^8 + (c^2*d^4*e^9 - 2*a*c*d^2*e^11 + a^2*e^13)*x^5 + 5*(c^2*d^5*e^8
- 2*a*c*d^3*e^10 + a^2*d*e^12)*x^4 + 10*(c^2*d^6*e^7 - 2*a*c*d^4*e^9 + a^2*d^2*e^11)*x^3 + 10*(c^2*d^7*e^6 - 2
*a*c*d^5*e^8 + a^2*d^3*e^10)*x^2 + 5*(c^2*d^8*e^5 - 2*a*c*d^6*e^7 + a^2*d^4*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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(15/2),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.08, size = 662, normalized size = 2.20 \begin {gather*} \frac {\sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, \left (15 c^{4} d^{4} e^{4} x^{4} \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )+60 c^{4} d^{5} 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 )+90 c^{4} d^{6} 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 )+60 c^{4} d^{7} 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^{4} d^{8} \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^{3} d^{3} e^{3} x^{3}-118 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a \,c^{2} d^{2} e^{4} x^{2}+73 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c^{3} d^{4} e^{2} x^{2}-136 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a^{2} c d \,e^{5} x +36 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a \,c^{2} d^{3} e^{3} x +55 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c^{3} d^{5} e x -48 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a^{3} e^{6}+8 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a^{2} c \,d^{2} e^{4}+10 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a \,c^{2} d^{4} e^{2}+15 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, c^{3} d^{6}\right )}{192 \left (e x +d \right )^{\frac {9}{2}} \sqrt {c d x +a e}\, \left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)/(e*x+d)^(15/2),x)

[Out]

1/192*(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^4*c^4
*d^4*e^4+60*arctanh((c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2)*e)*x^3*c^4*d^5*e^3+90*arctanh((c*d*x+a*e)^(1/2)/
((a*e^2-c*d^2)*e)^(1/2)*e)*x^2*c^4*d^6*e^2+60*arctanh((c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2)*e)*x*c^4*d^7*e
-15*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)*c^3*d^3*e^3*x^3+15*arctanh((c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(
1/2)*e)*c^4*d^8-118*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)*a*c^2*d^2*e^4*x^2+73*(c*d*x+a*e)^(1/2)*((a*e^2-c
*d^2)*e)^(1/2)*c^3*d^4*e^2*x^2-136*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)*a^2*c*d*e^5*x+36*(c*d*x+a*e)^(1/2
)*((a*e^2-c*d^2)*e)^(1/2)*a*c^2*d^3*e^3*x+55*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)*c^3*d^5*e*x-48*((a*e^2-
c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a^3*e^6+8*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a^2*c*d^2*e^4+10*((a*e^2
-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a*c^2*d^4*e^2+15*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*c^3*d^6)/(e*x+d)
^(9/2)/(c*d*x+a*e)^(1/2)/e^3/(a*e^2-c*d^2)/((a*e^2-c*d^2)*e)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {15}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(15/2),x, algorithm="maxima")

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/(e*x + d)^(15/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{{\left (d+e\,x\right )}^{15/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(d + e*x)^(15/2),x)

[Out]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(d + e*x)^(15/2), x)

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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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(15/2),x)

[Out]

Timed out

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