3.39 \(\int \frac{(g+h x) \sqrt{a+b x+c x^2}}{\left (a d+b d x+c d x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=136 \[ \frac{h \sqrt{a+b x+c x^2} \log \left (a+b x+c x^2\right )}{2 c d \sqrt{a d+b d x+c d x^2}}-\frac{\sqrt{a+b x+c x^2} (2 c g-b h) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c d \sqrt{b^2-4 a c} \sqrt{a d+b d x+c d x^2}} \]

[Out]

-(((2*c*g - b*h)*Sqrt[a + b*x + c*x^2]*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(
c*Sqrt[b^2 - 4*a*c]*d*Sqrt[a*d + b*d*x + c*d*x^2])) + (h*Sqrt[a + b*x + c*x^2]*L
og[a + b*x + c*x^2])/(2*c*d*Sqrt[a*d + b*d*x + c*d*x^2])

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Rubi [A]  time = 0.299394, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.132 \[ \frac{h \sqrt{a+b x+c x^2} \log \left (a+b x+c x^2\right )}{2 c d \sqrt{a d+b d x+c d x^2}}-\frac{\sqrt{a+b x+c x^2} (2 c g-b h) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c d \sqrt{b^2-4 a c} \sqrt{a d+b d x+c d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

-(((2*c*g - b*h)*Sqrt[a + b*x + c*x^2]*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(
c*Sqrt[b^2 - 4*a*c]*d*Sqrt[a*d + b*d*x + c*d*x^2])) + (h*Sqrt[a + b*x + c*x^2]*L
og[a + b*x + c*x^2])/(2*c*d*Sqrt[a*d + b*d*x + c*d*x^2])

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Rubi in Sympy [A]  time = 42.4655, size = 129, normalized size = 0.95 \[ \frac{h \sqrt{a d + b d x + c d x^{2}} \log{\left (a + b x + c x^{2} \right )}}{2 c d^{2} \sqrt{a + b x + c x^{2}}} + \frac{\left (b h - 2 c g\right ) \sqrt{a d + b d x + c d x^{2}} \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{c d^{2} \sqrt{- 4 a c + b^{2}} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

h*sqrt(a*d + b*d*x + c*d*x**2)*log(a + b*x + c*x**2)/(2*c*d**2*sqrt(a + b*x + c*
x**2)) + (b*h - 2*c*g)*sqrt(a*d + b*d*x + c*d*x**2)*atanh((b + 2*c*x)/sqrt(-4*a*
c + b**2))/(c*d**2*sqrt(-4*a*c + b**2)*sqrt(a + b*x + c*x**2))

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Mathematica [A]  time = 0.181597, size = 108, normalized size = 0.79 \[ \frac{(a+x (b+c x))^{3/2} \left ((4 c g-2 b h) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )+h \sqrt{4 a c-b^2} \log (a+x (b+c x))\right )}{2 c \sqrt{4 a c-b^2} (d (a+x (b+c x)))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((a + x*(b + c*x))^(3/2)*((4*c*g - 2*b*h)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]]
+ Sqrt[-b^2 + 4*a*c]*h*Log[a + x*(b + c*x)]))/(2*c*Sqrt[-b^2 + 4*a*c]*(d*(a + x
*(b + c*x)))^(3/2))

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Maple [A]  time = 0.025, size = 121, normalized size = 0.9 \[{\frac{1}{2\,c{d}^{2}}\sqrt{d \left ( c{x}^{2}+bx+a \right ) } \left ( h\ln \left ( c{x}^{2}+bx+a \right ) \sqrt{4\,ac-{b}^{2}}-2\,\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) bh+4\,\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) cg \right ){\frac{1}{\sqrt{c{x}^{2}+bx+a}}}{\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2/(c*x^2+b*x+a)^(1/2)*(d*(c*x^2+b*x+a))^(1/2)*(h*ln(c*x^2+b*x+a)*(4*a*c-b^2)^(
1/2)-2*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*h+4*arctan((2*c*x+b)/(4*a*c-b^2)^(1
/2))*c*g)/d^2/c/(4*a*c-b^2)^(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(sqrt(c*x^2 + b*x + a)*(h*x + g)/(c*d*x^2 + b*d*x + a*d)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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(sqrt(c*x^2 + b*x + a)*(h*x + g)/(c*d*x^2 + b*d*x + a*d)^(3/2),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((g + h*x)*sqrt(a + b*x + c*x**2)/(d*(a + b*x + c*x**2))**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2} + b x + a}{\left (h x + g\right )}}{{\left (c d x^{2} + b d x + a d\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(c*x^2 + b*x + a)*(h*x + g)/(c*d*x^2 + b*d*x + a*d)^(3/2), x)