3.1216 \(\int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^6} \, dx\)

Optimal. Leaf size=139 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{64 c^{7/2} d^6}-\frac{\sqrt{a+b x+c x^2}}{32 c^3 d^6 (b+2 c x)}-\frac{\left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^6 (b+2 c x)^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5} \]

[Out]

-Sqrt[a + b*x + c*x^2]/(32*c^3*d^6*(b + 2*c*x)) - (a + b*x + c*x^2)^(3/2)/(24*c^
2*d^6*(b + 2*c*x)^3) - (a + b*x + c*x^2)^(5/2)/(10*c*d^6*(b + 2*c*x)^5) + ArcTan
h[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])]/(64*c^(7/2)*d^6)

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Rubi [A]  time = 0.211301, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{64 c^{7/2} d^6}-\frac{\sqrt{a+b x+c x^2}}{32 c^3 d^6 (b+2 c x)}-\frac{\left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^6 (b+2 c x)^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^6,x]

[Out]

-Sqrt[a + b*x + c*x^2]/(32*c^3*d^6*(b + 2*c*x)) - (a + b*x + c*x^2)^(3/2)/(24*c^
2*d^6*(b + 2*c*x)^3) - (a + b*x + c*x^2)^(5/2)/(10*c*d^6*(b + 2*c*x)^5) + ArcTan
h[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])]/(64*c^(7/2)*d^6)

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Rubi in Sympy [A]  time = 47.2364, size = 126, normalized size = 0.91 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{10 c d^{6} \left (b + 2 c x\right )^{5}} - \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{24 c^{2} d^{6} \left (b + 2 c x\right )^{3}} - \frac{\sqrt{a + b x + c x^{2}}}{32 c^{3} d^{6} \left (b + 2 c x\right )} + \frac{\operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{64 c^{\frac{7}{2}} d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**6,x)

[Out]

-(a + b*x + c*x**2)**(5/2)/(10*c*d**6*(b + 2*c*x)**5) - (a + b*x + c*x**2)**(3/2
)/(24*c**2*d**6*(b + 2*c*x)**3) - sqrt(a + b*x + c*x**2)/(32*c**3*d**6*(b + 2*c*
x)) + atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(64*c**(7/2)*d**6)

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Mathematica [A]  time = 0.305007, size = 110, normalized size = 0.79 \[ \frac{\frac{\log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{64 c^{7/2}}-\frac{\sqrt{a+x (b+c x)} \left (-11 \left (b^2-4 a c\right ) (b+2 c x)^2+3 \left (b^2-4 a c\right )^2+23 (b+2 c x)^4\right )}{480 c^3 (b+2 c x)^5}}{d^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^6,x]

[Out]

(-(Sqrt[a + x*(b + c*x)]*(3*(b^2 - 4*a*c)^2 - 11*(b^2 - 4*a*c)*(b + 2*c*x)^2 + 2
3*(b + 2*c*x)^4))/(480*c^3*(b + 2*c*x)^5) + Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x
*(b + c*x)]]/(64*c^(7/2)))/d^6

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Maple [B]  time = 0.026, size = 1080, normalized size = 7.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^6,x)

[Out]

-1/80/d^6/c^5/(4*a*c-b^2)/(x+1/2*b/c)^5*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(7/2
)-1/30/d^6/c^3/(4*a*c-b^2)^2/(x+1/2*b/c)^3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(
7/2)-8/15/d^6/c/(4*a*c-b^2)^3/(x+1/2*b/c)*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(7
/2)+8/15/d^6/(4*a*c-b^2)^3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(5/2)*x+4/15/d^6/
c/(4*a*c-b^2)^3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(5/2)*b+2/3/d^6/(4*a*c-b^2)^
3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)*x*a-1/6/d^6/c/(4*a*c-b^2)^3*((x+1/2*
b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)*x*b^2+1/3/d^6/c/(4*a*c-b^2)^3*((x+1/2*b/c)^2*c
+1/4*(4*a*c-b^2)/c)^(3/2)*b*a-1/12/d^6/c^2/(4*a*c-b^2)^3*((x+1/2*b/c)^2*c+1/4*(4
*a*c-b^2)/c)^(3/2)*b^3+1/d^6/(4*a*c-b^2)^3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(
1/2)*x*a^2-1/2/d^6/c/(4*a*c-b^2)^3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)*x*a
*b^2+1/16/d^6/c^2/(4*a*c-b^2)^3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)*x*b^4+
1/2/d^6/c/(4*a*c-b^2)^3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)*b*a^2-1/4/d^6/
c^2/(4*a*c-b^2)^3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)*b^3*a+1/32/d^6/c^3/(
4*a*c-b^2)^3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)*b^5+1/d^6/c^(1/2)/(4*a*c-
b^2)^3*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^3-3/4
/d^6/c^(3/2)/(4*a*c-b^2)^3*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^2+3/16/d^6/c^(5/2)/(4*a*c-b^2)^3*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*a-1/64/d^6/c^(7/2)/(4*a*c-b^2)^3*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^6

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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)^(5/2)/(2*c*d*x + b*d)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.826095, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left (368 \, c^{4} x^{4} + 736 \, b c^{3} x^{3} + 15 \, b^{4} + 20 \, a b^{2} c + 48 \, a^{2} c^{2} + 4 \,{\left (127 \, b^{2} c^{2} + 44 \, a c^{3}\right )} x^{2} + 4 \,{\left (35 \, b^{3} c + 44 \, a b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} - 15 \,{\left (32 \, c^{5} x^{5} + 80 \, b c^{4} x^{4} + 80 \, b^{2} c^{3} x^{3} + 40 \, b^{3} c^{2} x^{2} + 10 \, b^{4} c x + b^{5}\right )} \log \left (-4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{1920 \,{\left (32 \, c^{8} d^{6} x^{5} + 80 \, b c^{7} d^{6} x^{4} + 80 \, b^{2} c^{6} d^{6} x^{3} + 40 \, b^{3} c^{5} d^{6} x^{2} + 10 \, b^{4} c^{4} d^{6} x + b^{5} c^{3} d^{6}\right )} \sqrt{c}}, -\frac{2 \,{\left (368 \, c^{4} x^{4} + 736 \, b c^{3} x^{3} + 15 \, b^{4} + 20 \, a b^{2} c + 48 \, a^{2} c^{2} + 4 \,{\left (127 \, b^{2} c^{2} + 44 \, a c^{3}\right )} x^{2} + 4 \,{\left (35 \, b^{3} c + 44 \, a b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} - 15 \,{\left (32 \, c^{5} x^{5} + 80 \, b c^{4} x^{4} + 80 \, b^{2} c^{3} x^{3} + 40 \, b^{3} c^{2} x^{2} + 10 \, b^{4} c x + b^{5}\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{960 \,{\left (32 \, c^{8} d^{6} x^{5} + 80 \, b c^{7} d^{6} x^{4} + 80 \, b^{2} c^{6} d^{6} x^{3} + 40 \, b^{3} c^{5} d^{6} x^{2} + 10 \, b^{4} c^{4} d^{6} x + b^{5} c^{3} d^{6}\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^6,x, algorithm="fricas")

[Out]

[-1/1920*(4*(368*c^4*x^4 + 736*b*c^3*x^3 + 15*b^4 + 20*a*b^2*c + 48*a^2*c^2 + 4*
(127*b^2*c^2 + 44*a*c^3)*x^2 + 4*(35*b^3*c + 44*a*b*c^2)*x)*sqrt(c*x^2 + b*x + a
)*sqrt(c) - 15*(32*c^5*x^5 + 80*b*c^4*x^4 + 80*b^2*c^3*x^3 + 40*b^3*c^2*x^2 + 10
*b^4*c*x + b^5)*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)))/((32*c^8*d^6*x^5 + 80*b*c^7*d^6*x^4 + 80*b^2*c^6*d^
6*x^3 + 40*b^3*c^5*d^6*x^2 + 10*b^4*c^4*d^6*x + b^5*c^3*d^6)*sqrt(c)), -1/960*(2
*(368*c^4*x^4 + 736*b*c^3*x^3 + 15*b^4 + 20*a*b^2*c + 48*a^2*c^2 + 4*(127*b^2*c^
2 + 44*a*c^3)*x^2 + 4*(35*b^3*c + 44*a*b*c^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c)
- 15*(32*c^5*x^5 + 80*b*c^4*x^4 + 80*b^2*c^3*x^3 + 40*b^3*c^2*x^2 + 10*b^4*c*x +
b^5)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/((32*c^8*d^6*x
^5 + 80*b*c^7*d^6*x^4 + 80*b^2*c^6*d^6*x^3 + 40*b^3*c^5*d^6*x^2 + 10*b^4*c^4*d^6
*x + b^5*c^3*d^6)*sqrt(-c))]

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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((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**6,x)

[Out]

Timed out

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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)^(5/2)/(2*c*d*x + b*d)^6,x, algorithm="giac")

[Out]

Exception raised: TypeError