3.90 \(\int \frac{\left (a+b x+c x^2\right )^{3/2}}{x^3 \left (d-f x^2\right )} \, dx\)

Optimal. Leaf size=614 \[ -\frac{a^{3/2} f \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{d^2}-\frac{b f \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{3/2} d^2}-\frac{\sqrt{a+b x+c x^2} \left (8 a c f+b^2 f+2 b c f x+8 c^2 d\right )}{8 c d^2}-\frac{b \left (12 a c f+b^2 (-f)+24 c^2 d\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{3/2} d^2}+\frac{f \left (8 a c+b^2+2 b c x\right ) \sqrt{a+b x+c x^2}}{8 c d^2}-\frac{3 \left (4 a c+b^2\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{8 \sqrt{a} d}-\frac{\left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 d^2 \sqrt{f}}+\frac{\left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2} \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 d^2 \sqrt{f}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{2 d x^2}-\frac{3 (b-2 c x) \sqrt{a+b x+c x^2}}{4 d x}+\frac{3 b \sqrt{c} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 d} \]

[Out]

(-3*(b - 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*d*x) + (f*(b^2 + 8*a*c + 2*b*c*x)*Sqrt
[a + b*x + c*x^2])/(8*c*d^2) - ((8*c^2*d + b^2*f + 8*a*c*f + 2*b*c*f*x)*Sqrt[a +
b*x + c*x^2])/(8*c*d^2) - (a + b*x + c*x^2)^(3/2)/(2*d*x^2) - (3*(b^2 + 4*a*c)*
ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(8*Sqrt[a]*d) - (a^(3/2)
*f*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/d^2 + (3*b*Sqrt[c]*Ar
cTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*d) - (b*(b^2 - 12*a*c)*
f*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*d^2) - (b*
(24*c^2*d - b^2*f + 12*a*c*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^
2])])/(16*c^(3/2)*d^2) - ((c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(b*Sqrt[
d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f]
+ a*f]*Sqrt[a + b*x + c*x^2])])/(2*d^2*Sqrt[f]) + ((c*d + b*Sqrt[d]*Sqrt[f] + a*
f)^(3/2)*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt
[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d^2*Sqrt[f])

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Rubi [A]  time = 3.19505, antiderivative size = 614, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 13, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.464 \[ -\frac{a^{3/2} f \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{d^2}-\frac{b f \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{3/2} d^2}-\frac{\sqrt{a+b x+c x^2} \left (8 a c f+b^2 f+2 b c f x+8 c^2 d\right )}{8 c d^2}-\frac{b \left (12 a c f+b^2 (-f)+24 c^2 d\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{3/2} d^2}+\frac{f \left (8 a c+b^2+2 b c x\right ) \sqrt{a+b x+c x^2}}{8 c d^2}-\frac{3 \left (4 a c+b^2\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{8 \sqrt{a} d}-\frac{\left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 d^2 \sqrt{f}}+\frac{\left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2} \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 d^2 \sqrt{f}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{2 d x^2}-\frac{3 (b-2 c x) \sqrt{a+b x+c x^2}}{4 d x}+\frac{3 b \sqrt{c} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(3/2)/(x^3*(d - f*x^2)),x]

[Out]

(-3*(b - 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*d*x) + (f*(b^2 + 8*a*c + 2*b*c*x)*Sqrt
[a + b*x + c*x^2])/(8*c*d^2) - ((8*c^2*d + b^2*f + 8*a*c*f + 2*b*c*f*x)*Sqrt[a +
b*x + c*x^2])/(8*c*d^2) - (a + b*x + c*x^2)^(3/2)/(2*d*x^2) - (3*(b^2 + 4*a*c)*
ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(8*Sqrt[a]*d) - (a^(3/2)
*f*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/d^2 + (3*b*Sqrt[c]*Ar
cTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*d) - (b*(b^2 - 12*a*c)*
f*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*d^2) - (b*
(24*c^2*d - b^2*f + 12*a*c*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^
2])])/(16*c^(3/2)*d^2) - ((c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(b*Sqrt[
d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f]
+ a*f]*Sqrt[a + b*x + c*x^2])])/(2*d^2*Sqrt[f]) + ((c*d + b*Sqrt[d]*Sqrt[f] + a*
f)^(3/2)*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt
[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d^2*Sqrt[f])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(3/2)/x**3/(-f*x**2+d),x)

[Out]

Timed out

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Mathematica [A]  time = 0.97355, size = 429, normalized size = 0.7 \[ \frac{\frac{\log (x) \left (4 a (2 a f+3 c d)+3 b^2 d\right )}{\sqrt{a}}-\frac{\left (4 a (2 a f+3 c d)+3 b^2 d\right ) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )}{\sqrt{a}}-\frac{4 \log \left (\sqrt{d} \sqrt{f}+f x\right ) \left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2}}{\sqrt{f}}+\frac{4 \left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2} \log \left (\sqrt{d} \left (2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}+2 a \sqrt{f}-b \sqrt{d}+b \sqrt{f} x-2 c \sqrt{d} x\right )\right )}{\sqrt{f}}-\frac{4 \log \left (\sqrt{d} \sqrt{f}-f x\right ) \left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2}}{\sqrt{f}}+\frac{4 \left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2} \log \left (\sqrt{d} \left (2 \left (\sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}+a \sqrt{f}+c \sqrt{d} x\right )+b \left (\sqrt{d}+\sqrt{f} x\right )\right )\right )}{\sqrt{f}}-\frac{2 d (2 a+5 b x) \sqrt{a+x (b+c x)}}{x^2}}{8 d^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(3/2)/(x^3*(d - f*x^2)),x]

[Out]

((-2*d*(2*a + 5*b*x)*Sqrt[a + x*(b + c*x)])/x^2 + ((3*b^2*d + 4*a*(3*c*d + 2*a*f
))*Log[x])/Sqrt[a] - (4*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*Log[Sqrt[d]*Sqrt[f
] - f*x])/Sqrt[f] - (4*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*Log[Sqrt[d]*Sqrt[f]
+ f*x])/Sqrt[f] - ((3*b^2*d + 4*a*(3*c*d + 2*a*f))*Log[2*a + b*x + 2*Sqrt[a]*Sq
rt[a + x*(b + c*x)]])/Sqrt[a] + (4*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*Log[Sqr
t[d]*(-(b*Sqrt[d]) + 2*a*Sqrt[f] - 2*c*Sqrt[d]*x + b*Sqrt[f]*x + 2*Sqrt[c*d - b*
Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)])])/Sqrt[f] + (4*(c*d + b*Sqrt[d]*Sq
rt[f] + a*f)^(3/2)*Log[Sqrt[d]*(b*(Sqrt[d] + Sqrt[f]*x) + 2*(a*Sqrt[f] + c*Sqrt[
d]*x + Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + x*(b + c*x)]))])/Sqrt[f])/(8
*d^2)

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Maple [B]  time = 0.029, size = 5056, normalized size = 8.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(3/2)/x^3/(-f*x^2+d),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}{{\left (f x^{2} - d\right )} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(c*x^2 + b*x + a)^(3/2)/((f*x^2 - d)*x^3),x, algorithm="maxima")

[Out]

-integrate((c*x^2 + b*x + a)^(3/2)/((f*x^2 - d)*x^3), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(c*x^2 + b*x + a)^(3/2)/((f*x^2 - d)*x^3),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{a \sqrt{a + b x + c x^{2}}}{- d x^{3} + f x^{5}}\, dx - \int \frac{b x \sqrt{a + b x + c x^{2}}}{- d x^{3} + f x^{5}}\, dx - \int \frac{c x^{2} \sqrt{a + b x + c x^{2}}}{- d x^{3} + f x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**(3/2)/x**3/(-f*x**2+d),x)

[Out]

-Integral(a*sqrt(a + b*x + c*x**2)/(-d*x**3 + f*x**5), x) - Integral(b*x*sqrt(a
+ b*x + c*x**2)/(-d*x**3 + f*x**5), x) - Integral(c*x**2*sqrt(a + b*x + c*x**2)/
(-d*x**3 + f*x**5), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(c*x^2 + b*x + a)^(3/2)/((f*x^2 - d)*x^3),x, algorithm="giac")

[Out]

Exception raised: RuntimeError