3.839 \(\int \frac{a+b x+c x^2}{(d+e x)^{5/2} \sqrt{f+g x}} \, dx\)

Optimal. Leaf size=160 \[ \frac{2 \sqrt{f+g x} \left (c \left (6 d e f-4 d^2 g\right )-e (-2 a e g-b d g+3 b e f)\right )}{3 e^2 \sqrt{d+e x} (e f-d g)^2}-\frac{2 \sqrt{f+g x} \left (a+\frac{d (c d-b e)}{e^2}\right )}{3 (d+e x)^{3/2} (e f-d g)}+\frac{2 c \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{e^{5/2} \sqrt{g}} \]

[Out]

(-2*(a + (d*(c*d - b*e))/e^2)*Sqrt[f + g*x])/(3*(e*f - d*g)*(d + e*x)^(3/2)) + (
2*(c*(6*d*e*f - 4*d^2*g) - e*(3*b*e*f - b*d*g - 2*a*e*g))*Sqrt[f + g*x])/(3*e^2*
(e*f - d*g)^2*Sqrt[d + e*x]) + (2*c*ArcTanh[(Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqr
t[f + g*x])])/(e^(5/2)*Sqrt[g])

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Rubi [A]  time = 0.426206, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ -\frac{2 \sqrt{f+g x} \left (a+\frac{d (c d-b e)}{e^2}\right )}{3 (d+e x)^{3/2} (e f-d g)}+\frac{2 \sqrt{f+g x} (2 c d (3 e f-2 d g)-e (-2 a e g-b d g+3 b e f))}{3 e^2 \sqrt{d+e x} (e f-d g)^2}+\frac{2 c \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{e^{5/2} \sqrt{g}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)/((d + e*x)^(5/2)*Sqrt[f + g*x]),x]

[Out]

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

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Rubi in Sympy [A]  time = 48.2009, size = 207, normalized size = 1.29 \[ \frac{2 c d^{2} \sqrt{f + g x}}{3 e^{2} \left (d + e x\right )^{\frac{3}{2}} \left (d g - e f\right )} - \frac{4 c d \sqrt{f + g x} \left (2 d g - 3 e f\right )}{3 e^{2} \sqrt{d + e x} \left (d g - e f\right )^{2}} + \frac{2 c \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{f + g x}}{\sqrt{g} \sqrt{d + e x}} \right )}}{e^{\frac{5}{2}} \sqrt{g}} + \frac{2 \sqrt{f + g x} \left (2 a e g + b d g - 3 b e f\right )}{3 e \sqrt{d + e x} \left (d g - e f\right )^{2}} + \frac{2 \sqrt{f + g x} \left (a e - b d\right )}{3 e \left (d + e x\right )^{\frac{3}{2}} \left (d g - e f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)/(e*x+d)**(5/2)/(g*x+f)**(1/2),x)

[Out]

2*c*d**2*sqrt(f + g*x)/(3*e**2*(d + e*x)**(3/2)*(d*g - e*f)) - 4*c*d*sqrt(f + g*
x)*(2*d*g - 3*e*f)/(3*e**2*sqrt(d + e*x)*(d*g - e*f)**2) + 2*c*atanh(sqrt(e)*sqr
t(f + g*x)/(sqrt(g)*sqrt(d + e*x)))/(e**(5/2)*sqrt(g)) + 2*sqrt(f + g*x)*(2*a*e*
g + b*d*g - 3*b*e*f)/(3*e*sqrt(d + e*x)*(d*g - e*f)**2) + 2*sqrt(f + g*x)*(a*e -
b*d)/(3*e*(d + e*x)**(3/2)*(d*g - e*f))

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Mathematica [A]  time = 0.346789, size = 157, normalized size = 0.98 \[ \frac{2 \sqrt{f+g x} \left (e^2 (a (3 d g-e f+2 e g x)+b (-2 d f+d g x-3 e f x))+c d \left (-3 d^2 g+d e (5 f-4 g x)+6 e^2 f x\right )\right )}{3 e^2 (d+e x)^{3/2} (e f-d g)^2}+\frac{c \log \left (2 \sqrt{e} \sqrt{g} \sqrt{d+e x} \sqrt{f+g x}+d g+e f+2 e g x\right )}{e^{5/2} \sqrt{g}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)/((d + e*x)^(5/2)*Sqrt[f + g*x]),x]

[Out]

(2*Sqrt[f + g*x]*(c*d*(-3*d^2*g + 6*e^2*f*x + d*e*(5*f - 4*g*x)) + e^2*(b*(-2*d*
f - 3*e*f*x + d*g*x) + a*(-(e*f) + 3*d*g + 2*e*g*x))))/(3*e^2*(e*f - d*g)^2*(d +
e*x)^(3/2)) + (c*Log[e*f + d*g + 2*e*g*x + 2*Sqrt[e]*Sqrt[g]*Sqrt[d + e*x]*Sqrt
[f + g*x]])/(e^(5/2)*Sqrt[g])

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Maple [B]  time = 0.038, size = 773, normalized size = 4.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)/(e*x+d)^(5/2)/(g*x+f)^(1/2),x)

[Out]

1/3*(g*x+f)^(1/2)*(3*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+e
*f)/(e*g)^(1/2))*x^2*c*d^2*e^2*g^2-6*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(
e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*x^2*c*d*e^3*f*g+3*ln(1/2*(2*e*g*x+2*((e*x+d)*(g
*x+f))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*x^2*c*e^4*f^2+6*ln(1/2*(2*e*g*x+2
*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*x*c*d^3*e*g^2-12*ln(1
/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*x*c*d^2*
e^2*f*g+6*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(
1/2))*x*c*d*e^3*f^2+3*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+
e*f)/(e*g)^(1/2))*c*d^4*g^2-6*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1
/2)+d*g+e*f)/(e*g)^(1/2))*c*d^3*e*f*g+3*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2
)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*d^2*e^2*f^2+4*x*a*e^3*g*((e*x+d)*(g*x+f))^
(1/2)*(e*g)^(1/2)+2*x*b*d*e^2*g*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)-6*x*b*e^3*f*
((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)-8*x*c*d^2*e*g*((e*x+d)*(g*x+f))^(1/2)*(e*g)^
(1/2)+12*x*c*d*e^2*f*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+6*a*d*e^2*g*((e*x+d)*(g
*x+f))^(1/2)*(e*g)^(1/2)-2*a*e^3*f*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)-4*b*d*e^2
*f*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)-6*c*d^3*g*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(
1/2)+10*c*d^2*e*f*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2))/(e*g)^(1/2)/(d*g-e*f)^2/(
(e*x+d)*(g*x+f))^(1/2)/e^2/(e*x+d)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/((e*x + d)^(5/2)*sqrt(g*x + f)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.62119, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left ({\left (5 \, c d^{2} e - 2 \, b d e^{2} - a e^{3}\right )} f - 3 \,{\left (c d^{3} - a d e^{2}\right )} g +{\left (3 \,{\left (2 \, c d e^{2} - b e^{3}\right )} f -{\left (4 \, c d^{2} e - b d e^{2} - 2 \, a e^{3}\right )} g\right )} x\right )} \sqrt{e g} \sqrt{e x + d} \sqrt{g x + f} + 3 \,{\left (c d^{2} e^{2} f^{2} - 2 \, c d^{3} e f g + c d^{4} g^{2} +{\left (c e^{4} f^{2} - 2 \, c d e^{3} f g + c d^{2} e^{2} g^{2}\right )} x^{2} + 2 \,{\left (c d e^{3} f^{2} - 2 \, c d^{2} e^{2} f g + c d^{3} e g^{2}\right )} x\right )} \log \left (4 \,{\left (2 \, e^{2} g^{2} x + e^{2} f g + d e g^{2}\right )} \sqrt{e x + d} \sqrt{g x + f} +{\left (8 \, e^{2} g^{2} x^{2} + e^{2} f^{2} + 6 \, d e f g + d^{2} g^{2} + 8 \,{\left (e^{2} f g + d e g^{2}\right )} x\right )} \sqrt{e g}\right )}{6 \,{\left (d^{2} e^{4} f^{2} - 2 \, d^{3} e^{3} f g + d^{4} e^{2} g^{2} +{\left (e^{6} f^{2} - 2 \, d e^{5} f g + d^{2} e^{4} g^{2}\right )} x^{2} + 2 \,{\left (d e^{5} f^{2} - 2 \, d^{2} e^{4} f g + d^{3} e^{3} g^{2}\right )} x\right )} \sqrt{e g}}, \frac{2 \,{\left ({\left (5 \, c d^{2} e - 2 \, b d e^{2} - a e^{3}\right )} f - 3 \,{\left (c d^{3} - a d e^{2}\right )} g +{\left (3 \,{\left (2 \, c d e^{2} - b e^{3}\right )} f -{\left (4 \, c d^{2} e - b d e^{2} - 2 \, a e^{3}\right )} g\right )} x\right )} \sqrt{-e g} \sqrt{e x + d} \sqrt{g x + f} + 3 \,{\left (c d^{2} e^{2} f^{2} - 2 \, c d^{3} e f g + c d^{4} g^{2} +{\left (c e^{4} f^{2} - 2 \, c d e^{3} f g + c d^{2} e^{2} g^{2}\right )} x^{2} + 2 \,{\left (c d e^{3} f^{2} - 2 \, c d^{2} e^{2} f g + c d^{3} e g^{2}\right )} x\right )} \arctan \left (\frac{{\left (2 \, e g x + e f + d g\right )} \sqrt{-e g}}{2 \, \sqrt{e x + d} \sqrt{g x + f} e g}\right )}{3 \,{\left (d^{2} e^{4} f^{2} - 2 \, d^{3} e^{3} f g + d^{4} e^{2} g^{2} +{\left (e^{6} f^{2} - 2 \, d e^{5} f g + d^{2} e^{4} g^{2}\right )} x^{2} + 2 \,{\left (d e^{5} f^{2} - 2 \, d^{2} e^{4} f g + d^{3} e^{3} g^{2}\right )} x\right )} \sqrt{-e g}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/((e*x + d)^(5/2)*sqrt(g*x + f)),x, algorithm="fricas")

[Out]

[1/6*(4*((5*c*d^2*e - 2*b*d*e^2 - a*e^3)*f - 3*(c*d^3 - a*d*e^2)*g + (3*(2*c*d*e
^2 - b*e^3)*f - (4*c*d^2*e - b*d*e^2 - 2*a*e^3)*g)*x)*sqrt(e*g)*sqrt(e*x + d)*sq
rt(g*x + f) + 3*(c*d^2*e^2*f^2 - 2*c*d^3*e*f*g + c*d^4*g^2 + (c*e^4*f^2 - 2*c*d*
e^3*f*g + c*d^2*e^2*g^2)*x^2 + 2*(c*d*e^3*f^2 - 2*c*d^2*e^2*f*g + c*d^3*e*g^2)*x
)*log(4*(2*e^2*g^2*x + e^2*f*g + d*e*g^2)*sqrt(e*x + d)*sqrt(g*x + f) + (8*e^2*g
^2*x^2 + e^2*f^2 + 6*d*e*f*g + d^2*g^2 + 8*(e^2*f*g + d*e*g^2)*x)*sqrt(e*g)))/((
d^2*e^4*f^2 - 2*d^3*e^3*f*g + d^4*e^2*g^2 + (e^6*f^2 - 2*d*e^5*f*g + d^2*e^4*g^2
)*x^2 + 2*(d*e^5*f^2 - 2*d^2*e^4*f*g + d^3*e^3*g^2)*x)*sqrt(e*g)), 1/3*(2*((5*c*
d^2*e - 2*b*d*e^2 - a*e^3)*f - 3*(c*d^3 - a*d*e^2)*g + (3*(2*c*d*e^2 - b*e^3)*f
- (4*c*d^2*e - b*d*e^2 - 2*a*e^3)*g)*x)*sqrt(-e*g)*sqrt(e*x + d)*sqrt(g*x + f) +
3*(c*d^2*e^2*f^2 - 2*c*d^3*e*f*g + c*d^4*g^2 + (c*e^4*f^2 - 2*c*d*e^3*f*g + c*d
^2*e^2*g^2)*x^2 + 2*(c*d*e^3*f^2 - 2*c*d^2*e^2*f*g + c*d^3*e*g^2)*x)*arctan(1/2*
(2*e*g*x + e*f + d*g)*sqrt(-e*g)/(sqrt(e*x + d)*sqrt(g*x + f)*e*g)))/((d^2*e^4*f
^2 - 2*d^3*e^3*f*g + d^4*e^2*g^2 + (e^6*f^2 - 2*d*e^5*f*g + d^2*e^4*g^2)*x^2 + 2
*(d*e^5*f^2 - 2*d^2*e^4*f*g + d^3*e^3*g^2)*x)*sqrt(-e*g))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{a + b x + c x^{2}}{\left (d + e x\right )^{\frac{5}{2}} \sqrt{f + g x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)/(e*x+d)**(5/2)/(g*x+f)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.600819, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/((e*x + d)^(5/2)*sqrt(g*x + f)),x, algorithm="giac")

[Out]

sage0*x