3.2347 \(\int \left (a+b x+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=149 \[ -\frac{5 \left (b^2-4 a c\right )^3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{7/2}}+\frac{5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2}}{512 c^3}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^2}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c} \]

[Out]

(5*(b^2 - 4*a*c)^2*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(512*c^3) - (5*(b^2 - 4*a*
c)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(192*c^2) + ((b + 2*c*x)*(a + b*x + c*x^
2)^(5/2))/(12*c) - (5*(b^2 - 4*a*c)^3*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*
x + c*x^2])])/(1024*c^(7/2))

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Rubi [A]  time = 0.132489, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{5 \left (b^2-4 a c\right )^3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{7/2}}+\frac{5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2}}{512 c^3}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^2}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(5/2),x]

[Out]

(5*(b^2 - 4*a*c)^2*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(512*c^3) - (5*(b^2 - 4*a*
c)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(192*c^2) + ((b + 2*c*x)*(a + b*x + c*x^
2)^(5/2))/(12*c) - (5*(b^2 - 4*a*c)^3*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*
x + c*x^2])])/(1024*c^(7/2))

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Rubi in Sympy [A]  time = 14.1787, size = 143, normalized size = 0.96 \[ \frac{\left (b + 2 c x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{12 c} - \frac{5 \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{192 c^{2}} + \frac{5 \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}}}{512 c^{3}} - \frac{5 \left (- 4 a c + b^{2}\right )^{3} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{1024 c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b + 2*c*x)*(a + b*x + c*x**2)**(5/2)/(12*c) - 5*(b + 2*c*x)*(-4*a*c + b**2)*(a
+ b*x + c*x**2)**(3/2)/(192*c**2) + 5*(b + 2*c*x)*(-4*a*c + b**2)**2*sqrt(a + b*
x + c*x**2)/(512*c**3) - 5*(-4*a*c + b**2)**3*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(
a + b*x + c*x**2)))/(1024*c**(7/2))

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Mathematica [A]  time = 0.205894, size = 147, normalized size = 0.99 \[ \frac{2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left (16 c^2 \left (33 a^2+26 a c x^2+8 c^2 x^4\right )+8 b^2 c \left (11 c x^2-20 a\right )+32 b c^2 x \left (13 a+8 c x^2\right )+15 b^4-40 b^3 c x\right )-15 \left (b^2-4 a c\right )^3 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{3072 c^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(15*b^4 - 40*b^3*c*x + 32*b*c^2*x*(
13*a + 8*c*x^2) + 8*b^2*c*(-20*a + 11*c*x^2) + 16*c^2*(33*a^2 + 26*a*c*x^2 + 8*c
^2*x^4)) - 15*(b^2 - 4*a*c)^3*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/
(3072*c^(7/2))

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Maple [B]  time = 0.004, size = 360, normalized size = 2.4 \[{\frac{2\,cx+b}{12\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,ax}{24} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{b}^{2}x}{96\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,ab}{48\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{b}^{3}}{192\,{c}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}x}{16}\sqrt{c{x}^{2}+bx+a}}-{\frac{5\,ax{b}^{2}}{32\,c}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,x{b}^{4}}{256\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,{a}^{2}b}{32\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{5\,a{b}^{3}}{64\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,{b}^{5}}{512\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,{a}^{3}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}-{\frac{15\,{a}^{2}{b}^{2}}{64}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{15\,a{b}^{4}}{256}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{5\,{b}^{6}}{1024}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(5/2),x)

[Out]

1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(c*x^2+b*x+a)^(3/2)*x*a-5/96/c*(c*x^2+
b*x+a)^(3/2)*x*b^2+5/48/c*(c*x^2+b*x+a)^(3/2)*b*a-5/192/c^2*(c*x^2+b*x+a)^(3/2)*
b^3+5/16*(c*x^2+b*x+a)^(1/2)*x*a^2-5/32/c*(c*x^2+b*x+a)^(1/2)*x*a*b^2+5/256/c^2*
(c*x^2+b*x+a)^(1/2)*x*b^4+5/32/c*(c*x^2+b*x+a)^(1/2)*b*a^2-5/64/c^2*(c*x^2+b*x+a
)^(1/2)*b^3*a+5/512/c^3*(c*x^2+b*x+a)^(1/2)*b^5+5/16/c^(1/2)*ln((1/2*b+c*x)/c^(1
/2)+(c*x^2+b*x+a)^(1/2))*a^3-15/64/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^
(1/2))*a^2*b^2+15/256/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^4*a-
5/1024/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.267747, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (256 \, c^{5} x^{5} + 640 \, b c^{4} x^{4} + 15 \, b^{5} - 160 \, a b^{3} c + 528 \, a^{2} b c^{2} + 16 \,{\left (27 \, b^{2} c^{3} + 52 \, a c^{4}\right )} x^{3} + 8 \,{\left (b^{3} c^{2} + 156 \, a b c^{3}\right )} x^{2} - 2 \,{\left (5 \, b^{4} c - 48 \, a b^{2} c^{2} - 528 \, a^{2} c^{3}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} - 15 \,{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \log \left (-4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{6144 \, c^{\frac{7}{2}}}, \frac{2 \,{\left (256 \, c^{5} x^{5} + 640 \, b c^{4} x^{4} + 15 \, b^{5} - 160 \, a b^{3} c + 528 \, a^{2} b c^{2} + 16 \,{\left (27 \, b^{2} c^{3} + 52 \, a c^{4}\right )} x^{3} + 8 \,{\left (b^{3} c^{2} + 156 \, a b c^{3}\right )} x^{2} - 2 \,{\left (5 \, b^{4} c - 48 \, a b^{2} c^{2} - 528 \, a^{2} c^{3}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} - 15 \,{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{3072 \, \sqrt{-c} c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6144*(4*(256*c^5*x^5 + 640*b*c^4*x^4 + 15*b^5 - 160*a*b^3*c + 528*a^2*b*c^2 +
16*(27*b^2*c^3 + 52*a*c^4)*x^3 + 8*(b^3*c^2 + 156*a*b*c^3)*x^2 - 2*(5*b^4*c - 4
8*a*b^2*c^2 - 528*a^2*c^3)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) - 15*(b^6 - 12*a*b^4
*c + 48*a^2*b^2*c^2 - 64*a^3*c^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)))/c^(7/2), 1/3072*(2*(256*c^5*x^5 +
640*b*c^4*x^4 + 15*b^5 - 160*a*b^3*c + 528*a^2*b*c^2 + 16*(27*b^2*c^3 + 52*a*c^
4)*x^3 + 8*(b^3*c^2 + 156*a*b*c^3)*x^2 - 2*(5*b^4*c - 48*a*b^2*c^2 - 528*a^2*c^3
)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) - 15*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64
*a^3*c^3)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/(sqrt(-c)*
c^3)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**(5/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.258568, size = 281, normalized size = 1.89 \[ \frac{1}{1536} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \, c^{2} x + 5 \, b c\right )} x + \frac{27 \, b^{2} c^{5} + 52 \, a c^{6}}{c^{5}}\right )} x + \frac{b^{3} c^{4} + 156 \, a b c^{5}}{c^{5}}\right )} x - \frac{5 \, b^{4} c^{3} - 48 \, a b^{2} c^{4} - 528 \, a^{2} c^{5}}{c^{5}}\right )} x + \frac{15 \, b^{5} c^{2} - 160 \, a b^{3} c^{3} + 528 \, a^{2} b c^{4}}{c^{5}}\right )} + \frac{5 \,{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{1024 \, c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1536*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*c^2*x + 5*b*c)*x + (27*b^2*c^5 + 52*
a*c^6)/c^5)*x + (b^3*c^4 + 156*a*b*c^5)/c^5)*x - (5*b^4*c^3 - 48*a*b^2*c^4 - 528
*a^2*c^5)/c^5)*x + (15*b^5*c^2 - 160*a*b^3*c^3 + 528*a^2*b*c^4)/c^5) + 5/1024*(b
^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))*sqrt(c) - b))/c^(7/2)