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

Optimal. Leaf size=249 \[ \frac{\sqrt{d+e x} \sqrt{e+f x} \left (4 e f \left (-2 a e f-b d f+3 b e^2\right )-c \left (-d^2 f^2-6 d e^2 f+15 e^4\right )\right )}{4 e f^3 \left (e^2-d f\right )}-\frac{\tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{d+e x}}{\sqrt{e} \sqrt{e+f x}}\right ) \left (4 e f \left (-2 a e f-b d f+3 b e^2\right )-c \left (-d^2 f^2-6 d e^2 f+15 e^4\right )\right )}{4 e^{3/2} f^{7/2}}+\frac{2 (d+e x)^{3/2} \left (a+\frac{e (c e-b f)}{f^2}\right )}{\left (e^2-d f\right ) \sqrt{e+f x}}+\frac{c (d+e x)^{3/2} \sqrt{e+f x}}{2 e f^2} \]

[Out]

(2*(a + (e*(c*e - b*f))/f^2)*(d + e*x)^(3/2))/((e^2 - d*f)*Sqrt[e + f*x]) + ((4*
e*f*(3*b*e^2 - b*d*f - 2*a*e*f) - c*(15*e^4 - 6*d*e^2*f - d^2*f^2))*Sqrt[d + e*x
]*Sqrt[e + f*x])/(4*e*f^3*(e^2 - d*f)) + (c*(d + e*x)^(3/2)*Sqrt[e + f*x])/(2*e*
f^2) - ((4*e*f*(3*b*e^2 - b*d*f - 2*a*e*f) - c*(15*e^4 - 6*d*e^2*f - d^2*f^2))*A
rcTanh[(Sqrt[f]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[e + f*x])])/(4*e^(3/2)*f^(7/2))

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Rubi [A]  time = 0.606924, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ \frac{\sqrt{d+e x} \sqrt{e+f x} \left (4 e f \left (-2 a e f-b d f+3 b e^2\right )-c \left (-d^2 f^2-6 d e^2 f+15 e^4\right )\right )}{4 e f^3 \left (e^2-d f\right )}-\frac{\tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{d+e x}}{\sqrt{e} \sqrt{e+f x}}\right ) \left (4 e f \left (-2 a e f-b d f+3 b e^2\right )-c \left (-d^2 f^2-6 d e^2 f+15 e^4\right )\right )}{4 e^{3/2} f^{7/2}}+\frac{2 (d+e x)^{3/2} \left (a+\frac{e (c e-b f)}{f^2}\right )}{\left (e^2-d f\right ) \sqrt{e+f x}}+\frac{c (d+e x)^{3/2} \sqrt{e+f x}}{2 e f^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + (e*(c*e - b*f))/f^2)*(d + e*x)^(3/2))/((e^2 - d*f)*Sqrt[e + f*x]) + ((4*
e*f*(3*b*e^2 - b*d*f - 2*a*e*f) - c*(15*e^4 - 6*d*e^2*f - d^2*f^2))*Sqrt[d + e*x
]*Sqrt[e + f*x])/(4*e*f^3*(e^2 - d*f)) + (c*(d + e*x)^(3/2)*Sqrt[e + f*x])/(2*e*
f^2) - ((4*e*f*(3*b*e^2 - b*d*f - 2*a*e*f) - c*(15*e^4 - 6*d*e^2*f - d^2*f^2))*A
rcTanh[(Sqrt[f]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[e + f*x])])/(4*e^(3/2)*f^(7/2))

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Rubi in Sympy [A]  time = 69.5464, size = 311, normalized size = 1.25 \[ \frac{2 c e^{2} \left (d + e x\right )^{\frac{3}{2}}}{f^{2} \sqrt{e + f x} \left (- d f + e^{2}\right )} + \frac{c \left (d + e x\right )^{\frac{3}{2}} \sqrt{e + f x}}{2 e f^{2}} - \frac{c \sqrt{d + e x} \sqrt{e + f x} \left (- d^{2} f^{2} - 6 d e^{2} f + 15 e^{4}\right )}{4 e f^{3} \left (- d f + e^{2}\right )} + \frac{c \left (- d^{2} f^{2} - 6 d e^{2} f + 15 e^{4}\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{e + f x}}{\sqrt{f} \sqrt{d + e x}} \right )}}{4 e^{\frac{3}{2}} f^{\frac{7}{2}}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (a f - b e\right )}{f \sqrt{e + f x} \left (- d f + e^{2}\right )} + \frac{2 \sqrt{d + e x} \sqrt{e + f x} \left (- a e f + \frac{b \left (- d f + 3 e^{2}\right )}{2}\right )}{f^{2} \left (- d f + e^{2}\right )} - \frac{\left (- 2 a e f - b d f + 3 b e^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{e + f x}}{\sqrt{f} \sqrt{d + e x}} \right )}}{\sqrt{e} f^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c*e**2*(d + e*x)**(3/2)/(f**2*sqrt(e + f*x)*(-d*f + e**2)) + c*(d + e*x)**(3/2
)*sqrt(e + f*x)/(2*e*f**2) - c*sqrt(d + e*x)*sqrt(e + f*x)*(-d**2*f**2 - 6*d*e**
2*f + 15*e**4)/(4*e*f**3*(-d*f + e**2)) + c*(-d**2*f**2 - 6*d*e**2*f + 15*e**4)*
atanh(sqrt(e)*sqrt(e + f*x)/(sqrt(f)*sqrt(d + e*x)))/(4*e**(3/2)*f**(7/2)) + 2*(
d + e*x)**(3/2)*(a*f - b*e)/(f*sqrt(e + f*x)*(-d*f + e**2)) + 2*sqrt(d + e*x)*sq
rt(e + f*x)*(-a*e*f + b*(-d*f + 3*e**2)/2)/(f**2*(-d*f + e**2)) - (-2*a*e*f - b*
d*f + 3*b*e**2)*atanh(sqrt(e)*sqrt(e + f*x)/(sqrt(f)*sqrt(d + e*x)))/(sqrt(e)*f*
*(5/2))

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Mathematica [A]  time = 0.264183, size = 180, normalized size = 0.72 \[ \frac{\log \left (2 \sqrt{e} \sqrt{f} \sqrt{d+e x} \sqrt{e+f x}+d f+e^2+2 e f x\right ) \left (4 e f \left (2 a e f+b d f-3 b e^2\right )+c \left (-d^2 f^2-6 d e^2 f+15 e^4\right )\right )}{8 e^{3/2} f^{7/2}}+\frac{\sqrt{d+e x} \left (4 e f (-2 a f+3 b e+b f x)+c \left (e f \left (d+2 f x^2\right )+d f^2 x-15 e^3-5 e^2 f x\right )\right )}{4 e f^3 \sqrt{e+f x}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d + e*x]*(4*e*f*(3*b*e - 2*a*f + b*f*x) + c*(-15*e^3 - 5*e^2*f*x + d*f^2*x
+ e*f*(d + 2*f*x^2))))/(4*e*f^3*Sqrt[e + f*x]) + ((4*e*f*(-3*b*e^2 + b*d*f + 2*
a*e*f) + c*(15*e^4 - 6*d*e^2*f - d^2*f^2))*Log[e^2 + d*f + 2*e*f*x + 2*Sqrt[e]*S
qrt[f]*Sqrt[d + e*x]*Sqrt[e + f*x]])/(8*e^(3/2)*f^(7/2))

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Maple [B]  time = 0.048, size = 834, normalized size = 3.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(e*x+d)^(1/2)*(8*ln(1/2*(2*e*f*x+2*((e*x+d)*(f*x+e))^(1/2)*(e*f)^(1/2)+d*f+e
^2)/(e*f)^(1/2))*x*a*e^2*f^3+4*ln(1/2*(2*e*f*x+2*((e*x+d)*(f*x+e))^(1/2)*(e*f)^(
1/2)+d*f+e^2)/(e*f)^(1/2))*x*b*d*e*f^3-12*ln(1/2*(2*e*f*x+2*((e*x+d)*(f*x+e))^(1
/2)*(e*f)^(1/2)+d*f+e^2)/(e*f)^(1/2))*x*b*e^3*f^2-ln(1/2*(2*e*f*x+2*((e*x+d)*(f*
x+e))^(1/2)*(e*f)^(1/2)+d*f+e^2)/(e*f)^(1/2))*x*c*d^2*f^3-6*ln(1/2*(2*e*f*x+2*((
e*x+d)*(f*x+e))^(1/2)*(e*f)^(1/2)+d*f+e^2)/(e*f)^(1/2))*x*c*d*e^2*f^2+15*ln(1/2*
(2*e*f*x+2*((e*x+d)*(f*x+e))^(1/2)*(e*f)^(1/2)+d*f+e^2)/(e*f)^(1/2))*x*c*e^4*f+4
*x^2*c*e*f^2*((e*x+d)*(f*x+e))^(1/2)*(e*f)^(1/2)+8*ln(1/2*(2*e*f*x+2*((e*x+d)*(f
*x+e))^(1/2)*(e*f)^(1/2)+d*f+e^2)/(e*f)^(1/2))*a*e^3*f^2+4*ln(1/2*(2*e*f*x+2*((e
*x+d)*(f*x+e))^(1/2)*(e*f)^(1/2)+d*f+e^2)/(e*f)^(1/2))*b*d*e^2*f^2-12*ln(1/2*(2*
e*f*x+2*((e*x+d)*(f*x+e))^(1/2)*(e*f)^(1/2)+d*f+e^2)/(e*f)^(1/2))*b*e^4*f-ln(1/2
*(2*e*f*x+2*((e*x+d)*(f*x+e))^(1/2)*(e*f)^(1/2)+d*f+e^2)/(e*f)^(1/2))*c*d^2*e*f^
2-6*ln(1/2*(2*e*f*x+2*((e*x+d)*(f*x+e))^(1/2)*(e*f)^(1/2)+d*f+e^2)/(e*f)^(1/2))*
c*d*e^3*f+15*ln(1/2*(2*e*f*x+2*((e*x+d)*(f*x+e))^(1/2)*(e*f)^(1/2)+d*f+e^2)/(e*f
)^(1/2))*c*e^5+8*x*b*e*f^2*((e*x+d)*(f*x+e))^(1/2)*(e*f)^(1/2)+2*x*c*d*f^2*((e*x
+d)*(f*x+e))^(1/2)*(e*f)^(1/2)-10*x*c*e^2*f*((e*x+d)*(f*x+e))^(1/2)*(e*f)^(1/2)-
16*a*e*f^2*((e*x+d)*(f*x+e))^(1/2)*(e*f)^(1/2)+24*b*e^2*f*((e*x+d)*(f*x+e))^(1/2
)*(e*f)^(1/2)+2*c*d*e*f*((e*x+d)*(f*x+e))^(1/2)*(e*f)^(1/2)-30*c*e^3*((e*x+d)*(f
*x+e))^(1/2)*(e*f)^(1/2))/(e*f)^(1/2)/e/((e*x+d)*(f*x+e))^(1/2)/f^3/(f*x+e)^(1/2
)

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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)*sqrt(e*x + d)/(f*x + e)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.715297, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (2 \, c e f^{2} x^{2} - 15 \, c e^{3} - 8 \, a e f^{2} +{\left (c d e + 12 \, b e^{2}\right )} f -{\left (5 \, c e^{2} f -{\left (c d + 4 \, b e\right )} f^{2}\right )} x\right )} \sqrt{e f} \sqrt{e x + d} \sqrt{f x + e} +{\left (15 \, c e^{5} -{\left (c d^{2} e - 4 \, b d e^{2} - 8 \, a e^{3}\right )} f^{2} - 6 \,{\left (c d e^{3} + 2 \, b e^{4}\right )} f +{\left (15 \, c e^{4} f -{\left (c d^{2} - 4 \, b d e - 8 \, a e^{2}\right )} f^{3} - 6 \,{\left (c d e^{2} + 2 \, b e^{3}\right )} f^{2}\right )} x\right )} \log \left (4 \,{\left (2 \, e^{2} f^{2} x + e^{3} f + d e f^{2}\right )} \sqrt{e x + d} \sqrt{f x + e} +{\left (8 \, e^{2} f^{2} x^{2} + e^{4} + 6 \, d e^{2} f + d^{2} f^{2} + 8 \,{\left (e^{3} f + d e f^{2}\right )} x\right )} \sqrt{e f}\right )}{16 \,{\left (e f^{4} x + e^{2} f^{3}\right )} \sqrt{e f}}, \frac{2 \,{\left (2 \, c e f^{2} x^{2} - 15 \, c e^{3} - 8 \, a e f^{2} +{\left (c d e + 12 \, b e^{2}\right )} f -{\left (5 \, c e^{2} f -{\left (c d + 4 \, b e\right )} f^{2}\right )} x\right )} \sqrt{-e f} \sqrt{e x + d} \sqrt{f x + e} +{\left (15 \, c e^{5} -{\left (c d^{2} e - 4 \, b d e^{2} - 8 \, a e^{3}\right )} f^{2} - 6 \,{\left (c d e^{3} + 2 \, b e^{4}\right )} f +{\left (15 \, c e^{4} f -{\left (c d^{2} - 4 \, b d e - 8 \, a e^{2}\right )} f^{3} - 6 \,{\left (c d e^{2} + 2 \, b e^{3}\right )} f^{2}\right )} x\right )} \arctan \left (\frac{{\left (2 \, e f x + e^{2} + d f\right )} \sqrt{-e f}}{2 \, \sqrt{e x + d} \sqrt{f x + e} e f}\right )}{8 \,{\left (e f^{4} x + e^{2} f^{3}\right )} \sqrt{-e f}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(4*(2*c*e*f^2*x^2 - 15*c*e^3 - 8*a*e*f^2 + (c*d*e + 12*b*e^2)*f - (5*c*e^2
*f - (c*d + 4*b*e)*f^2)*x)*sqrt(e*f)*sqrt(e*x + d)*sqrt(f*x + e) + (15*c*e^5 - (
c*d^2*e - 4*b*d*e^2 - 8*a*e^3)*f^2 - 6*(c*d*e^3 + 2*b*e^4)*f + (15*c*e^4*f - (c*
d^2 - 4*b*d*e - 8*a*e^2)*f^3 - 6*(c*d*e^2 + 2*b*e^3)*f^2)*x)*log(4*(2*e^2*f^2*x
+ e^3*f + d*e*f^2)*sqrt(e*x + d)*sqrt(f*x + e) + (8*e^2*f^2*x^2 + e^4 + 6*d*e^2*
f + d^2*f^2 + 8*(e^3*f + d*e*f^2)*x)*sqrt(e*f)))/((e*f^4*x + e^2*f^3)*sqrt(e*f))
, 1/8*(2*(2*c*e*f^2*x^2 - 15*c*e^3 - 8*a*e*f^2 + (c*d*e + 12*b*e^2)*f - (5*c*e^2
*f - (c*d + 4*b*e)*f^2)*x)*sqrt(-e*f)*sqrt(e*x + d)*sqrt(f*x + e) + (15*c*e^5 -
(c*d^2*e - 4*b*d*e^2 - 8*a*e^3)*f^2 - 6*(c*d*e^3 + 2*b*e^4)*f + (15*c*e^4*f - (c
*d^2 - 4*b*d*e - 8*a*e^2)*f^3 - 6*(c*d*e^2 + 2*b*e^3)*f^2)*x)*arctan(1/2*(2*e*f*
x + e^2 + d*f)*sqrt(-e*f)/(sqrt(e*x + d)*sqrt(f*x + e)*e*f)))/((e*f^4*x + e^2*f^
3)*sqrt(-e*f))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.300739, size = 320, normalized size = 1.29 \[ \frac{{\left ({\left (x e + d\right )}{\left (\frac{2 \,{\left (x e + d\right )} c e^{\left (-1\right )}}{f} - \frac{{\left (3 \, c d f^{4} e^{2} - 4 \, b f^{4} e^{3} + 5 \, c f^{3} e^{4}\right )} e^{\left (-3\right )}}{f^{5}}\right )} + \frac{{\left (c d^{2} f^{4} e^{2} - 4 \, b d f^{4} e^{3} + 6 \, c d f^{3} e^{4} - 8 \, a f^{4} e^{4} + 12 \, b f^{3} e^{5} - 15 \, c f^{2} e^{6}\right )} e^{\left (-3\right )}}{f^{5}}\right )} \sqrt{x e + d}}{4 \, \sqrt{{\left (x e + d\right )} f e - d f e + e^{3}}} + \frac{{\left (c d^{2} f^{2} - 4 \, b d f^{2} e + 6 \, c d f e^{2} - 8 \, a f^{2} e^{2} + 12 \, b f e^{3} - 15 \, c e^{4}\right )} e^{\left (-\frac{3}{2}\right )}{\rm ln}\left ({\left | -\sqrt{x e + d} \sqrt{f} e^{\frac{1}{2}} + \sqrt{{\left (x e + d\right )} f e - d f e + e^{3}} \right |}\right )}{4 \, f^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*((x*e + d)*(2*(x*e + d)*c*e^(-1)/f - (3*c*d*f^4*e^2 - 4*b*f^4*e^3 + 5*c*f^3*
e^4)*e^(-3)/f^5) + (c*d^2*f^4*e^2 - 4*b*d*f^4*e^3 + 6*c*d*f^3*e^4 - 8*a*f^4*e^4
+ 12*b*f^3*e^5 - 15*c*f^2*e^6)*e^(-3)/f^5)*sqrt(x*e + d)/sqrt((x*e + d)*f*e - d*
f*e + e^3) + 1/4*(c*d^2*f^2 - 4*b*d*f^2*e + 6*c*d*f*e^2 - 8*a*f^2*e^2 + 12*b*f*e
^3 - 15*c*e^4)*e^(-3/2)*ln(abs(-sqrt(x*e + d)*sqrt(f)*e^(1/2) + sqrt((x*e + d)*f
*e - d*f*e + e^3)))/f^(7/2)