3.7.76 \(\int \frac {(d+e x)^{5/2}}{(f+g x) (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [676]

Optimal. Leaf size=188 \[ -\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 g \sqrt {d+e x}}{(c d f-a e g)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 g^{3/2} \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{(c d f-a e g)^{5/2}} \]

[Out]

-2/3*(e*x+d)^(3/2)/(-a*e*g+c*d*f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+2*g^(3/2)*arctan(g^(1/2)*(a*d*e+(a*e
^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)^(1/2)/(e*x+d)^(1/2))/(-a*e*g+c*d*f)^(5/2)+2*g*(e*x+d)^(1/2)/(-a*e*
g+c*d*f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)

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Rubi [A]
time = 0.18, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {882, 888, 211} \begin {gather*} \frac {2 g^{3/2} \text {ArcTan}\left (\frac {\sqrt {g} \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 f-a e g}}\right )}{(c d f-a e g)^{5/2}}+\frac {2 g \sqrt {d+e x}}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}-\frac {2 (d+e x)^{3/2}}{3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(d + e*x)^(3/2))/(3*(c*d*f - a*e*g)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (2*g*Sqrt[d + e*x])/(
(c*d*f - a*e*g)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (2*g^(3/2)*ArcTan[(Sqrt[g]*Sqrt[a*d*e + (c*d^
2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d*f - a*e*g]*Sqrt[d + e*x])])/(c*d*f - a*e*g)^(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 882

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Simp[e^2*(d + e*x)^(m - 1)*(f + g*x)^(n + 1)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(c*e*f + c*d*g - b*e*g))), x]
+ Dist[e^2*g*((m - n - 2)/((p + 1)*(c*e*f + c*d*g - b*e*g))), Int[(d + e*x)^(m - 1)*(f + g*x)^n*(a + b*x + c*
x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[
c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0] && LtQ[p, -1] && RationalQ[n]

Rule 888

Int[Sqrt[(d_) + (e_.)*(x_)]/(((f_.) + (g_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[
2*e^2, Subst[Int[1/(c*(e*f + d*g) - b*e*g + e^2*g*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; Fre
eQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{5/2}}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {g \int \frac {(d+e x)^{3/2}}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{c d f-a e g}\\ &=-\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 g \sqrt {d+e x}}{(c d f-a e g)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {g^2 \int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{(c d f-a e g)^2}\\ &=-\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 g \sqrt {d+e x}}{(c d f-a e g)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (2 e^2 g^2\right ) \text {Subst}\left (\int \frac {1}{-e \left (c d^2+a e^2\right ) g+c d e (e f+d g)+e^2 g 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 )}{(c d f-a e g)^2}\\ &=-\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 g \sqrt {d+e x}}{(c d f-a e g)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 g^{3/2} \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{(c d f-a e g)^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.22, size = 129, normalized size = 0.69 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (\sqrt {c d f-a e g} (4 a e g-c d (f-3 g x))+3 g^{3/2} (a e+c d x)^{3/2} \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )\right )}{3 (c d f-a e g)^{5/2} ((a e+c d x) (d+e x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(Sqrt[c*d*f - a*e*g]*(4*a*e*g - c*d*(f - 3*g*x)) + 3*g^(3/2)*(a*e + c*d*x)^(3/2)*ArcTan[(Sq
rt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d*f - a*e*g]]))/(3*(c*d*f - a*e*g)^(5/2)*((a*e + c*d*x)*(d + e*x))^(3/2))

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Maple [A]
time = 0.15, size = 209, normalized size = 1.11

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (3 \sqrt {c d x +a e}\, \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c d \,g^{2} x +3 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a e \,g^{2} \sqrt {c d x +a e}-3 \sqrt {\left (a e g -c d f \right ) g}\, c d g x -4 \sqrt {\left (a e g -c d f \right ) g}\, a e g +\sqrt {\left (a e g -c d f \right ) g}\, c d f \right )}{3 \sqrt {e x +d}\, \left (c d x +a e \right )^{2} \left (a e g -c d f \right )^{2} \sqrt {\left (a e g -c d f \right ) g}}\)	\(209\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*((c*d*x+a*e)*(e*x+d))^(1/2)*(3*(c*d*x+a*e)^(1/2)*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c*d
*g^2*x+3*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*a*e*g^2*(c*d*x+a*e)^(1/2)-3*((a*e*g-c*d*f)*g)^(1
/2)*c*d*g*x-4*((a*e*g-c*d*f)*g)^(1/2)*a*e*g+((a*e*g-c*d*f)*g)^(1/2)*c*d*f)/(e*x+d)^(1/2)/(c*d*x+a*e)^2/(a*e*g-
c*d*f)^2/((a*e*g-c*d*f)*g)^(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((e*x+d)^(5/2)/(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (176) = 352\).
time = 4.66, size = 1029, normalized size = 5.47 \begin {gather*} \left [\frac {3 \, {\left (c^{2} d^{3} g x^{2} + a^{2} g x e^{3} + {\left (2 \, a c d g x^{2} + a^{2} d g\right )} e^{2} + {\left (c^{2} d^{2} g x^{3} + 2 \, a c d^{2} g x\right )} e\right )} \sqrt {-\frac {g}{c d f - a g e}} \log \left (-\frac {c d^{2} g x - c d^{2} f + 2 \, a g x e^{2} + 2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (c d f - a g e\right )} \sqrt {x e + d} \sqrt {-\frac {g}{c d f - a g e}} + {\left (c d g x^{2} - c d f x + 2 \, a d g\right )} e}{d g x + d f + {\left (g x^{2} + f x\right )} e}\right ) + 2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (3 \, c d g x - c d f + 4 \, a g e\right )} \sqrt {x e + d}}{3 \, {\left (c^{4} d^{5} f^{2} x^{2} + a^{4} g^{2} x e^{5} + {\left (2 \, a^{3} c d g^{2} x^{2} - 2 \, a^{3} c d f g x + a^{4} d g^{2}\right )} e^{4} + {\left (a^{2} c^{2} d^{2} g^{2} x^{3} - 4 \, a^{2} c^{2} d^{2} f g x^{2} - 2 \, a^{3} c d^{2} f g + {\left (a^{2} c^{2} d^{2} f^{2} + 2 \, a^{3} c d^{2} g^{2}\right )} x\right )} e^{3} - {\left (2 \, a c^{3} d^{3} f g x^{3} + 4 \, a^{2} c^{2} d^{3} f g x - a^{2} c^{2} d^{3} f^{2} - {\left (2 \, a c^{3} d^{3} f^{2} + a^{2} c^{2} d^{3} g^{2}\right )} x^{2}\right )} e^{2} + {\left (c^{4} d^{4} f^{2} x^{3} - 2 \, a c^{3} d^{4} f g x^{2} + 2 \, a c^{3} d^{4} f^{2} x\right )} e\right )}}, \frac {2 \, {\left (3 \, {\left (c^{2} d^{3} g x^{2} + a^{2} g x e^{3} + {\left (2 \, a c d g x^{2} + a^{2} d g\right )} e^{2} + {\left (c^{2} d^{2} g x^{3} + 2 \, a c d^{2} g x\right )} e\right )} \sqrt {\frac {g}{c d f - a g e}} \arctan \left (-\frac {\sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (c d f - a g e\right )} \sqrt {x e + d} \sqrt {\frac {g}{c d f - a g e}}}{c d^{2} g x + a g x e^{2} + {\left (c d g x^{2} + a d g\right )} e}\right ) + \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (3 \, c d g x - c d f + 4 \, a g e\right )} \sqrt {x e + d}\right )}}{3 \, {\left (c^{4} d^{5} f^{2} x^{2} + a^{4} g^{2} x e^{5} + {\left (2 \, a^{3} c d g^{2} x^{2} - 2 \, a^{3} c d f g x + a^{4} d g^{2}\right )} e^{4} + {\left (a^{2} c^{2} d^{2} g^{2} x^{3} - 4 \, a^{2} c^{2} d^{2} f g x^{2} - 2 \, a^{3} c d^{2} f g + {\left (a^{2} c^{2} d^{2} f^{2} + 2 \, a^{3} c d^{2} g^{2}\right )} x\right )} e^{3} - {\left (2 \, a c^{3} d^{3} f g x^{3} + 4 \, a^{2} c^{2} d^{3} f g x - a^{2} c^{2} d^{3} f^{2} - {\left (2 \, a c^{3} d^{3} f^{2} + a^{2} c^{2} d^{3} g^{2}\right )} x^{2}\right )} e^{2} + {\left (c^{4} d^{4} f^{2} x^{3} - 2 \, a c^{3} d^{4} f g x^{2} + 2 \, a c^{3} d^{4} f^{2} x\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/3*(3*(c^2*d^3*g*x^2 + a^2*g*x*e^3 + (2*a*c*d*g*x^2 + a^2*d*g)*e^2 + (c^2*d^2*g*x^3 + 2*a*c*d^2*g*x)*e)*sqrt
(-g/(c*d*f - a*g*e))*log(-(c*d^2*g*x - c*d^2*f + 2*a*g*x*e^2 + 2*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*(
c*d*f - a*g*e)*sqrt(x*e + d)*sqrt(-g/(c*d*f - a*g*e)) + (c*d*g*x^2 - c*d*f*x + 2*a*d*g)*e)/(d*g*x + d*f + (g*x
^2 + f*x)*e)) + 2*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*(3*c*d*g*x - c*d*f + 4*a*g*e)*sqrt(x*e + d))/(c^
4*d^5*f^2*x^2 + a^4*g^2*x*e^5 + (2*a^3*c*d*g^2*x^2 - 2*a^3*c*d*f*g*x + a^4*d*g^2)*e^4 + (a^2*c^2*d^2*g^2*x^3 -
4*a^2*c^2*d^2*f*g*x^2 - 2*a^3*c*d^2*f*g + (a^2*c^2*d^2*f^2 + 2*a^3*c*d^2*g^2)*x)*e^3 - (2*a*c^3*d^3*f*g*x^3 +
4*a^2*c^2*d^3*f*g*x - a^2*c^2*d^3*f^2 - (2*a*c^3*d^3*f^2 + a^2*c^2*d^3*g^2)*x^2)*e^2 + (c^4*d^4*f^2*x^3 - 2*a
*c^3*d^4*f*g*x^2 + 2*a*c^3*d^4*f^2*x)*e), 2/3*(3*(c^2*d^3*g*x^2 + a^2*g*x*e^3 + (2*a*c*d*g*x^2 + a^2*d*g)*e^2
+ (c^2*d^2*g*x^3 + 2*a*c*d^2*g*x)*e)*sqrt(g/(c*d*f - a*g*e))*arctan(-sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*
e)*(c*d*f - a*g*e)*sqrt(x*e + d)*sqrt(g/(c*d*f - a*g*e))/(c*d^2*g*x + a*g*x*e^2 + (c*d*g*x^2 + a*d*g)*e)) + sq
rt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*(3*c*d*g*x - c*d*f + 4*a*g*e)*sqrt(x*e + d))/(c^4*d^5*f^2*x^2 + a^4*
g^2*x*e^5 + (2*a^3*c*d*g^2*x^2 - 2*a^3*c*d*f*g*x + a^4*d*g^2)*e^4 + (a^2*c^2*d^2*g^2*x^3 - 4*a^2*c^2*d^2*f*g*x
^2 - 2*a^3*c*d^2*f*g + (a^2*c^2*d^2*f^2 + 2*a^3*c*d^2*g^2)*x)*e^3 - (2*a*c^3*d^3*f*g*x^3 + 4*a^2*c^2*d^3*f*g*x
- a^2*c^2*d^3*f^2 - (2*a*c^3*d^3*f^2 + a^2*c^2*d^3*g^2)*x^2)*e^2 + (c^4*d^4*f^2*x^3 - 2*a*c^3*d^4*f*g*x^2 + 2
*a*c^3*d^4*f^2*x)*e)]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 661 vs. \(2 (176) = 352\).
time = 1.91, size = 661, normalized size = 3.52 \begin {gather*} \frac {2}{3} \, {\left (\frac {3 \, g^{2} \arctan \left (\frac {\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} g e^{\left (-1\right )}}{\sqrt {c d f g - a g^{2} e}}\right ) e^{\left (-1\right )}}{{\left (c^{2} d^{2} f^{2} e - 2 \, a c d f g e^{2} + a^{2} g^{2} e^{3}\right )} \sqrt {c d f g - a g^{2} e}} - \frac {c d f e^{2} - a g e^{3} - 3 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )} g}{{\left (c^{2} d^{2} f^{2} e - 2 \, a c d f g e^{2} + a^{2} g^{2} e^{3}\right )} {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}}}\right )} e^{2} - \frac {2 \, {\left (3 \, \sqrt {-c d^{2} e + a e^{3}} c d^{2} g^{2} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g e^{\left (-1\right )}}{\sqrt {c d f g - a g^{2} e}}\right ) - 3 \, \sqrt {-c d^{2} e + a e^{3}} a g^{2} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g e^{\left (-1\right )}}{\sqrt {c d f g - a g^{2} e}}\right ) e^{2} + 3 \, \sqrt {c d f g - a g^{2} e} c d^{2} g e + \sqrt {c d f g - a g^{2} e} c d f e^{2} - 4 \, \sqrt {c d f g - a g^{2} e} a g e^{3}\right )}}{3 \, {\left (\sqrt {c d f g - a g^{2} e} \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{4} f^{2} - 2 \, \sqrt {c d f g - a g^{2} e} \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{3} f g e - \sqrt {c d f g - a g^{2} e} \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{2} f^{2} e^{2} + \sqrt {c d f g - a g^{2} e} \sqrt {-c d^{2} e + a e^{3}} a^{2} c d^{2} g^{2} e^{2} + 2 \, \sqrt {c d f g - a g^{2} e} \sqrt {-c d^{2} e + a e^{3}} a^{2} c d f g e^{3} - \sqrt {c d f g - a g^{2} e} \sqrt {-c d^{2} e + a e^{3}} a^{3} g^{2} e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)/(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")

[Out]

2/3*(3*g^2*arctan(sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*g*e^(-1)/sqrt(c*d*f*g - a*g^2*e))*e^(-1)/((c^2*d^2*f
^2*e - 2*a*c*d*f*g*e^2 + a^2*g^2*e^3)*sqrt(c*d*f*g - a*g^2*e)) - (c*d*f*e^2 - a*g*e^3 - 3*((x*e + d)*c*d*e - c
*d^2*e + a*e^3)*g)/((c^2*d^2*f^2*e - 2*a*c*d*f*g*e^2 + a^2*g^2*e^3)*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2))
)*e^2 - 2/3*(3*sqrt(-c*d^2*e + a*e^3)*c*d^2*g^2*arctan(sqrt(-c*d^2*e + a*e^3)*g*e^(-1)/sqrt(c*d*f*g - a*g^2*e)
) - 3*sqrt(-c*d^2*e + a*e^3)*a*g^2*arctan(sqrt(-c*d^2*e + a*e^3)*g*e^(-1)/sqrt(c*d*f*g - a*g^2*e))*e^2 + 3*sqr
t(c*d*f*g - a*g^2*e)*c*d^2*g*e + sqrt(c*d*f*g - a*g^2*e)*c*d*f*e^2 - 4*sqrt(c*d*f*g - a*g^2*e)*a*g*e^3)/(sqrt(
c*d*f*g - a*g^2*e)*sqrt(-c*d^2*e + a*e^3)*c^3*d^4*f^2 - 2*sqrt(c*d*f*g - a*g^2*e)*sqrt(-c*d^2*e + a*e^3)*a*c^2
*d^3*f*g*e - sqrt(c*d*f*g - a*g^2*e)*sqrt(-c*d^2*e + a*e^3)*a*c^2*d^2*f^2*e^2 + sqrt(c*d*f*g - a*g^2*e)*sqrt(-
c*d^2*e + a*e^3)*a^2*c*d^2*g^2*e^2 + 2*sqrt(c*d*f*g - a*g^2*e)*sqrt(-c*d^2*e + a*e^3)*a^2*c*d*f*g*e^3 - sqrt(c
*d*f*g - a*g^2*e)*sqrt(-c*d^2*e + a*e^3)*a^3*g^2*e^4)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^{5/2}}{\left (f+g\,x\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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