3.614 \(\int \frac{(d+e x)^{3/2}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, dx\)

Optimal. Leaf size=625 \[ \frac{2 \sqrt{d+e x} (e f-d g)}{\sqrt{f+g x} \left (a g^2+c f^2\right )}-\frac{2 \sqrt{e} (e f-d g) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{\sqrt{g} \left (a g^2+c f^2\right )}-\frac{\sqrt{e} \left (-\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{\sqrt{-a} \sqrt{c} \sqrt{g} \left (a g^2+c f^2\right )}+\frac{\sqrt{e} \left (\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{\sqrt{-a} \sqrt{c} \sqrt{g} \left (a g^2+c f^2\right )}+\frac{\sqrt{\sqrt{c} d-\sqrt{-a} e} \left (-\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{-a} \sqrt{c} \sqrt{\sqrt{c} f-\sqrt{-a} g} \left (a g^2+c f^2\right )}-\frac{\sqrt{\sqrt{-a} e+\sqrt{c} d} \left (\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{-a} \sqrt{c} \sqrt{\sqrt{-a} g+\sqrt{c} f} \left (a g^2+c f^2\right )} \]

[Out]

(2*(e*f - d*g)*Sqrt[d + e*x])/((c*f^2 + a*g^2)*Sqrt[f + g*x]) - (2*Sqrt[e]*(e*f
- d*g)*ArcTanh[(Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/(Sqrt[g]*(c*f^2
+ a*g^2)) - (Sqrt[e]*(c*d*f + a*e*g - Sqrt[-a]*Sqrt[c]*(e*f - d*g))*ArcTanh[(Sq
rt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/(Sqrt[-a]*Sqrt[c]*Sqrt[g]*(c*f^2
+ a*g^2)) + (Sqrt[e]*(c*d*f + a*e*g + Sqrt[-a]*Sqrt[c]*(e*f - d*g))*ArcTanh[(Sqr
t[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/(Sqrt[-a]*Sqrt[c]*Sqrt[g]*(c*f^2 +
a*g^2)) + (Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*(c*d*f + a*e*g - Sqrt[-a]*Sqrt[c]*(e*f
- d*g))*ArcTanh[(Sqrt[Sqrt[c]*f - Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d - S
qrt[-a]*e]*Sqrt[f + g*x])])/(Sqrt[-a]*Sqrt[c]*Sqrt[Sqrt[c]*f - Sqrt[-a]*g]*(c*f^
2 + a*g^2)) - (Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*(c*d*f + a*e*g + Sqrt[-a]*Sqrt[c]*(e
*f - d*g))*ArcTanh[(Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d
+ Sqrt[-a]*e]*Sqrt[f + g*x])])/(Sqrt[-a]*Sqrt[c]*Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*(c
*f^2 + a*g^2))

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Rubi [A]  time = 4.86496, antiderivative size = 625, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{2 \sqrt{d+e x} (e f-d g)}{\sqrt{f+g x} \left (a g^2+c f^2\right )}-\frac{2 \sqrt{e} (e f-d g) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{\sqrt{g} \left (a g^2+c f^2\right )}-\frac{\sqrt{e} \left (-\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{\sqrt{-a} \sqrt{c} \sqrt{g} \left (a g^2+c f^2\right )}+\frac{\sqrt{e} \left (\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{\sqrt{-a} \sqrt{c} \sqrt{g} \left (a g^2+c f^2\right )}+\frac{\sqrt{\sqrt{c} d-\sqrt{-a} e} \left (-\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{-a} \sqrt{c} \sqrt{\sqrt{c} f-\sqrt{-a} g} \left (a g^2+c f^2\right )}-\frac{\sqrt{\sqrt{-a} e+\sqrt{c} d} \left (\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{-a} \sqrt{c} \sqrt{\sqrt{-a} g+\sqrt{c} f} \left (a g^2+c f^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(2*(e*f - d*g)*Sqrt[d + e*x])/((c*f^2 + a*g^2)*Sqrt[f + g*x]) - (2*Sqrt[e]*(e*f
- d*g)*ArcTanh[(Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/(Sqrt[g]*(c*f^2
+ a*g^2)) - (Sqrt[e]*(c*d*f + a*e*g - Sqrt[-a]*Sqrt[c]*(e*f - d*g))*ArcTanh[(Sq
rt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/(Sqrt[-a]*Sqrt[c]*Sqrt[g]*(c*f^2
+ a*g^2)) + (Sqrt[e]*(c*d*f + a*e*g + Sqrt[-a]*Sqrt[c]*(e*f - d*g))*ArcTanh[(Sqr
t[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/(Sqrt[-a]*Sqrt[c]*Sqrt[g]*(c*f^2 +
a*g^2)) + (Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*(c*d*f + a*e*g - Sqrt[-a]*Sqrt[c]*(e*f
- d*g))*ArcTanh[(Sqrt[Sqrt[c]*f - Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d - S
qrt[-a]*e]*Sqrt[f + g*x])])/(Sqrt[-a]*Sqrt[c]*Sqrt[Sqrt[c]*f - Sqrt[-a]*g]*(c*f^
2 + a*g^2)) - (Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*(c*d*f + a*e*g + Sqrt[-a]*Sqrt[c]*(e
*f - d*g))*ArcTanh[(Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d
+ Sqrt[-a]*e]*Sqrt[f + g*x])])/(Sqrt[-a]*Sqrt[c]*Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*(c
*f^2 + a*g^2))

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

[Out]

Timed out

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Mathematica [C]  time = 3.91013, size = 558, normalized size = 0.89 \[ \frac{\frac{\left (\sqrt{c} d-i \sqrt{a} e\right )^{3/2} \left (\sqrt{a} g-i \sqrt{c} f\right ) \log \left (-\frac{i \sqrt{a} \sqrt{c} \sqrt{\sqrt{c} f-i \sqrt{a} g} \left (2 \sqrt{d+e x} \sqrt{f+g x} \sqrt{\sqrt{c} d-i \sqrt{a} e} \sqrt{\sqrt{c} f-i \sqrt{a} g}-i \sqrt{a} (d g+e f+2 e g x)+\sqrt{c} (2 d f+d g x+e f x)\right )}{\left (\sqrt{c} x+i \sqrt{a}\right ) \left (\sqrt{c} d-i \sqrt{a} e\right )^{5/2}}\right )}{\sqrt{a} \sqrt{c} \sqrt{\sqrt{c} f-i \sqrt{a} g}}+\frac{\left (\sqrt{c} d+i \sqrt{a} e\right )^{3/2} \left (\sqrt{a} g+i \sqrt{c} f\right ) \log \left (\frac{i \sqrt{a} \sqrt{c} \sqrt{\sqrt{c} f+i \sqrt{a} g} \left (2 \sqrt{d+e x} \sqrt{f+g x} \sqrt{\sqrt{c} d+i \sqrt{a} e} \sqrt{\sqrt{c} f+i \sqrt{a} g}+i \sqrt{a} (d g+e f+2 e g x)+\sqrt{c} (2 d f+d g x+e f x)\right )}{\left (\sqrt{c} x-i \sqrt{a}\right ) \left (\sqrt{c} d+i \sqrt{a} e\right )^{5/2}}\right )}{\sqrt{a} \sqrt{c} \sqrt{\sqrt{c} f+i \sqrt{a} g}}+\frac{4 \sqrt{d+e x} (e f-d g)}{\sqrt{f+g x}}}{2 \left (a g^2+c f^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.108, size = 8264, normalized size = 13.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out