3.21.51 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{7/2}} \, dx\) [2051]

Optimal. Leaf size=240 \[ \frac {2 \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e^3 \sqrt {d+e x}}+\frac {2 \left (a-\frac {c d^2}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 (d+e x)^{3/2}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}-\frac {2 \left (c d^2-a e^2\right )^{5/2} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{e^{7/2}} \]

[Out]

2/3*(a-c*d^2/e^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2)+2/5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^
(5/2)/e/(e*x+d)^(5/2)-2*(-a*e^2+c*d^2)^(5/2)*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))/e^(7/2)+2*(-a*e^2+c*d^2)^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 = 0.14, antiderivative size = 240, 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 = {678, 674, 211} \begin {gather*} -\frac {2 \left (c d^2-a e^2\right )^{5/2} \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 )}{e^{7/2}}+\frac {2 \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)^{5/2}}+\frac {2 \left (a-\frac {c d^2}{e^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 (d+e x)^{3/2}}+\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^3 \sqrt {d+e x}} \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)^(7/2),x]

[Out]

(2*(c*d^2 - a*e^2)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(e^3*Sqrt[d + e*x]) + (2*(a - (c*d^2)/e^2)*(
a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3*(d + e*x)^(3/2)) + (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)
^(5/2))/(5*e*(d + e*x)^(5/2)) - (2*(c*d^2 - a*e^2)^(5/2)*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])])/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 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)^{7/2}} \, dx &=\frac {2 \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)^{5/2}}-\frac {\left (2 c d^2 e-e \left (c d^2+a e^2\right )\right ) \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)^{5/2}} \, dx}{e^2}\\ &=\frac {2 \left (a-\frac {c d^2}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 (d+e x)^{3/2}}+\frac {2 \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)^{5/2}}+\frac {\left (c d^2-a e^2\right )^2 \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}{e^2}\\ &=\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e^3 \sqrt {d+e x}}+\frac {2 \left (a-\frac {c d^2}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 (d+e x)^{3/2}}+\frac {2 \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)^{5/2}}-\frac {\left (c d^2-a e^2\right )^3 \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}{e^3}\\ &=\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e^3 \sqrt {d+e x}}+\frac {2 \left (a-\frac {c d^2}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 (d+e x)^{3/2}}+\frac {2 \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)^{5/2}}-\frac {\left (2 \left (c d^2-a e^2\right )^3\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 )}{e^2}\\ &=\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e^3 \sqrt {d+e x}}+\frac {2 \left (a-\frac {c d^2}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 (d+e x)^{3/2}}+\frac {2 \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)^{5/2}}-\frac {2 \left (c d^2-a e^2\right )^{5/2} \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 )}{e^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.20, size = 170, normalized size = 0.71 \begin {gather*} \frac {2 \sqrt {a e+c d x} \sqrt {d+e x} \left (\sqrt {e} \sqrt {a e+c d x} \left (23 a^2 e^4+a c d e^2 (-35 d+11 e x)+c^2 d^2 \left (15 d^2-5 d e x+3 e^2 x^2\right )\right )-15 \left (c d^2-a e^2\right )^{5/2} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )\right )}{15 e^{7/2} \sqrt {(a e+c d x) (d+e x)}} \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)^(7/2),x]

[Out]

(2*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(Sqrt[e]*Sqrt[a*e + c*d*x]*(23*a^2*e^4 + a*c*d*e^2*(-35*d + 11*e*x) + c^2*d
^2*(15*d^2 - 5*d*e*x + 3*e^2*x^2)) - 15*(c*d^2 - a*e^2)^(5/2)*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[c*d^2 -
a*e^2]]))/(15*e^(7/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(426\) vs. \(2(212)=424\).
time = 0.83, size = 427, normalized size = 1.78

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (15 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a^{3} e^{6}-45 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a^{2} c \,d^{2} e^{4}+45 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a \,c^{2} d^{4} e^{2}-15 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{3} d^{6}-3 c^{2} d^{2} e^{2} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}-11 a c d \,e^{3} x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+5 c^{2} d^{3} e x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}-23 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a^{2} e^{4}+35 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a c \,d^{2} e^{2}-15 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, c^{2} d^{4}\right )}{15 \sqrt {e x +d}\, \sqrt {c d x +a e}\, e^{3} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\)	\(427\)

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)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-2/15*((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^3*e^6-45*arctanh(
e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a^2*c*d^2*e^4+45*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^6-3*c^2*d^2*e^2*x^2*(c*d*x+a*
e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)-11*a*c*d*e^3*x*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+5*c^2*d^3*e*x*(c*d*x
+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)-23*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a^2*e^4+35*((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)^(1/2)/(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)^(7/2),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)^(7/2), x)

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Fricas [A]
time = 3.19, size = 512, normalized size = 2.13 \begin {gather*} \left [\frac {15 \, {\left (c^{2} d^{4} x e + c^{2} d^{5} - 2 \, a c d^{2} x e^{3} - 2 \, a c d^{3} e^{2} + a^{2} x e^{5} + a^{2} d e^{4}\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 (5 \, c^{2} d^{3} x e - 15 \, c^{2} d^{4} - 11 \, a c d x e^{3} - 23 \, a^{2} e^{4} - {\left (3 \, c^{2} d^{2} x^{2} - 35 \, 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}}{15 \, {\left (x e^{4} + d e^{3}\right )}}, \frac {2 \, {\left (15 \, {\left (c^{2} d^{4} x e + c^{2} d^{5} - 2 \, a c d^{2} x e^{3} - 2 \, a c d^{3} e^{2} + a^{2} x e^{5} + a^{2} d e^{4}\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 (5 \, c^{2} d^{3} x e - 15 \, c^{2} d^{4} - 11 \, a c d x e^{3} - 23 \, a^{2} e^{4} - {\left (3 \, c^{2} d^{2} x^{2} - 35 \, 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}\right )}}{15 \, {\left (x e^{4} + d 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)^(7/2),x, algorithm="fricas")

[Out]

[1/15*(15*(c^2*d^4*x*e + c^2*d^5 - 2*a*c*d^2*x*e^3 - 2*a*c*d^3*e^2 + a^2*x*e^5 + a^2*d*e^4)*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*(5*c^2*d^3*x*e - 15*c^2*d^4 - 11*a*
c*d*x*e^3 - 23*a^2*e^4 - (3*c^2*d^2*x^2 - 35*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*e^4 + d*e^3), 2/15*(15*(c^2*d^4*x*e + c^2*d^5 - 2*a*c*d^2*x*e^3 - 2*a*c*d^3*e^2 + a^2*x*e^5 + a^2*d
*e^4)*sqrt(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) - (5*c^2*d^3*x*e - 15*c^2*d^4 - 11*a*c*d*x*e^3 - 23*a^2*e^4 - (3*c^2*d^2*x^2 - 35*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*e^4 + d*e^3)]

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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)**(7/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 607 vs. \(2 (213) = 426\).
time = 1.28, size = 607, normalized size = 2.53 \begin {gather*} \frac {2}{15} \, {\left ({\left (15 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{2} d^{4} e^{14} - 30 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a c d^{2} e^{16} - 5 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c d^{2} e^{13} + 15 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} e^{18} + 5 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a e^{15} + 3 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} e^{12}\right )} e^{\left (-15\right )} - \frac {15 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\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}}}\right )} e^{\left (-3\right )} + \frac {2 \, {\left (15 \, c^{3} d^{6} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right ) e - 45 \, a c^{2} d^{4} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right ) e^{3} - 23 \, \sqrt {c d^{2} e - a e^{3}} \sqrt {-c d^{2} e + a e^{3}} c^{2} d^{4} + 45 \, a^{2} c d^{2} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right ) e^{5} + 46 \, \sqrt {c d^{2} e - a e^{3}} \sqrt {-c d^{2} e + a e^{3}} a c d^{2} e^{2} - 15 \, a^{3} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right ) e^{7} - 23 \, \sqrt {c d^{2} e - a e^{3}} \sqrt {-c d^{2} e + a e^{3}} a^{2} e^{4}\right )} e^{\left (-4\right )}}{15 \, \sqrt {c d^{2} e - a e^{3}}} \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)^(7/2),x, algorithm="giac")

[Out]

2/15*((15*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*c^2*d^4*e^14 - 30*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*a*
c*d^2*e^16 - 5*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c*d^2*e^13 + 15*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^
3)*a^2*e^18 + 5*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*e^15 + 3*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)
*e^12)*e^(-15) - 15*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*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))*e^(-3) + 2/15*(15*c^3*d^6*arctan(sqrt(-c*d^2*e + a*
e^3)/sqrt(c*d^2*e - a*e^3))*e - 45*a*c^2*d^4*arctan(sqrt(-c*d^2*e + a*e^3)/sqrt(c*d^2*e - a*e^3))*e^3 - 23*sqr
t(c*d^2*e - a*e^3)*sqrt(-c*d^2*e + a*e^3)*c^2*d^4 + 45*a^2*c*d^2*arctan(sqrt(-c*d^2*e + a*e^3)/sqrt(c*d^2*e -
a*e^3))*e^5 + 46*sqrt(c*d^2*e - a*e^3)*sqrt(-c*d^2*e + a*e^3)*a*c*d^2*e^2 - 15*a^3*arctan(sqrt(-c*d^2*e + a*e^
3)/sqrt(c*d^2*e - a*e^3))*e^7 - 23*sqrt(c*d^2*e - a*e^3)*sqrt(-c*d^2*e + a*e^3)*a^2*e^4)*e^(-4)/sqrt(c*d^2*e -
a*e^3)

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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 )}^{7/2}} \,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)^(7/2),x)

[Out]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(d + e*x)^(7/2), x)

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