3.853 \(\int \frac{1}{(d+e x)^{3/2} \sqrt{f+g x} \left (a+b x+c x^2\right )} \, dx\)

Optimal. Leaf size=429 \[ -\frac{8 c^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{2 c f-g \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{f+g x} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{b^2-4 a c} \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )^{3/2} \sqrt{2 c f-g \left (b-\sqrt{b^2-4 a c}\right )}}+\frac{8 c^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{f+g x} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{b^2-4 a c} \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )^{3/2} \sqrt{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{4 c e \sqrt{f+g x}}{\sqrt{b^2-4 a c} \sqrt{d+e x} (e f-d g) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}-\frac{4 c e \sqrt{f+g x}}{\sqrt{b^2-4 a c} \sqrt{d+e x} (e f-d g) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )} \]

[Out]

(4*c*e*Sqrt[f + g*x])/(Sqrt[b^2 - 4*a*c]*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*(e*
f - d*g)*Sqrt[d + e*x]) - (4*c*e*Sqrt[f + g*x])/(Sqrt[b^2 - 4*a*c]*(2*c*d - (b +
Sqrt[b^2 - 4*a*c])*e)*(e*f - d*g)*Sqrt[d + e*x]) - (8*c^2*ArcTanh[(Sqrt[2*c*f -
(b - Sqrt[b^2 - 4*a*c])*g]*Sqrt[d + e*x])/(Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])
*e]*Sqrt[f + g*x])])/(Sqrt[b^2 - 4*a*c]*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)^(3/2
)*Sqrt[2*c*f - (b - Sqrt[b^2 - 4*a*c])*g]) + (8*c^2*ArcTanh[(Sqrt[2*c*f - (b + S
qrt[b^2 - 4*a*c])*g]*Sqrt[d + e*x])/(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*Sqr
t[f + g*x])])/(Sqrt[b^2 - 4*a*c]*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)^(3/2)*Sqrt[
2*c*f - (b + Sqrt[b^2 - 4*a*c])*g])

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Rubi [A]  time = 3.35416, antiderivative size = 429, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129 \[ -\frac{8 c^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{2 c f-g \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{f+g x} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{b^2-4 a c} \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )^{3/2} \sqrt{2 c f-g \left (b-\sqrt{b^2-4 a c}\right )}}+\frac{8 c^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{f+g x} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{b^2-4 a c} \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )^{3/2} \sqrt{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{4 c e \sqrt{f+g x}}{\sqrt{b^2-4 a c} \sqrt{d+e x} (e f-d g) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}-\frac{4 c e \sqrt{f+g x}}{\sqrt{b^2-4 a c} \sqrt{d+e x} (e f-d g) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(3/2)*Sqrt[f + g*x]*(a + b*x + c*x^2)),x]

[Out]

(4*c*e*Sqrt[f + g*x])/(Sqrt[b^2 - 4*a*c]*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*(e*
f - d*g)*Sqrt[d + e*x]) - (4*c*e*Sqrt[f + g*x])/(Sqrt[b^2 - 4*a*c]*(2*c*d - (b +
Sqrt[b^2 - 4*a*c])*e)*(e*f - d*g)*Sqrt[d + e*x]) - (8*c^2*ArcTanh[(Sqrt[2*c*f -
(b - Sqrt[b^2 - 4*a*c])*g]*Sqrt[d + e*x])/(Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])
*e]*Sqrt[f + g*x])])/(Sqrt[b^2 - 4*a*c]*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)^(3/2
)*Sqrt[2*c*f - (b - Sqrt[b^2 - 4*a*c])*g]) + (8*c^2*ArcTanh[(Sqrt[2*c*f - (b + S
qrt[b^2 - 4*a*c])*g]*Sqrt[d + e*x])/(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*Sqr
t[f + g*x])])/(Sqrt[b^2 - 4*a*c]*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)^(3/2)*Sqrt[
2*c*f - (b + Sqrt[b^2 - 4*a*c])*g])

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

[Out]

Timed out

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Mathematica [B]  time = 2.84465, size = 1011, normalized size = 2.36 \[ -\frac{2 \sqrt{f+g x} e^2}{\left (c d^2+e (a e-b d)\right ) (e f-d g) \sqrt{d+e x}}+\frac{c \left (\left (b+\sqrt{b^2-4 a c}\right ) e-2 c d\right ) \log \left (-b-2 c x+\sqrt{b^2-4 a c}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \left (e (b d-a e)-c d^2\right ) \sqrt{2 d f c^2+\left (\sqrt{b^2-4 a c} e f+\sqrt{b^2-4 a c} d g-2 a e g-b (e f+d g)\right ) c+b \left (b-\sqrt{b^2-4 a c}\right ) e g}}+\frac{c \left (2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e\right ) \log \left (b+2 c x+\sqrt{b^2-4 a c}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \left (e (b d-a e)-c d^2\right ) \sqrt{2 d f c^2-\left (b e f+\sqrt{b^2-4 a c} e f+b d g+\sqrt{b^2-4 a c} d g+2 a e g\right ) c+b \left (b+\sqrt{b^2-4 a c}\right ) e g}}-\frac{c \left (2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e\right ) \log \left (b e f+\sqrt{b^2-4 a c} e f+b d g+\sqrt{b^2-4 a c} d g+2 b e g x+2 \sqrt{b^2-4 a c} e g x-2 c (2 d f+e x f+d g x)-2 \sqrt{4 d f c^2-2 \left (b e f+\sqrt{b^2-4 a c} e f+b d g+\sqrt{b^2-4 a c} d g+2 a e g\right ) c+2 b \left (b+\sqrt{b^2-4 a c}\right ) e g} \sqrt{d+e x} \sqrt{f+g x}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \left (e (b d-a e)-c d^2\right ) \sqrt{2 d f c^2-\left (b e f+\sqrt{b^2-4 a c} e f+b d g+\sqrt{b^2-4 a c} d g+2 a e g\right ) c+b \left (b+\sqrt{b^2-4 a c}\right ) e g}}-\frac{c \left (\left (b+\sqrt{b^2-4 a c}\right ) e-2 c d\right ) \log \left (-b e f+\sqrt{b^2-4 a c} e f-b d g+\sqrt{b^2-4 a c} d g-2 b e g x+2 \sqrt{b^2-4 a c} e g x+2 c (2 d f+e x f+d g x)+2 \sqrt{2} \sqrt{2 d f c^2+\left (\sqrt{b^2-4 a c} e f+\sqrt{b^2-4 a c} d g-2 a e g-b (e f+d g)\right ) c+b \left (b-\sqrt{b^2-4 a c}\right ) e g} \sqrt{d+e x} \sqrt{f+g x}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \left (e (b d-a e)-c d^2\right ) \sqrt{2 d f c^2+\left (\sqrt{b^2-4 a c} e f+\sqrt{b^2-4 a c} d g-2 a e g-b (e f+d g)\right ) c+b \left (b-\sqrt{b^2-4 a c}\right ) e g}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(3/2)*Sqrt[f + g*x]*(a + b*x + c*x^2)),x]

[Out]

(-2*e^2*Sqrt[f + g*x])/((c*d^2 + e*(-(b*d) + a*e))*(e*f - d*g)*Sqrt[d + e*x]) +
(c*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x])/(Sq
rt[2]*Sqrt[b^2 - 4*a*c]*(-(c*d^2) + e*(b*d - a*e))*Sqrt[2*c^2*d*f + b*(b - Sqrt[
b^2 - 4*a*c])*e*g + c*(Sqrt[b^2 - 4*a*c]*e*f + Sqrt[b^2 - 4*a*c]*d*g - 2*a*e*g -
b*(e*f + d*g))]) + (c*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*Log[b + Sqrt[b^2 - 4
*a*c] + 2*c*x])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*(-(c*d^2) + e*(b*d - a*e))*Sqrt[2*c^2
*d*f + b*(b + Sqrt[b^2 - 4*a*c])*e*g - c*(b*e*f + Sqrt[b^2 - 4*a*c]*e*f + b*d*g
+ Sqrt[b^2 - 4*a*c]*d*g + 2*a*e*g)]) - (c*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*L
og[b*e*f + Sqrt[b^2 - 4*a*c]*e*f + b*d*g + Sqrt[b^2 - 4*a*c]*d*g + 2*b*e*g*x + 2
*Sqrt[b^2 - 4*a*c]*e*g*x - 2*Sqrt[4*c^2*d*f + 2*b*(b + Sqrt[b^2 - 4*a*c])*e*g -
2*c*(b*e*f + Sqrt[b^2 - 4*a*c]*e*f + b*d*g + Sqrt[b^2 - 4*a*c]*d*g + 2*a*e*g)]*S
qrt[d + e*x]*Sqrt[f + g*x] - 2*c*(2*d*f + e*f*x + d*g*x)])/(Sqrt[2]*Sqrt[b^2 - 4
*a*c]*(-(c*d^2) + e*(b*d - a*e))*Sqrt[2*c^2*d*f + b*(b + Sqrt[b^2 - 4*a*c])*e*g
- c*(b*e*f + Sqrt[b^2 - 4*a*c]*e*f + b*d*g + Sqrt[b^2 - 4*a*c]*d*g + 2*a*e*g)])
- (c*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*Log[-(b*e*f) + Sqrt[b^2 - 4*a*c]*e*f -
b*d*g + Sqrt[b^2 - 4*a*c]*d*g - 2*b*e*g*x + 2*Sqrt[b^2 - 4*a*c]*e*g*x + 2*Sqrt[
2]*Sqrt[2*c^2*d*f + b*(b - Sqrt[b^2 - 4*a*c])*e*g + c*(Sqrt[b^2 - 4*a*c]*e*f + S
qrt[b^2 - 4*a*c]*d*g - 2*a*e*g - b*(e*f + d*g))]*Sqrt[d + e*x]*Sqrt[f + g*x] + 2
*c*(2*d*f + e*f*x + d*g*x)])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*(-(c*d^2) + e*(b*d - a*e
))*Sqrt[2*c^2*d*f + b*(b - Sqrt[b^2 - 4*a*c])*e*g + c*(Sqrt[b^2 - 4*a*c]*e*f + S
qrt[b^2 - 4*a*c]*d*g - 2*a*e*g - b*(e*f + d*g))])

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Maple [B]  time = 0.266, size = 47349, normalized size = 110.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x + a\right )}{\left (e x + d\right )}^{\frac{3}{2}} \sqrt{g x + f}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)*(e*x + d)^(3/2)*sqrt(g*x + f)),x, algorithm="maxima")

[Out]

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

[Out]

Timed out