3.8.5 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2} (f+g x)} \, dx\) [705]

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

[Out]

-2/3*(-a*e*g+c*d*f)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/g^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)/g/(e*x+d)^(5/2)-2*(-a*e*g+c*d*f)^(5/2)*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))/g^(7/2)+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)/g^3/(e*x+d)
^(1/2)

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Rubi [A]
time = 0.30, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {878, 888, 211} \begin {gather*} -\frac {2 (c d f-a e g)^{5/2} \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 )}{g^{7/2}}+\frac {2 \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}{g^3 \sqrt {d+e x}}-\frac {2 \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)}{3 g^2 (d+e x)^{3/2}}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 g (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)^(5/2)*(f + g*x)),x]

[Out]

(2*(c*d*f - a*e*g)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(g^3*Sqrt[d + e*x]) - (2*(c*d*f - a*e*g)*(a*
d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3*g^2*(d + e*x)^(3/2)) + (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^
2)^(5/2))/(5*g*(d + e*x)^(5/2)) - (2*(c*d*f - a*e*g)^(5/2)*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])])/g^(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 878

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Simp[(-(d + e*x)^m)*(f + g*x)^(n + 1)*((a + b*x + c*x^2)^p/(g*(m - n - 1))), x] - Dist[m*((c*e*f + c*d*g - b*e
*g)/(e^2*g*(m - n - 1))), 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] &&  !Intege
rQ[p] && EqQ[m + p, 0] && GtQ[p, 0] && NeQ[m - n - 1, 0] &&  !IGtQ[n, 0] &&  !(IntegerQ[n + p] && LtQ[n + p +
2, 0]) && 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 {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)} \, 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 g (d+e x)^{5/2}}-\frac {\left (c d e^2 f+c d^2 e g-e \left (c d^2+a e^2\right ) g\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)^{3/2} (f+g x)} \, dx}{e^2 g}\\ &=-\frac {2 (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}}{3 g^2 (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 g (d+e x)^{5/2}}+\frac {(c d f-a e g)^2 \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)} \, dx}{g^2}\\ &=\frac {2 (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}}{g^3 \sqrt {d+e x}}-\frac {2 (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}}{3 g^2 (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 g (d+e x)^{5/2}}-\frac {(c d f-a e g)^3 \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}{g^3}\\ &=\frac {2 (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}}{g^3 \sqrt {d+e x}}-\frac {2 (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}}{3 g^2 (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 g (d+e x)^{5/2}}-\frac {\left (2 e^2 (c d f-a e g)^3\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 )}{g^3}\\ &=\frac {2 (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}}{g^3 \sqrt {d+e x}}-\frac {2 (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}}{3 g^2 (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 g (d+e x)^{5/2}}-\frac {2 (c d f-a e g)^{5/2} \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 )}{g^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.23, size = 168, normalized size = 0.71 \begin {gather*} \frac {2 \sqrt {a e+c d x} \sqrt {d+e x} \left (\sqrt {g} \sqrt {a e+c d x} \left (23 a^2 e^2 g^2+a c d e g (-35 f+11 g x)+c^2 d^2 \left (15 f^2-5 f g x+3 g^2 x^2\right )\right )-15 (c d f-a e g)^{5/2} \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )\right )}{15 g^{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)^(5/2)*(f + g*x)),x]

[Out]

(2*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(Sqrt[g]*Sqrt[a*e + c*d*x]*(23*a^2*e^2*g^2 + a*c*d*e*g*(-35*f + 11*g*x) + c
^2*d^2*(15*f^2 - 5*f*g*x + 3*g^2*x^2)) - 15*(c*d*f - a*e*g)^(5/2)*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d*
f - a*e*g]]))/(15*g^(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. \(420\) vs. \(2(208)=416\).
time = 0.15, size = 421, normalized size = 1.78

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (15 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a^{3} e^{3} g^{3}-45 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a^{2} c d \,e^{2} f \,g^{2}+45 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a \,c^{2} d^{2} e \,f^{2} g -15 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} f^{3}-3 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c^{2} d^{2} g^{2} x^{2}-11 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a c d e \,g^{2} x +5 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c^{2} d^{2} f g x -23 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a^{2} e^{2} g^{2}+35 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a c d e f g -15 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c^{2} d^{2} f^{2}\right )}{15 \sqrt {e x +d}\, \sqrt {c d x +a e}\, g^{3} \sqrt {\left (a e g -c d f \right ) g}}\)	\(421\)

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)^(5/2)/(g*x+f),x,method=_RETURNVERBOSE)

[Out]

-2/15*((c*d*x+a*e)*(e*x+d))^(1/2)*(15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*a^3*e^3*g^3-45*arct
anh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*a^2*c*d*e^2*f*g^2+45*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*
f)*g)^(1/2))*a*c^2*d^2*e*f^2*g-15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^3*d^3*f^3-3*((a*e*g-c
*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*c^2*d^2*g^2*x^2-11*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a*c*d*e*g^2*x+5*
((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*c^2*d^2*f*g*x-23*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a^2*e^2*g
^2+35*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a*c*d*e*f*g-15*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*c^2*d
^2*f^2)/(e*x+d)^(1/2)/(c*d*x+a*e)^(1/2)/g^3/((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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.68, size = 596, normalized size = 2.53 \begin {gather*} \left [\frac {15 \, {\left (c^{2} d^{3} f^{2} + a^{2} g^{2} x e^{3} - {\left (2 \, a c d f g x - a^{2} d g^{2}\right )} e^{2} + {\left (c^{2} d^{2} f^{2} x - 2 \, a c d^{2} f g\right )} e\right )} \sqrt {-\frac {c d f - a g e}{g}} \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} \sqrt {x e + d} g \sqrt {-\frac {c d f - a g e}{g}} + {\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 \, {\left (3 \, c^{2} d^{2} g^{2} x^{2} - 5 \, c^{2} d^{2} f g x + 15 \, c^{2} d^{2} f^{2} + 23 \, a^{2} g^{2} e^{2} + {\left (11 \, a c d g^{2} x - 35 \, a c d f g\right )} e\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 (g^{3} x e + d g^{3}\right )}}, \frac {2 \, {\left (15 \, {\left (c^{2} d^{3} f^{2} + a^{2} g^{2} x e^{3} - {\left (2 \, a c d f g x - a^{2} d g^{2}\right )} e^{2} + {\left (c^{2} d^{2} f^{2} x - 2 \, a c d^{2} f g\right )} e\right )} \sqrt {\frac {c d f - a g e}{g}} \arctan \left (\frac {\sqrt {x e + d} \sqrt {\frac {c d f - a g e}{g}}}{\sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}\right ) + {\left (3 \, c^{2} d^{2} g^{2} x^{2} - 5 \, c^{2} d^{2} f g x + 15 \, c^{2} d^{2} f^{2} + 23 \, a^{2} g^{2} e^{2} + {\left (11 \, a c d g^{2} x - 35 \, a c d f g\right )} e\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 (g^{3} x e + d g^{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)^(5/2)/(g*x+f),x, algorithm="fricas")

[Out]

[1/15*(15*(c^2*d^3*f^2 + a^2*g^2*x*e^3 - (2*a*c*d*f*g*x - a^2*d*g^2)*e^2 + (c^2*d^2*f^2*x - 2*a*c*d^2*f*g)*e)*
sqrt(-(c*d*f - a*g*e)/g)*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)*sqrt(x*e + d)*g*sqrt(-(c*d*f - a*g*e)/g) + (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*(3*c^2*d^2*g^2*x^2 - 5*c^2*d^2*f*g*x + 15*c^2*d^2*f^2 + 23*a^2*g^2*e^2 + (11*a*c*d*g^2*x - 35*a*c*d*f*
g)*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(g^3*x*e + d*g^3), 2/15*(15*(c^2*d^3*f^2 + a^
2*g^2*x*e^3 - (2*a*c*d*f*g*x - a^2*d*g^2)*e^2 + (c^2*d^2*f^2*x - 2*a*c*d^2*f*g)*e)*sqrt((c*d*f - a*g*e)/g)*arc
tan(sqrt(x*e + d)*sqrt((c*d*f - a*g*e)/g)/sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)) + (3*c^2*d^2*g^2*x^2 -
5*c^2*d^2*f*g*x + 15*c^2*d^2*f^2 + 23*a^2*g^2*e^2 + (11*a*c*d*g^2*x - 35*a*c*d*f*g)*e)*sqrt(c*d^2*x + a*x*e^2
+ (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(g^3*x*e + d*g^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)**(5/2)/(g*x+f),x)

[Out]

Timed out

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Giac [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)^(5/2)/(g*x+f),x, algorithm="giac")

[Out]

Timed out

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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 (f+g\,x\right )\,{\left (d+e\,x\right )}^{5/2}} \,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)/((f + g*x)*(d + e*x)^(5/2)),x)

[Out]

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

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