3.21.53 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{11/2}} \, dx\) [2053]

Optimal. Leaf size=236 \[ \frac {15 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e^3 \sqrt {d+e x}}-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 e^2 (d+e x)^{5/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{2 e (d+e x)^{9/2}}-\frac {15 c^2 d^2 \sqrt {c d^2-a e^2} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{4 e^{7/2}} \]

[Out]

-5/4*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/e^2/(e*x+d)^(5/2)-1/2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)
/e/(e*x+d)^(9/2)-15/4*c^2*d^2*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)^(1/2)/e^(7/2)+15/4*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)^(1/2)

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Rubi [A]
time = 0.11, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {676, 678, 674, 211} \begin {gather*} -\frac {15 c^2 d^2 \sqrt {c d^2-a e^2} \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 )}{4 e^{7/2}}+\frac {15 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e^3 \sqrt {d+e x}}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{2 e (d+e x)^{9/2}}-\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}}{4 e^2 (d+e x)^{5/2}} \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)^(11/2),x]

[Out]

(15*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*e^3*Sqrt[d + e*x]) - (5*c*d*(a*d*e + (c*d^2 + a*e^
2)*x + c*d*e*x^2)^(3/2))/(4*e^2*(d + e*x)^(5/2)) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(2*e*(d + e*x
)^(9/2)) - (15*c^2*d^2*Sqrt[c*d^2 - a*e^2]*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])])/(4*e^(7/2))

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 678

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 + 2*p + 1))), x] - Dist[p*((2*c*d - b*e)/(e^2*(m + 2*p + 1))), Int[(d + e*x)^(m + 1)*
(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] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && NeQ[m + 2*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)^{11/2}} \, dx &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{2 e (d+e x)^{9/2}}+\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)^{7/2}} \, dx}{4 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}}{4 e^2 (d+e x)^{5/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{2 e (d+e x)^{9/2}}+\frac {\left (15 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)^{3/2}} \, dx}{8 e^2}\\ &=\frac {15 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e^3 \sqrt {d+e x}}-\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}}{4 e^2 (d+e x)^{5/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{2 e (d+e x)^{9/2}}-\frac {\left (15 c^2 d^2 \left (c d^2-a e^2\right )\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}{8 e^3}\\ &=\frac {15 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e^3 \sqrt {d+e x}}-\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}}{4 e^2 (d+e x)^{5/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{2 e (d+e x)^{9/2}}-\frac {\left (15 c^2 d^2 \left (c d^2-a e^2\right )\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 )}{4 e^2}\\ &=\frac {15 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e^3 \sqrt {d+e x}}-\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}}{4 e^2 (d+e x)^{5/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{2 e (d+e x)^{9/2}}-\frac {15 c^2 d^2 \sqrt {c d^2-a e^2} \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 )}{4 e^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.46, size = 184, normalized size = 0.78 \begin {gather*} \frac {\sqrt {(a e+c d x) (d+e x)} \left (\sqrt {e} \sqrt {a e+c d x} \left (-2 a^2 e^4-a c d e^2 (5 d+9 e x)+c^2 d^2 \left (15 d^2+25 d e x+8 e^2 x^2\right )\right )-15 c^2 d^2 \sqrt {c d^2-a e^2} (d+e x)^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )\right )}{4 e^{7/2} \sqrt {a e+c d x} (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)^(11/2),x]

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[e]*Sqrt[a*e + c*d*x]*(-2*a^2*e^4 - a*c*d*e^2*(5*d + 9*e*x) + c^2*d^2*(15*
d^2 + 25*d*e*x + 8*e^2*x^2)) - 15*c^2*d^2*Sqrt[c*d^2 - a*e^2]*(d + e*x)^2*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x])/S
qrt[c*d^2 - a*e^2]]))/(4*e^(7/2)*Sqrt[a*e + c*d*x]*(d + e*x)^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(516\) vs. \(2(204)=408\).
time = 0.75, size = 517, normalized size = 2.19

method	result	size
default	\(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (15 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a \,c^{2} d^{2} e^{4} x^{2}-15 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{3} d^{4} e^{2} x^{2}+30 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a \,c^{2} d^{3} e^{3} x -30 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{3} d^{5} e x +15 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a \,c^{2} d^{4} e^{2}-15 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{3} d^{6}-8 c^{2} d^{2} e^{2} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+9 a c d \,e^{3} x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}-25 c^{2} d^{3} e x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a^{2} e^{4}+5 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a c \,d^{2} e^{2}-15 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, c^{2} d^{4}\right )}{4 \left (e x +d \right )^{\frac {5}{2}} \sqrt {c d x +a e}\, e^{3} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\)	\(517\)

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)^(11/2),x,method=_RETURNVERBOSE)

[Out]

-1/4*((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))*a*c^2*d^2*e^4*x^2-15
*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^3*d^4*e^2*x^2+30*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c
*d^2)*e)^(1/2))*a*c^2*d^3*e^3*x-30*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^3*d^5*e*x+15*arctanh
(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^2*d^4*e^2-15*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(
1/2))*c^3*d^6-8*c^2*d^2*e^2*x^2*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+9*a*c*d*e^3*x*(c*d*x+a*e)^(1/2)*((a*
e^2-c*d^2)*e)^(1/2)-25*c^2*d^3*e*x*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+2*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+
a*e)^(1/2)*a^2*e^4+5*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a*c*d^2*e^2-15*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a
*e)^(1/2)*c^2*d^4)/(e*x+d)^(5/2)/(c*d*x+a*e)^(1/2)/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)^(11/2),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)^(11/2), x)

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Fricas [A]
time = 3.09, size = 535, normalized size = 2.27 \begin {gather*} \left [\frac {15 \, {\left (c^{2} d^{2} x^{3} e^{3} + 3 \, c^{2} d^{3} x^{2} e^{2} + 3 \, c^{2} d^{4} x e + c^{2} d^{5}\right )} \sqrt {-{\left (c d^{2} - a e^{2}\right )} e^{\left (-1\right )}} \log \left (\frac {c d^{3} - 2 \, a x e^{3} + 2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d} \sqrt {-{\left (c d^{2} - a e^{2}\right )} e^{\left (-1\right )}} e - {\left (c d x^{2} + 2 \, a d\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) + 2 \, {\left (25 \, c^{2} d^{3} x e + 15 \, c^{2} d^{4} - 9 \, a c d x e^{3} - 2 \, a^{2} e^{4} + {\left (8 \, c^{2} d^{2} x^{2} - 5 \, a c d^{2}\right )} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{8 \, {\left (x^{3} e^{6} + 3 \, d x^{2} e^{5} + 3 \, d^{2} x e^{4} + d^{3} e^{3}\right )}}, \frac {15 \, {\left (c^{2} d^{2} x^{3} e^{3} + 3 \, c^{2} d^{3} x^{2} e^{2} + 3 \, c^{2} d^{4} x e + c^{2} d^{5}\right )} \sqrt {c d^{2} - a e^{2}} \arctan \left (\frac {\sqrt {c d^{2} - a e^{2}} \sqrt {x e + d} e^{\left (-\frac {1}{2}\right )}}{\sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}\right ) e^{\left (-\frac {1}{2}\right )} + {\left (25 \, c^{2} d^{3} x e + 15 \, c^{2} d^{4} - 9 \, a c d x e^{3} - 2 \, a^{2} e^{4} + {\left (8 \, c^{2} d^{2} x^{2} - 5 \, a c d^{2}\right )} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{4 \, {\left (x^{3} e^{6} + 3 \, d x^{2} e^{5} + 3 \, d^{2} x e^{4} + d^{3} e^{3}\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)^(11/2),x, algorithm="fricas")

[Out]

[1/8*(15*(c^2*d^2*x^3*e^3 + 3*c^2*d^3*x^2*e^2 + 3*c^2*d^4*x*e + c^2*d^5)*sqrt(-(c*d^2 - a*e^2)*e^(-1))*log((c*
d^3 - 2*a*x*e^3 + 2*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d)*sqrt(-(c*d^2 - a*e^2)*e^(-1))*e
- (c*d*x^2 + 2*a*d)*e^2)/(x^2*e^2 + 2*d*x*e + d^2)) + 2*(25*c^2*d^3*x*e + 15*c^2*d^4 - 9*a*c*d*x*e^3 - 2*a^2*e
^4 + (8*c^2*d^2*x^2 - 5*a*c*d^2)*e^2)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(x^3*e^6 + 3*
d*x^2*e^5 + 3*d^2*x*e^4 + d^3*e^3), 1/4*(15*(c^2*d^2*x^3*e^3 + 3*c^2*d^3*x^2*e^2 + 3*c^2*d^4*x*e + c^2*d^5)*sq
rt(c*d^2 - a*e^2)*arctan(sqrt(c*d^2 - a*e^2)*sqrt(x*e + d)*e^(-1/2)/sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e
))*e^(-1/2) + (25*c^2*d^3*x*e + 15*c^2*d^4 - 9*a*c*d*x*e^3 - 2*a^2*e^4 + (8*c^2*d^2*x^2 - 5*a*c*d^2)*e^2)*sqrt
(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(x^3*e^6 + 3*d*x^2*e^5 + 3*d^2*x*e^4 + d^3*e^3)]

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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)**(11/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 8856 deep

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Giac [A]
time = 1.21, size = 329, normalized size = 1.39 \begin {gather*} \frac {{\left (8 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{3} d^{3} - \frac {15 \, {\left (c^{4} d^{5} e - a c^{3} d^{3} e^{3}\right )} \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 )}{\sqrt {c d^{2} e - a e^{3}}} + \frac {{\left (7 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{5} d^{7} e^{2} - 14 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a c^{4} d^{5} e^{4} + 9 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{4} d^{5} e + 7 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{3} d^{3} e^{6} - 9 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{3} d^{3} e^{3}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2} c^{2} d^{2}}\right )} e^{\left (-4\right )}}{4 \, 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)^(11/2),x, algorithm="giac")

[Out]

1/4*(8*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*c^3*d^3 - 15*(c^4*d^5*e - a*c^3*d^3*e^3)*arctan(sqrt((x*e + d)*
c*d*e - c*d^2*e + a*e^3)/sqrt(c*d^2*e - a*e^3))/sqrt(c*d^2*e - a*e^3) + (7*sqrt((x*e + d)*c*d*e - c*d^2*e + a*
e^3)*c^5*d^7*e^2 - 14*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*a*c^4*d^5*e^4 + 9*((x*e + d)*c*d*e - c*d^2*e + a
*e^3)^(3/2)*c^4*d^5*e + 7*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*a^2*c^3*d^3*e^6 - 9*((x*e + d)*c*d*e - c*d^2
*e + a*e^3)^(3/2)*a*c^3*d^3*e^3)*e^(-2)/((x*e + d)^2*c^2*d^2))*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 )}^{11/2}} \,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)^(11/2),x)

[Out]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(d + e*x)^(11/2), x)

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