3.21.56 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{17/2}} \, dx\) [2056]

Optimal. Leaf size=366 \[ -\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 e^3 (d+e x)^{7/2}}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 e^3 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {3 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 e^3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac {3 c^5 d^5 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{128 e^{7/2} \left (c d^2-a e^2\right )^{5/2}} \]

[Out]

-1/8*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/e^2/(e*x+d)^(11/2)-1/5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2
)/e/(e*x+d)^(15/2)+3/128*c^5*d^5*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))/e^(7/2)/(-a*e^2+c*d^2)^(5/2)-1/16*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e^3/(e*x+d)^(7
/2)+1/64*c^3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e^3/(-a*e^2+c*d^2)/(e*x+d)^(5/2)+3/128*c^4*d^4*(a*d*e
+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e^3/(-a*e^2+c*d^2)^2/(e*x+d)^(3/2)

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Rubi [A]
time = 0.19, antiderivative size = 366, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {676, 686, 674, 211} \begin {gather*} \frac {3 c^5 d^5 \text {ArcTan}\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 )}{128 e^{7/2} \left (c d^2-a e^2\right )^{5/2}}+\frac {3 c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 e^3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac {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)^{5/2} \left (c d^2-a e^2\right )}-\frac {c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{16 e^3 (d+e x)^{7/2}}-\frac {c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}} \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)^(17/2),x]

[Out]

-1/16*(c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(e^3*(d + e*x)^(7/2)) + (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)^(5/2)) + (3*c^4*d^4*Sqrt[a*d*e + (c*d^2 + a*e^2)
*x + c*d*e*x^2])/(128*e^3*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2)) - (c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3
/2))/(8*e^2*(d + e*x)^(11/2)) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(5*e*(d + e*x)^(15/2)) + (3*c^5*
d^5*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])])/(128*e^
(7/2)*(c*d^2 - a*e^2)^(5/2))

Rule 211

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

Rule 674

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 676

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 686

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))),
Int[(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] && E
qQ[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)^{17/2}} \, dx &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac {(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)^{13/2}} \, dx}{2 e}\\ &=-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac {\left (3 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)^{9/2}} \, dx}{16 e^2}\\ &=-\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 e^3 (d+e x)^{7/2}}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac {\left (c^3 d^3\right ) \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}{32 e^3}\\ &=-\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 e^3 (d+e x)^{7/2}}+\frac {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)^{5/2}}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac {\left (3 c^4 d^4\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}{128 e^3 \left (c d^2-a e^2\right )}\\ &=-\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 e^3 (d+e x)^{7/2}}+\frac {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)^{5/2}}+\frac {3 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 e^3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac {\left (3 c^5 d^5\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}{256 e^3 \left (c d^2-a e^2\right )^2}\\ &=-\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 e^3 (d+e x)^{7/2}}+\frac {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)^{5/2}}+\frac {3 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 e^3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac {\left (3 c^5 d^5\right ) \text {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 )}{128 e^2 \left (c d^2-a e^2\right )^2}\\ &=-\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 e^3 (d+e x)^{7/2}}+\frac {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)^{5/2}}+\frac {3 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 e^3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac {3 c^5 d^5 \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 )}{128 e^{7/2} \left (c d^2-a e^2\right )^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 1.51, size = 300, normalized size = 0.82 \begin {gather*} \frac {c^5 d^5 ((a e+c d x) (d+e x))^{5/2} \left (-\frac {\sqrt {e} \left (128 a^4 e^8+16 a^3 c d e^6 (-11 d+21 e x)+8 a^2 c^2 d^2 e^4 \left (d^2-64 d e x+31 e^2 x^2\right )+2 a c^3 d^3 e^2 \left (5 d^3+23 d^2 e x-233 d e^2 x^2+5 e^3 x^3\right )+c^4 d^4 \left (15 d^4+70 d^3 e x+128 d^2 e^2 x^2-70 d e^3 x^3-15 e^4 x^4\right )\right )}{c^5 d^5 \left (c d^2-a e^2\right )^2 (a e+c d x)^2 (d+e x)^5}+\frac {15 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2} (a e+c d x)^{5/2}}\right )}{640 e^{7/2} (d+e x)^{5/2}} \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)^(17/2),x]

[Out]

(c^5*d^5*((a*e + c*d*x)*(d + e*x))^(5/2)*(-((Sqrt[e]*(128*a^4*e^8 + 16*a^3*c*d*e^6*(-11*d + 21*e*x) + 8*a^2*c^
2*d^2*e^4*(d^2 - 64*d*e*x + 31*e^2*x^2) + 2*a*c^3*d^3*e^2*(5*d^3 + 23*d^2*e*x - 233*d*e^2*x^2 + 5*e^3*x^3) + c
^4*d^4*(15*d^4 + 70*d^3*e*x + 128*d^2*e^2*x^2 - 70*d*e^3*x^3 - 15*e^4*x^4)))/(c^5*d^5*(c*d^2 - a*e^2)^2*(a*e +
c*d*x)^2*(d + e*x)^5)) + (15*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[c*d^2 - a*e^2]])/((c*d^2 - a*e^2)^(5/2)*
(a*e + c*d*x)^(5/2))))/(640*e^(7/2)*(d + e*x)^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(899\) vs. \(2(322)=644\).
time = 0.76, size = 900, normalized size = 2.46

method	result	size
default	\(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (15 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{5} d^{5} e^{5} x^{5}+75 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{5} d^{6} e^{4} x^{4}+150 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{5} d^{7} e^{3} x^{3}+150 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{5} d^{8} e^{2} x^{2}-15 c^{4} d^{4} e^{4} x^{4} \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+75 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{5} d^{9} e x +10 a \,c^{3} d^{3} e^{5} x^{3} \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}-70 c^{4} d^{5} e^{3} x^{3} \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+15 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{5} d^{10}+248 a^{2} c^{2} d^{2} e^{6} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}-466 a \,c^{3} d^{4} e^{4} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+128 c^{4} d^{6} e^{2} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+336 a^{3} c d \,e^{7} x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}-512 a^{2} c^{2} d^{3} e^{5} x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+46 a \,c^{3} d^{5} e^{3} x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+70 c^{4} d^{7} e x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+128 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a^{4} e^{8}-176 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a^{3} c \,d^{2} e^{6}+8 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a^{2} c^{2} d^{4} e^{4}+10 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a \,c^{3} d^{6} e^{2}+15 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, c^{4} d^{8}\right )}{640 \left (e x +d \right )^{\frac {11}{2}} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, e^{3} \left (e^{2} a -c \,d^{2}\right )^{2} \sqrt {c d x +a e}}\)	\(900\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(17/2),x,method=_RETURNVERBOSE)

[Out]

-1/640*((c*d*x+a*e)*(e*x+d))^(1/2)*(15*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^5*d^5*e^5*x^5+75
*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^5*d^6*e^4*x^4+150*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-
c*d^2)*e)^(1/2))*c^5*d^7*e^3*x^3+150*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^5*d^8*e^2*x^2-15*c
^4*d^4*e^4*x^4*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+75*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2
))*c^5*d^9*e*x+10*a*c^3*d^3*e^5*x^3*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)-70*c^4*d^5*e^3*x^3*(c*d*x+a*e)^(
1/2)*((a*e^2-c*d^2)*e)^(1/2)+15*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^5*d^10+248*a^2*c^2*d^2*
e^6*x^2*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)-466*a*c^3*d^4*e^4*x^2*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1
/2)+128*c^4*d^6*e^2*x^2*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+336*a^3*c*d*e^7*x*(c*d*x+a*e)^(1/2)*((a*e^2-
c*d^2)*e)^(1/2)-512*a^2*c^2*d^3*e^5*x*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+46*a*c^3*d^5*e^3*x*(c*d*x+a*e)
^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+70*c^4*d^7*e*x*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+128*((a*e^2-c*d^2)*e)^
(1/2)*(c*d*x+a*e)^(1/2)*a^4*e^8-176*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a^3*c*d^2*e^6+8*((a*e^2-c*d^2)*e
)^(1/2)*(c*d*x+a*e)^(1/2)*a^2*c^2*d^4*e^4+10*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a*c^3*d^6*e^2+15*((a*e^
2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*c^4*d^8)/(e*x+d)^(11/2)/((a*e^2-c*d^2)*e)^(1/2)/e^3/(a*e^2-c*d^2)^2/(c*d*x
+a*e)^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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)^(17/2),x, algorithm="maxima")

[Out]

integrate((c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/(x*e + d)^(17/2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 822 vs. \(2 (323) = 646\).
time = 4.92, size = 1664, normalized size = 4.55 \begin {gather*} \left [-\frac {15 \, {\left (c^{5} d^{5} x^{6} e^{6} + 6 \, c^{5} d^{6} x^{5} e^{5} + 15 \, c^{5} d^{7} x^{4} e^{4} + 20 \, c^{5} d^{8} x^{3} e^{3} + 15 \, c^{5} d^{9} x^{2} e^{2} + 6 \, c^{5} d^{10} x e + c^{5} d^{11}\right )} \sqrt {-c d^{2} e + a e^{3}} \log \left (\frac {c d^{3} - 2 \, a x e^{3} - {\left (c d x^{2} + 2 \, a d\right )} e^{2} + 2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {-c d^{2} e + a e^{3}} \sqrt {x e + d}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) + 2 \, {\left (70 \, c^{5} d^{9} x e^{2} + 15 \, c^{5} d^{10} e - 336 \, a^{4} c d x e^{10} - 128 \, a^{5} e^{11} - 8 \, {\left (31 \, a^{3} c^{2} d^{2} x^{2} - 38 \, a^{4} c d^{2}\right )} e^{9} - 2 \, {\left (5 \, a^{2} c^{3} d^{3} x^{3} - 424 \, a^{3} c^{2} d^{3} x\right )} e^{8} + {\left (15 \, a c^{4} d^{4} x^{4} + 714 \, a^{2} c^{3} d^{4} x^{2} - 184 \, a^{3} c^{2} d^{4}\right )} e^{7} + 2 \, {\left (40 \, a c^{4} d^{5} x^{3} - 279 \, a^{2} c^{3} d^{5} x\right )} e^{6} - {\left (15 \, c^{5} d^{6} x^{4} + 594 \, a c^{4} d^{6} x^{2} + 2 \, a^{2} c^{3} d^{6}\right )} e^{5} - 2 \, {\left (35 \, c^{5} d^{7} x^{3} + 12 \, a c^{4} d^{7} x\right )} e^{4} + {\left (128 \, c^{5} d^{8} x^{2} - 5 \, a c^{4} d^{8}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{1280 \, {\left (6 \, c^{3} d^{11} x e^{5} + c^{3} d^{12} e^{4} - a^{3} x^{6} e^{16} - 6 \, a^{3} d x^{5} e^{15} + 3 \, {\left (a^{2} c d^{2} x^{6} - 5 \, a^{3} d^{2} x^{4}\right )} e^{14} + 2 \, {\left (9 \, a^{2} c d^{3} x^{5} - 10 \, a^{3} d^{3} x^{3}\right )} e^{13} - 3 \, {\left (a c^{2} d^{4} x^{6} - 15 \, a^{2} c d^{4} x^{4} + 5 \, a^{3} d^{4} x^{2}\right )} e^{12} - 6 \, {\left (3 \, a c^{2} d^{5} x^{5} - 10 \, a^{2} c d^{5} x^{3} + a^{3} d^{5} x\right )} e^{11} + {\left (c^{3} d^{6} x^{6} - 45 \, a c^{2} d^{6} x^{4} + 45 \, a^{2} c d^{6} x^{2} - a^{3} d^{6}\right )} e^{10} + 6 \, {\left (c^{3} d^{7} x^{5} - 10 \, a c^{2} d^{7} x^{3} + 3 \, a^{2} c d^{7} x\right )} e^{9} + 3 \, {\left (5 \, c^{3} d^{8} x^{4} - 15 \, a c^{2} d^{8} x^{2} + a^{2} c d^{8}\right )} e^{8} + 2 \, {\left (10 \, c^{3} d^{9} x^{3} - 9 \, a c^{2} d^{9} x\right )} e^{7} + 3 \, {\left (5 \, c^{3} d^{10} x^{2} - a c^{2} d^{10}\right )} e^{6}\right )}}, -\frac {15 \, {\left (c^{5} d^{5} x^{6} e^{6} + 6 \, c^{5} d^{6} x^{5} e^{5} + 15 \, c^{5} d^{7} x^{4} e^{4} + 20 \, c^{5} d^{8} x^{3} e^{3} + 15 \, c^{5} d^{9} x^{2} e^{2} + 6 \, c^{5} d^{10} x e + c^{5} d^{11}\right )} \sqrt {c d^{2} e - a e^{3}} \arctan \left (\frac {\sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {c d^{2} e - a e^{3}} \sqrt {x e + d}}{c d^{2} x e + a x e^{3} + {\left (c d x^{2} + a d\right )} e^{2}}\right ) + {\left (70 \, c^{5} d^{9} x e^{2} + 15 \, c^{5} d^{10} e - 336 \, a^{4} c d x e^{10} - 128 \, a^{5} e^{11} - 8 \, {\left (31 \, a^{3} c^{2} d^{2} x^{2} - 38 \, a^{4} c d^{2}\right )} e^{9} - 2 \, {\left (5 \, a^{2} c^{3} d^{3} x^{3} - 424 \, a^{3} c^{2} d^{3} x\right )} e^{8} + {\left (15 \, a c^{4} d^{4} x^{4} + 714 \, a^{2} c^{3} d^{4} x^{2} - 184 \, a^{3} c^{2} d^{4}\right )} e^{7} + 2 \, {\left (40 \, a c^{4} d^{5} x^{3} - 279 \, a^{2} c^{3} d^{5} x\right )} e^{6} - {\left (15 \, c^{5} d^{6} x^{4} + 594 \, a c^{4} d^{6} x^{2} + 2 \, a^{2} c^{3} d^{6}\right )} e^{5} - 2 \, {\left (35 \, c^{5} d^{7} x^{3} + 12 \, a c^{4} d^{7} x\right )} e^{4} + {\left (128 \, c^{5} d^{8} x^{2} - 5 \, a c^{4} d^{8}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{640 \, {\left (6 \, c^{3} d^{11} x e^{5} + c^{3} d^{12} e^{4} - a^{3} x^{6} e^{16} - 6 \, a^{3} d x^{5} e^{15} + 3 \, {\left (a^{2} c d^{2} x^{6} - 5 \, a^{3} d^{2} x^{4}\right )} e^{14} + 2 \, {\left (9 \, a^{2} c d^{3} x^{5} - 10 \, a^{3} d^{3} x^{3}\right )} e^{13} - 3 \, {\left (a c^{2} d^{4} x^{6} - 15 \, a^{2} c d^{4} x^{4} + 5 \, a^{3} d^{4} x^{2}\right )} e^{12} - 6 \, {\left (3 \, a c^{2} d^{5} x^{5} - 10 \, a^{2} c d^{5} x^{3} + a^{3} d^{5} x\right )} e^{11} + {\left (c^{3} d^{6} x^{6} - 45 \, a c^{2} d^{6} x^{4} + 45 \, a^{2} c d^{6} x^{2} - a^{3} d^{6}\right )} e^{10} + 6 \, {\left (c^{3} d^{7} x^{5} - 10 \, a c^{2} d^{7} x^{3} + 3 \, a^{2} c d^{7} x\right )} e^{9} + 3 \, {\left (5 \, c^{3} d^{8} x^{4} - 15 \, a c^{2} d^{8} x^{2} + a^{2} c d^{8}\right )} e^{8} + 2 \, {\left (10 \, c^{3} d^{9} x^{3} - 9 \, a c^{2} d^{9} x\right )} e^{7} + 3 \, {\left (5 \, c^{3} d^{10} x^{2} - a c^{2} d^{10}\right )} e^{6}\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)^(17/2),x, algorithm="fricas")

[Out]

[-1/1280*(15*(c^5*d^5*x^6*e^6 + 6*c^5*d^6*x^5*e^5 + 15*c^5*d^7*x^4*e^4 + 20*c^5*d^8*x^3*e^3 + 15*c^5*d^9*x^2*e
^2 + 6*c^5*d^10*x*e + c^5*d^11)*sqrt(-c*d^2*e + a*e^3)*log((c*d^3 - 2*a*x*e^3 - (c*d*x^2 + 2*a*d)*e^2 + 2*sqrt
(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(-c*d^2*e + a*e^3)*sqrt(x*e + d))/(x^2*e^2 + 2*d*x*e + d^2)) + 2*(
70*c^5*d^9*x*e^2 + 15*c^5*d^10*e - 336*a^4*c*d*x*e^10 - 128*a^5*e^11 - 8*(31*a^3*c^2*d^2*x^2 - 38*a^4*c*d^2)*e
^9 - 2*(5*a^2*c^3*d^3*x^3 - 424*a^3*c^2*d^3*x)*e^8 + (15*a*c^4*d^4*x^4 + 714*a^2*c^3*d^4*x^2 - 184*a^3*c^2*d^4
)*e^7 + 2*(40*a*c^4*d^5*x^3 - 279*a^2*c^3*d^5*x)*e^6 - (15*c^5*d^6*x^4 + 594*a*c^4*d^6*x^2 + 2*a^2*c^3*d^6)*e^
5 - 2*(35*c^5*d^7*x^3 + 12*a*c^4*d^7*x)*e^4 + (128*c^5*d^8*x^2 - 5*a*c^4*d^8)*e^3)*sqrt(c*d^2*x + a*x*e^2 + (c
*d*x^2 + a*d)*e)*sqrt(x*e + d))/(6*c^3*d^11*x*e^5 + c^3*d^12*e^4 - a^3*x^6*e^16 - 6*a^3*d*x^5*e^15 + 3*(a^2*c*
d^2*x^6 - 5*a^3*d^2*x^4)*e^14 + 2*(9*a^2*c*d^3*x^5 - 10*a^3*d^3*x^3)*e^13 - 3*(a*c^2*d^4*x^6 - 15*a^2*c*d^4*x^
4 + 5*a^3*d^4*x^2)*e^12 - 6*(3*a*c^2*d^5*x^5 - 10*a^2*c*d^5*x^3 + a^3*d^5*x)*e^11 + (c^3*d^6*x^6 - 45*a*c^2*d^
6*x^4 + 45*a^2*c*d^6*x^2 - a^3*d^6)*e^10 + 6*(c^3*d^7*x^5 - 10*a*c^2*d^7*x^3 + 3*a^2*c*d^7*x)*e^9 + 3*(5*c^3*d
^8*x^4 - 15*a*c^2*d^8*x^2 + a^2*c*d^8)*e^8 + 2*(10*c^3*d^9*x^3 - 9*a*c^2*d^9*x)*e^7 + 3*(5*c^3*d^10*x^2 - a*c^
2*d^10)*e^6), -1/640*(15*(c^5*d^5*x^6*e^6 + 6*c^5*d^6*x^5*e^5 + 15*c^5*d^7*x^4*e^4 + 20*c^5*d^8*x^3*e^3 + 15*c
^5*d^9*x^2*e^2 + 6*c^5*d^10*x*e + c^5*d^11)*sqrt(c*d^2*e - a*e^3)*arctan(sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a
*d)*e)*sqrt(c*d^2*e - a*e^3)*sqrt(x*e + d)/(c*d^2*x*e + a*x*e^3 + (c*d*x^2 + a*d)*e^2)) + (70*c^5*d^9*x*e^2 +
15*c^5*d^10*e - 336*a^4*c*d*x*e^10 - 128*a^5*e^11 - 8*(31*a^3*c^2*d^2*x^2 - 38*a^4*c*d^2)*e^9 - 2*(5*a^2*c^3*d
^3*x^3 - 424*a^3*c^2*d^3*x)*e^8 + (15*a*c^4*d^4*x^4 + 714*a^2*c^3*d^4*x^2 - 184*a^3*c^2*d^4)*e^7 + 2*(40*a*c^4
*d^5*x^3 - 279*a^2*c^3*d^5*x)*e^6 - (15*c^5*d^6*x^4 + 594*a*c^4*d^6*x^2 + 2*a^2*c^3*d^6)*e^5 - 2*(35*c^5*d^7*x
^3 + 12*a*c^4*d^7*x)*e^4 + (128*c^5*d^8*x^2 - 5*a*c^4*d^8)*e^3)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sq
rt(x*e + d))/(6*c^3*d^11*x*e^5 + c^3*d^12*e^4 - a^3*x^6*e^16 - 6*a^3*d*x^5*e^15 + 3*(a^2*c*d^2*x^6 - 5*a^3*d^2
*x^4)*e^14 + 2*(9*a^2*c*d^3*x^5 - 10*a^3*d^3*x^3)*e^13 - 3*(a*c^2*d^4*x^6 - 15*a^2*c*d^4*x^4 + 5*a^3*d^4*x^2)*
e^12 - 6*(3*a*c^2*d^5*x^5 - 10*a^2*c*d^5*x^3 + a^3*d^5*x)*e^11 + (c^3*d^6*x^6 - 45*a*c^2*d^6*x^4 + 45*a^2*c*d^
6*x^2 - a^3*d^6)*e^10 + 6*(c^3*d^7*x^5 - 10*a*c^2*d^7*x^3 + 3*a^2*c*d^7*x)*e^9 + 3*(5*c^3*d^8*x^4 - 15*a*c^2*d
^8*x^2 + a^2*c*d^8)*e^8 + 2*(10*c^3*d^9*x^3 - 9*a*c^2*d^9*x)*e^7 + 3*(5*c^3*d^10*x^2 - a*c^2*d^10)*e^6)]

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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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(17/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 710 vs. \(2 (323) = 646\).
time = 1.87, size = 710, normalized size = 1.94 \begin {gather*} \frac {{\left (\frac {15 \, c^{6} d^{6} \arctan \left (\frac {\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right ) e}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {c d^{2} e - a e^{3}}} - \frac {{\left (15 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{10} d^{14} e^{5} - 60 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a c^{9} d^{12} e^{7} + 70 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{9} d^{12} e^{4} + 90 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{8} d^{10} e^{9} - 210 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{8} d^{10} e^{6} + 128 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c^{8} d^{10} e^{3} - 60 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a^{3} c^{7} d^{8} e^{11} + 210 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} c^{7} d^{8} e^{8} - 256 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a c^{7} d^{8} e^{5} - 70 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} c^{7} d^{8} e^{2} + 15 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a^{4} c^{6} d^{6} e^{13} - 70 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{3} c^{6} d^{6} e^{10} + 128 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a^{2} c^{6} d^{6} e^{7} + 70 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} a c^{6} d^{6} e^{4} - 15 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {9}{2}} c^{6} d^{6} e\right )} e^{\left (-5\right )}}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (x e + d\right )}^{5} c^{5} d^{5}}\right )} e^{\left (-4\right )}}{640 \, c d} \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)^(17/2),x, algorithm="giac")

[Out]

1/640*(15*c^6*d^6*arctan(sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)/sqrt(c*d^2*e - a*e^3))*e/((c^2*d^4 - 2*a*c*d^
2*e^2 + a^2*e^4)*sqrt(c*d^2*e - a*e^3)) - (15*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*c^10*d^14*e^5 - 60*sqrt(
(x*e + d)*c*d*e - c*d^2*e + a*e^3)*a*c^9*d^12*e^7 + 70*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^9*d^12*e^4
+ 90*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*a^2*c^8*d^10*e^9 - 210*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*
a*c^8*d^10*e^6 + 128*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*c^8*d^10*e^3 - 60*sqrt((x*e + d)*c*d*e - c*d^2*
e + a*e^3)*a^3*c^7*d^8*e^11 + 210*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*c^7*d^8*e^8 - 256*((x*e + d)*c
*d*e - c*d^2*e + a*e^3)^(5/2)*a*c^7*d^8*e^5 - 70*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*c^7*d^8*e^2 + 15*sq
rt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*a^4*c^6*d^6*e^13 - 70*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^3*c^6*
d^6*e^10 + 128*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*c^6*d^6*e^7 + 70*((x*e + d)*c*d*e - c*d^2*e + a*e
^3)^(7/2)*a*c^6*d^6*e^4 - 15*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(9/2)*c^6*d^6*e)*e^(-5)/((c^2*d^4 - 2*a*c*d^2
*e^2 + a^2*e^4)*(x*e + d)^5*c^5*d^5))*e^(-4)/(c*d)

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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 )}^{17/2}} \,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)^(17/2),x)

[Out]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(d + e*x)^(17/2), x)

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