3.21.55 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{15/2}} \, dx\) [2055]

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

[Out]

-5/24*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/e^2/(e*x+d)^(9/2)-1/4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2
)/e/(e*x+d)^(13/2)+5/64*c^4*d^4*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)^(3/2)-5/32*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)^(5/
2)+5/64*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)^(3/2)

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Rubi [A]
time = 0.16, 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 = {676, 686, 674, 211} \begin {gather*} \frac {5 c^4 d^4 \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 )}{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}} \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 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)^{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 ) \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 )}{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 [A]
time = 1.08, size = 245, normalized size = 0.81 \begin {gather*} \frac {c^4 d^4 ((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {e} \left (48 a^3 e^6-8 a^2 c d e^4 (d-17 e x)-2 a c^2 d^2 e^2 \left (5 d^2+18 d e x-59 e^2 x^2\right )-c^3 d^3 \left (15 d^3+55 d^2 e x+73 d e^2 x^2-15 e^3 x^3\right )\right )}{c^4 d^4 \left (c d^2-a e^2\right ) (a e+c d x)^2 (d+e x)^4}+\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 )^{3/2} (a e+c d x)^{5/2}}\right )}{192 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)^(15/2),x]

[Out]

(c^4*d^4*((a*e + c*d*x)*(d + e*x))^(5/2)*((Sqrt[e]*(48*a^3*e^6 - 8*a^2*c*d*e^4*(d - 17*e*x) - 2*a*c^2*d^2*e^2*
(5*d^2 + 18*d*e*x - 59*e^2*x^2) - c^3*d^3*(15*d^3 + 55*d^2*e*x + 73*d*e^2*x^2 - 15*e^3*x^3)))/(c^4*d^4*(c*d^2
- a*e^2)*(a*e + c*d*x)^2*(d + e*x)^4) + (15*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[c*d^2 - a*e^2]])/((c*d^2 -
a*e^2)^(3/2)*(a*e + c*d*x)^(5/2))))/(192*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. \(651\) vs. \(2(263)=526\).
time = 0.73, size = 652, normalized size = 2.17

method	result	size
default	\(\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (15 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{4} d^{4} e^{4} x^{4}+60 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{4} d^{5} e^{3} x^{3}+90 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{4} d^{6} e^{2} x^{2}+60 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{4} d^{7} e x -15 c^{3} d^{3} e^{3} x^{3} \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+15 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{4} d^{8}-118 a \,c^{2} d^{2} e^{4} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+73 c^{3} d^{4} e^{2} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}-136 a^{2} c d \,e^{5} x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+36 a \,c^{2} d^{3} e^{3} x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+55 c^{3} d^{5} e x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}-48 \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{3} e^{6}+8 \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} c \,d^{2} e^{4}+10 \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{2} d^{4} e^{2}+15 \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{3} d^{6}\right )}{192 \left (e x +d \right )^{\frac {9}{2}} \sqrt {c d x +a e}\, \left (e^{2} a -c \,d^{2}\right ) e^{3} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\)	\(652\)

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)^(15/2),x,method=_RETURNVERBOSE)

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 584 vs. \(2 (264) = 528\).
time = 3.72, size = 1188, normalized size = 3.95 \begin {gather*} \left [-\frac {15 \, {\left (c^{4} d^{4} x^{5} e^{5} + 5 \, c^{4} d^{5} x^{4} e^{4} + 10 \, c^{4} d^{6} x^{3} e^{3} + 10 \, c^{4} d^{7} x^{2} e^{2} + 5 \, c^{4} d^{8} x e + c^{4} d^{9}\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 (55 \, c^{4} d^{7} x e^{2} + 15 \, c^{4} d^{8} e + 136 \, a^{3} c d x e^{8} + 48 \, a^{4} e^{9} + 2 \, {\left (59 \, a^{2} c^{2} d^{2} x^{2} - 28 \, a^{3} c d^{2}\right )} e^{7} + {\left (15 \, a c^{3} d^{3} x^{3} - 172 \, a^{2} c^{2} d^{3} x\right )} e^{6} - {\left (191 \, a c^{3} d^{4} x^{2} + 2 \, a^{2} c^{2} d^{4}\right )} e^{5} - {\left (15 \, c^{4} d^{5} x^{3} + 19 \, a c^{3} d^{5} x\right )} e^{4} + {\left (73 \, c^{4} d^{6} x^{2} - 5 \, a c^{3} d^{6}\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}}{384 \, {\left (5 \, c^{2} d^{8} x e^{5} + c^{2} d^{9} e^{4} + a^{2} x^{5} e^{13} + 5 \, a^{2} d x^{4} e^{12} - 2 \, {\left (a c d^{2} x^{5} - 5 \, a^{2} d^{2} x^{3}\right )} e^{11} - 10 \, {\left (a c d^{3} x^{4} - a^{2} d^{3} x^{2}\right )} e^{10} + {\left (c^{2} d^{4} x^{5} - 20 \, a c d^{4} x^{3} + 5 \, a^{2} d^{4} x\right )} e^{9} + {\left (5 \, c^{2} d^{5} x^{4} - 20 \, a c d^{5} x^{2} + a^{2} d^{5}\right )} e^{8} + 10 \, {\left (c^{2} d^{6} x^{3} - a c d^{6} x\right )} e^{7} + 2 \, {\left (5 \, c^{2} d^{7} x^{2} - a c d^{7}\right )} e^{6}\right )}}, -\frac {15 \, {\left (c^{4} d^{4} x^{5} e^{5} + 5 \, c^{4} d^{5} x^{4} e^{4} + 10 \, c^{4} d^{6} x^{3} e^{3} + 10 \, c^{4} d^{7} x^{2} e^{2} + 5 \, c^{4} d^{8} x e + c^{4} d^{9}\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 (55 \, c^{4} d^{7} x e^{2} + 15 \, c^{4} d^{8} e + 136 \, a^{3} c d x e^{8} + 48 \, a^{4} e^{9} + 2 \, {\left (59 \, a^{2} c^{2} d^{2} x^{2} - 28 \, a^{3} c d^{2}\right )} e^{7} + {\left (15 \, a c^{3} d^{3} x^{3} - 172 \, a^{2} c^{2} d^{3} x\right )} e^{6} - {\left (191 \, a c^{3} d^{4} x^{2} + 2 \, a^{2} c^{2} d^{4}\right )} e^{5} - {\left (15 \, c^{4} d^{5} x^{3} + 19 \, a c^{3} d^{5} x\right )} e^{4} + {\left (73 \, c^{4} d^{6} x^{2} - 5 \, a c^{3} d^{6}\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}}{192 \, {\left (5 \, c^{2} d^{8} x e^{5} + c^{2} d^{9} e^{4} + a^{2} x^{5} e^{13} + 5 \, a^{2} d x^{4} e^{12} - 2 \, {\left (a c d^{2} x^{5} - 5 \, a^{2} d^{2} x^{3}\right )} e^{11} - 10 \, {\left (a c d^{3} x^{4} - a^{2} d^{3} x^{2}\right )} e^{10} + {\left (c^{2} d^{4} x^{5} - 20 \, a c d^{4} x^{3} + 5 \, a^{2} d^{4} x\right )} e^{9} + {\left (5 \, c^{2} d^{5} x^{4} - 20 \, a c d^{5} x^{2} + a^{2} d^{5}\right )} e^{8} + 10 \, {\left (c^{2} d^{6} x^{3} - a c d^{6} x\right )} e^{7} + 2 \, {\left (5 \, c^{2} d^{7} x^{2} - a c d^{7}\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)^(15/2),x, algorithm="fricas")

[Out]

[-1/384*(15*(c^4*d^4*x^5*e^5 + 5*c^4*d^5*x^4*e^4 + 10*c^4*d^6*x^3*e^3 + 10*c^4*d^7*x^2*e^2 + 5*c^4*d^8*x*e + c
^4*d^9)*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*(55*c^4*d^7*x*e^2 + 15*c^
4*d^8*e + 136*a^3*c*d*x*e^8 + 48*a^4*e^9 + 2*(59*a^2*c^2*d^2*x^2 - 28*a^3*c*d^2)*e^7 + (15*a*c^3*d^3*x^3 - 172
*a^2*c^2*d^3*x)*e^6 - (191*a*c^3*d^4*x^2 + 2*a^2*c^2*d^4)*e^5 - (15*c^4*d^5*x^3 + 19*a*c^3*d^5*x)*e^4 + (73*c^
4*d^6*x^2 - 5*a*c^3*d^6)*e^3)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(5*c^2*d^8*x*e^5 + c^
2*d^9*e^4 + a^2*x^5*e^13 + 5*a^2*d*x^4*e^12 - 2*(a*c*d^2*x^5 - 5*a^2*d^2*x^3)*e^11 - 10*(a*c*d^3*x^4 - a^2*d^3
*x^2)*e^10 + (c^2*d^4*x^5 - 20*a*c*d^4*x^3 + 5*a^2*d^4*x)*e^9 + (5*c^2*d^5*x^4 - 20*a*c*d^5*x^2 + a^2*d^5)*e^8
+ 10*(c^2*d^6*x^3 - a*c*d^6*x)*e^7 + 2*(5*c^2*d^7*x^2 - a*c*d^7)*e^6), -1/192*(15*(c^4*d^4*x^5*e^5 + 5*c^4*d^
5*x^4*e^4 + 10*c^4*d^6*x^3*e^3 + 10*c^4*d^7*x^2*e^2 + 5*c^4*d^8*x*e + c^4*d^9)*sqrt(c*d^2*e - a*e^3)*arctan(sq
rt(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)) + (55*c^4*d^7*x*e^2 + 15*c^4*d^8*e + 136*a^3*c*d*x*e^8 + 48*a^4*e^9 + 2*(59*a^2*c^2*d^2*x^2 - 28*
a^3*c*d^2)*e^7 + (15*a*c^3*d^3*x^3 - 172*a^2*c^2*d^3*x)*e^6 - (191*a*c^3*d^4*x^2 + 2*a^2*c^2*d^4)*e^5 - (15*c^
4*d^5*x^3 + 19*a*c^3*d^5*x)*e^4 + (73*c^4*d^6*x^2 - 5*a*c^3*d^6)*e^3)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)
*e)*sqrt(x*e + d))/(5*c^2*d^8*x*e^5 + c^2*d^9*e^4 + a^2*x^5*e^13 + 5*a^2*d*x^4*e^12 - 2*(a*c*d^2*x^5 - 5*a^2*d
^2*x^3)*e^11 - 10*(a*c*d^3*x^4 - a^2*d^3*x^2)*e^10 + (c^2*d^4*x^5 - 20*a*c*d^4*x^3 + 5*a^2*d^4*x)*e^9 + (5*c^2
*d^5*x^4 - 20*a*c*d^5*x^2 + a^2*d^5)*e^8 + 10*(c^2*d^6*x^3 - a*c*d^6*x)*e^7 + 2*(5*c^2*d^7*x^2 - a*c*d^7)*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)**(15/2),x)

[Out]

Timed out

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Giac [A]
time = 1.89, size = 496, normalized size = 1.65 \begin {gather*} \frac {{\left (\frac {15 \, c^{5} d^{5} \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}{\sqrt {c d^{2} e - a e^{3}} {\left (c d^{2} - a e^{2}\right )}} - \frac {{\left (15 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{8} d^{11} e^{4} - 45 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a c^{7} d^{9} e^{6} + 55 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{7} d^{9} e^{3} + 45 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{6} d^{7} e^{8} - 110 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{6} d^{7} e^{5} + 73 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c^{6} d^{7} e^{2} - 15 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a^{3} c^{5} d^{5} e^{10} + 55 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} c^{5} d^{5} e^{7} - 73 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a c^{5} d^{5} e^{4} - 15 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} c^{5} d^{5} e\right )} e^{\left (-4\right )}}{{\left (c d^{2} - a e^{2}\right )} {\left (x e + d\right )}^{4} c^{4} d^{4}}\right )} e^{\left (-4\right )}}{192 \, 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)^(15/2),x, algorithm="giac")

[Out]

1/192*(15*c^5*d^5*arctan(sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)/sqrt(c*d^2*e - a*e^3))*e/(sqrt(c*d^2*e - a*e^
3)*(c*d^2 - a*e^2)) - (15*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*c^8*d^11*e^4 - 45*sqrt((x*e + d)*c*d*e - c*d
^2*e + a*e^3)*a*c^7*d^9*e^6 + 55*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^7*d^9*e^3 + 45*sqrt((x*e + d)*c*d
*e - c*d^2*e + a*e^3)*a^2*c^6*d^7*e^8 - 110*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*c^6*d^7*e^5 + 73*((x*e
+ d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*c^6*d^7*e^2 - 15*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*a^3*c^5*d^5*e^10
+ 55*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*c^5*d^5*e^7 - 73*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)
*a*c^5*d^5*e^4 - 15*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*c^5*d^5*e)*e^(-4)/((c*d^2 - a*e^2)*(x*e + d)^4*c
^4*d^4))*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 )}^{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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