3.1199 \(\int \frac{\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^4} \, dx\)

Optimal. Leaf size=107 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{5/2} d^4}-\frac{\sqrt{a+b x+c x^2}}{8 c^2 d^4 (b+2 c x)}-\frac{\left (a+b x+c x^2\right )^{3/2}}{6 c d^4 (b+2 c x)^3} \]

[Out]

-Sqrt[a + b*x + c*x^2]/(8*c^2*d^4*(b + 2*c*x)) - (a + b*x + c*x^2)^(3/2)/(6*c*d^
4*(b + 2*c*x)^3) + ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])]/(16*c^
(5/2)*d^4)

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Rubi [A]  time = 0.145845, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{5/2} d^4}-\frac{\sqrt{a+b x+c x^2}}{8 c^2 d^4 (b+2 c x)}-\frac{\left (a+b x+c x^2\right )^{3/2}}{6 c d^4 (b+2 c x)^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^4,x]

[Out]

-Sqrt[a + b*x + c*x^2]/(8*c^2*d^4*(b + 2*c*x)) - (a + b*x + c*x^2)^(3/2)/(6*c*d^
4*(b + 2*c*x)^3) + ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])]/(16*c^
(5/2)*d^4)

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Rubi in Sympy [A]  time = 31.4813, size = 95, normalized size = 0.89 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{6 c d^{4} \left (b + 2 c x\right )^{3}} - \frac{\sqrt{a + b x + c x^{2}}}{8 c^{2} d^{4} \left (b + 2 c x\right )} + \frac{\operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{16 c^{\frac{5}{2}} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(3/2)/(2*c*d*x+b*d)**4,x)

[Out]

-(a + b*x + c*x**2)**(3/2)/(6*c*d**4*(b + 2*c*x)**3) - sqrt(a + b*x + c*x**2)/(8
*c**2*d**4*(b + 2*c*x)) + atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/
(16*c**(5/2)*d**4)

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Mathematica [A]  time = 0.249233, size = 91, normalized size = 0.85 \[ \frac{\frac{\log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{16 c^{5/2}}-\frac{\sqrt{a+x (b+c x)} \left (4 c \left (a+4 c x^2\right )+3 b^2+16 b c x\right )}{24 c^2 (b+2 c x)^3}}{d^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^4,x]

[Out]

(-(Sqrt[a + x*(b + c*x)]*(3*b^2 + 16*b*c*x + 4*c*(a + 4*c*x^2)))/(24*c^2*(b + 2*
c*x)^3) + Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]]/(16*c^(5/2)))/d^4

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Maple [B]  time = 0.018, size = 629, normalized size = 5.9 \[ -{\frac{1}{12\,{c}^{3}{d}^{4} \left ( 4\,ac-{b}^{2} \right ) } \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{b}{2\,c}} \right ) ^{-3}}-{\frac{2}{3\,c{d}^{4} \left ( 4\,ac-{b}^{2} \right ) ^{2}} \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}}+{\frac{2\,x}{3\,{d}^{4} \left ( 4\,ac-{b}^{2} \right ) ^{2}} \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{3}{2}}}}+{\frac{b}{3\,c{d}^{4} \left ( 4\,ac-{b}^{2} \right ) ^{2}} \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{3}{2}}}}+{\frac{ax}{{d}^{4} \left ( 4\,ac-{b}^{2} \right ) ^{2}}\sqrt{ \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}}}}-{\frac{{b}^{2}x}{4\,c{d}^{4} \left ( 4\,ac-{b}^{2} \right ) ^{2}}\sqrt{ \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}}}}+{\frac{ab}{2\,c{d}^{4} \left ( 4\,ac-{b}^{2} \right ) ^{2}}\sqrt{ \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}}}}-{\frac{{b}^{3}}{8\,{c}^{2}{d}^{4} \left ( 4\,ac-{b}^{2} \right ) ^{2}}\sqrt{ \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}}}}+{\frac{{a}^{2}}{{d}^{4} \left ( 4\,ac-{b}^{2} \right ) ^{2}}\ln \left ( \sqrt{c} \left ( x+{\frac{b}{2\,c}} \right ) +\sqrt{ \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}}} \right ){\frac{1}{\sqrt{c}}}}-{\frac{a{b}^{2}}{2\,{d}^{4} \left ( 4\,ac-{b}^{2} \right ) ^{2}}\ln \left ( \sqrt{c} \left ( x+{\frac{b}{2\,c}} \right ) +\sqrt{ \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}}} \right ){c}^{-{\frac{3}{2}}}}+{\frac{{b}^{4}}{16\,{d}^{4} \left ( 4\,ac-{b}^{2} \right ) ^{2}}\ln \left ( \sqrt{c} \left ( x+{\frac{b}{2\,c}} \right ) +\sqrt{ \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}}} \right ){c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^4,x)

[Out]

-1/12/d^4/c^3/(4*a*c-b^2)/(x+1/2*b/c)^3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(5/2
)-2/3/d^4/c/(4*a*c-b^2)^2/(x+1/2*b/c)*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(5/2)+
2/3/d^4/(4*a*c-b^2)^2*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)*x+1/3/d^4/c/(4*a
*c-b^2)^2*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)*b+1/d^4/(4*a*c-b^2)^2*((x+1/
2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)*x*a-1/4/d^4/c/(4*a*c-b^2)^2*((x+1/2*b/c)^2*c
+1/4*(4*a*c-b^2)/c)^(1/2)*x*b^2+1/2/d^4/c/(4*a*c-b^2)^2*((x+1/2*b/c)^2*c+1/4*(4*
a*c-b^2)/c)^(1/2)*b*a-1/8/d^4/c^2/(4*a*c-b^2)^2*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)
/c)^(1/2)*b^3+1/d^4/c^(1/2)/(4*a*c-b^2)^2*ln(c^(1/2)*(x+1/2*b/c)+((x+1/2*b/c)^2*
c+1/4*(4*a*c-b^2)/c)^(1/2))*a^2-1/2/d^4/c^(3/2)/(4*a*c-b^2)^2*ln(c^(1/2)*(x+1/2*
b/c)+((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2))*b^2*a+1/16/d^4/c^(5/2)/(4*a*c-b^
2)^2*ln(c^(1/2)*(x+1/2*b/c)+((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2))*b^4

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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)^(3/2)/(2*c*d*x + b*d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.345962, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left (16 \, c^{2} x^{2} + 16 \, b c x + 3 \, b^{2} + 4 \, a c\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} - 3 \,{\left (8 \, c^{3} x^{3} + 12 \, b c^{2} x^{2} + 6 \, b^{2} c x + b^{3}\right )} \log \left (-4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{96 \,{\left (8 \, c^{5} d^{4} x^{3} + 12 \, b c^{4} d^{4} x^{2} + 6 \, b^{2} c^{3} d^{4} x + b^{3} c^{2} d^{4}\right )} \sqrt{c}}, -\frac{2 \,{\left (16 \, c^{2} x^{2} + 16 \, b c x + 3 \, b^{2} + 4 \, a c\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} - 3 \,{\left (8 \, c^{3} x^{3} + 12 \, b c^{2} x^{2} + 6 \, b^{2} c x + b^{3}\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{48 \,{\left (8 \, c^{5} d^{4} x^{3} + 12 \, b c^{4} d^{4} x^{2} + 6 \, b^{2} c^{3} d^{4} x + b^{3} c^{2} d^{4}\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/2)/(2*c*d*x + b*d)^4,x, algorithm="fricas")

[Out]

[-1/96*(4*(16*c^2*x^2 + 16*b*c*x + 3*b^2 + 4*a*c)*sqrt(c*x^2 + b*x + a)*sqrt(c)
- 3*(8*c^3*x^3 + 12*b*c^2*x^2 + 6*b^2*c*x + b^3)*log(-4*(2*c^2*x + b*c)*sqrt(c*x
^2 + b*x + a) - (8*c^2*x^2 + 8*b*c*x + b^2 + 4*a*c)*sqrt(c)))/((8*c^5*d^4*x^3 +
12*b*c^4*d^4*x^2 + 6*b^2*c^3*d^4*x + b^3*c^2*d^4)*sqrt(c)), -1/48*(2*(16*c^2*x^2
+ 16*b*c*x + 3*b^2 + 4*a*c)*sqrt(c*x^2 + b*x + a)*sqrt(-c) - 3*(8*c^3*x^3 + 12*
b*c^2*x^2 + 6*b^2*c*x + b^3)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x +
a)*c)))/((8*c^5*d^4*x^3 + 12*b*c^4*d^4*x^2 + 6*b^2*c^3*d^4*x + b^3*c^2*d^4)*sqr
t(-c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{a \sqrt{a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx + \int \frac{b x \sqrt{a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx + \int \frac{c x^{2} \sqrt{a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx}{d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**(3/2)/(2*c*d*x+b*d)**4,x)

[Out]

(Integral(a*sqrt(a + b*x + c*x**2)/(b**4 + 8*b**3*c*x + 24*b**2*c**2*x**2 + 32*b
*c**3*x**3 + 16*c**4*x**4), x) + Integral(b*x*sqrt(a + b*x + c*x**2)/(b**4 + 8*b
**3*c*x + 24*b**2*c**2*x**2 + 32*b*c**3*x**3 + 16*c**4*x**4), x) + Integral(c*x*
*2*sqrt(a + b*x + c*x**2)/(b**4 + 8*b**3*c*x + 24*b**2*c**2*x**2 + 32*b*c**3*x**
3 + 16*c**4*x**4), x))/d**4

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/2)/(2*c*d*x + b*d)^4,x, algorithm="giac")

[Out]

Exception raised: TypeError