3.8.89 \(\int \frac {(d+e x)^{3/2}}{(f+g x)^2 \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [789]

Optimal. Leaf size=170 \[ -\frac {(e f-d g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g (c d f-a e g) \sqrt {d+e x} (f+g x)}-\frac {\left (2 a e^2 g-c d (e f+d g)\right ) \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{g^{3/2} (c d f-a e g)^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.16, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {892, 888, 211} \begin {gather*} -\frac {\left (2 a e^2 g-c d (d g+e f)\right ) \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^{3/2} (c d f-a e g)^{3/2}}-\frac {(e f-d g) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt {d+e x} (f+g x) (c d f-a e g)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(3/2)/((f + g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]

[Out]

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

Rule 892

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Simp[e^2*(e*f - d*g)*(d + e*x)^(m - 2)*(f + g*x)^(n + 1)*((a + b*x + c*x^2)^(p + 1)/(g*(n + 1)*(c*e*f + c*d*g
- b*e*g))), x] - Dist[e*((b*e*g*(n + 1) + c*e*f*(p + 1) - c*d*g*(2*n + p + 3))/(g*(n + 1)*(c*e*f + c*d*g - b*e
*g))), Int[(d + e*x)^(m - 1)*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p
}, 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 - 1, 0] && LtQ[n, -1] && IntegerQ[2*p]

Rubi steps

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

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Mathematica [A]
time = 0.49, size = 154, normalized size = 0.91 \begin {gather*} \frac {\sqrt {d+e x} \left (-\frac {\sqrt {g} (-e f+d g) (a e+c d x)}{(-c d f+a e g) (f+g x)}+\frac {\left (-2 a e^2 g+c d (e f+d g)\right ) \sqrt {a e+c d x} \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )}{(c d f-a e g)^{3/2}}\right )}{g^{3/2} \sqrt {(a e+c d x) (d+e x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(3/2)/((f + g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]

[Out]

(Sqrt[d + e*x]*(-((Sqrt[g]*(-(e*f) + d*g)*(a*e + c*d*x))/((-(c*d*f) + a*e*g)*(f + g*x))) + ((-2*a*e^2*g + c*d*
(e*f + d*g))*Sqrt[a*e + c*d*x]*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d*f - a*e*g]])/(c*d*f - a*e*g)^(3/2))
)/(g^(3/2)*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. \(336\) vs. \(2(154)=308\).
time = 0.14, size = 337, normalized size = 1.98

method	result	size
default	\(\frac {\left (-2 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a \,e^{2} g^{2} x +\arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c \,d^{2} g^{2} x +\arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c d e f g x -2 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a \,e^{2} f g +\arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c \,d^{2} f g +\arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c d e \,f^{2}-\sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, d g +\sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, e f \right ) \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}}{\sqrt {e x +d}\, \sqrt {c d x +a e}\, \left (a e g -c d f \right ) g \left (g x +f \right ) \sqrt {\left (a e g -c d f \right ) g}}\)	\(337\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)/(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

[Out]

integrate((x*e + d)^(3/2)/(sqrt(c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 441 vs. \(2 (159) = 318\).
time = 2.65, size = 921, normalized size = 5.42 \begin {gather*} \left [-\frac {{\left (c d^{3} g^{2} x + c d^{3} f g - 2 \, {\left (a g^{2} x^{2} + a f g x\right )} e^{3} + {\left (c d f g x^{2} - 2 \, a d f g + {\left (c d f^{2} - 2 \, a d g^{2}\right )} x\right )} e^{2} + {\left (c d^{2} g^{2} x^{2} + 2 \, c d^{2} f g x + c d^{2} f^{2}\right )} e\right )} \sqrt {-c d f g + a g^{2} e} \log \left (-\frac {c d^{2} g x - c d^{2} f + 2 \, a g x e^{2} + {\left (c d g x^{2} - c d f x + 2 \, a d g\right )} e - 2 \, \sqrt {-c d f g + a g^{2} e} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{d g x + d f + {\left (g x^{2} + f x\right )} e}\right ) - 2 \, {\left (c d^{2} f g^{2} + a f g^{2} e^{2} - {\left (c d f^{2} g + a d g^{3}\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}}{2 \, {\left (c^{2} d^{3} f^{2} g^{3} x + c^{2} d^{3} f^{3} g^{2} + {\left (a^{2} g^{5} x^{2} + a^{2} f g^{4} x\right )} e^{3} - {\left (2 \, a c d f g^{4} x^{2} - a^{2} d f g^{4} + {\left (2 \, a c d f^{2} g^{3} - a^{2} d g^{5}\right )} x\right )} e^{2} + {\left (c^{2} d^{2} f^{2} g^{3} x^{2} - 2 \, a c d^{2} f^{2} g^{3} + {\left (c^{2} d^{2} f^{3} g^{2} - 2 \, a c d^{2} f g^{4}\right )} x\right )} e\right )}}, -\frac {{\left (c d^{3} g^{2} x + c d^{3} f g - 2 \, {\left (a g^{2} x^{2} + a f g x\right )} e^{3} + {\left (c d f g x^{2} - 2 \, a d f g + {\left (c d f^{2} - 2 \, a d g^{2}\right )} x\right )} e^{2} + {\left (c d^{2} g^{2} x^{2} + 2 \, c d^{2} f g x + c d^{2} f^{2}\right )} e\right )} \sqrt {c d f g - a g^{2} e} \arctan \left (\frac {\sqrt {c d f g - a g^{2} e} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{c d^{2} g x + a g x e^{2} + {\left (c d g x^{2} + a d g\right )} e}\right ) - {\left (c d^{2} f g^{2} + a f g^{2} e^{2} - {\left (c d f^{2} g + a d g^{3}\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}}{c^{2} d^{3} f^{2} g^{3} x + c^{2} d^{3} f^{3} g^{2} + {\left (a^{2} g^{5} x^{2} + a^{2} f g^{4} x\right )} e^{3} - {\left (2 \, a c d f g^{4} x^{2} - a^{2} d f g^{4} + {\left (2 \, a c d f^{2} g^{3} - a^{2} d g^{5}\right )} x\right )} e^{2} + {\left (c^{2} d^{2} f^{2} g^{3} x^{2} - 2 \, a c d^{2} f^{2} g^{3} + {\left (c^{2} d^{2} f^{3} g^{2} - 2 \, a c d^{2} f g^{4}\right )} x\right )} e}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{\frac {3}{2}}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (f + g x\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(3/2)/(g*x+f)**2/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Integral((d + e*x)**(3/2)/(sqrt((d + e*x)*(a*e + c*d*x))*(f + g*x)**2), x)

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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((e*x+d)^(3/2)/(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^{3/2}}{{\left (f+g\,x\right )}^2\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(3/2)/((f + g*x)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)),x)

[Out]

int((d + e*x)^(3/2)/((f + g*x)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)), x)

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