3.1211 \(\int \frac{\left (a+b x+c x^2\right )^{5/2}}{b d+2 c d x} \, dx\)

Optimal. Leaf size=149 \[ -\frac{\left (b^2-4 a c\right )^{5/2} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{64 c^{7/2} d}+\frac{\left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}{32 c^3 d}-\frac{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}{24 c^2 d}+\frac{\left (a+b x+c x^2\right )^{5/2}}{10 c d} \]

[Out]

((b^2 - 4*a*c)^2*Sqrt[a + b*x + c*x^2])/(32*c^3*d) - ((b^2 - 4*a*c)*(a + b*x + c
*x^2)^(3/2))/(24*c^2*d) + (a + b*x + c*x^2)^(5/2)/(10*c*d) - ((b^2 - 4*a*c)^(5/2
)*ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/(64*c^(7/2)*d)

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Rubi [A]  time = 0.314872, antiderivative size = 149, 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 \[ -\frac{\left (b^2-4 a c\right )^{5/2} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{64 c^{7/2} d}+\frac{\left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}{32 c^3 d}-\frac{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}{24 c^2 d}+\frac{\left (a+b x+c x^2\right )^{5/2}}{10 c d} \]

Antiderivative was successfully verified.

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

[Out]

((b^2 - 4*a*c)^2*Sqrt[a + b*x + c*x^2])/(32*c^3*d) - ((b^2 - 4*a*c)*(a + b*x + c
*x^2)^(3/2))/(24*c^2*d) + (a + b*x + c*x^2)^(5/2)/(10*c*d) - ((b^2 - 4*a*c)^(5/2
)*ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/(64*c^(7/2)*d)

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Rubi in Sympy [A]  time = 68.4605, size = 131, normalized size = 0.88 \[ \frac{\left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{10 c d} - \frac{\left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{24 c^{2} d} + \frac{\left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}}}{32 c^{3} d} - \frac{\left (- 4 a c + b^{2}\right )^{\frac{5}{2}} \operatorname{atan}{\left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} \right )}}{64 c^{\frac{7}{2}} d} \]

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),x)

[Out]

(a + b*x + c*x**2)**(5/2)/(10*c*d) - (-4*a*c + b**2)*(a + b*x + c*x**2)**(3/2)/(
24*c**2*d) + (-4*a*c + b**2)**2*sqrt(a + b*x + c*x**2)/(32*c**3*d) - (-4*a*c + b
**2)**(5/2)*atan(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2))/(64*c**(7
/2)*d)

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Mathematica [A]  time = 0.364455, size = 213, normalized size = 1.43 \[ \frac{2 \sqrt{c} \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)} \left (16 c^2 \left (23 a^2+11 a c x^2+3 c^2 x^4\right )+28 b^2 c \left (c x^2-5 a\right )+16 b c^2 x \left (11 a+6 c x^2\right )+15 b^4-20 b^3 c x\right )+15 \left (b^2-4 a c\right )^3 \log \left (2 c \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)}+4 a c^{3/2}+b^2 \left (-\sqrt{c}\right )\right )-15 \left (b^2-4 a c\right )^3 \log (b+2 c x)}{960 c^{7/2} d \sqrt{4 a c-b^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c]*Sqrt[-b^2 + 4*a*c]*Sqrt[a + x*(b + c*x)]*(15*b^4 - 20*b^3*c*x + 28*b^
2*c*(-5*a + c*x^2) + 16*b*c^2*x*(11*a + 6*c*x^2) + 16*c^2*(23*a^2 + 11*a*c*x^2 +
3*c^2*x^4)) - 15*(b^2 - 4*a*c)^3*Log[b + 2*c*x] + 15*(b^2 - 4*a*c)^3*Log[-(b^2*
Sqrt[c]) + 4*a*c^(3/2) + 2*c*Sqrt[-b^2 + 4*a*c]*Sqrt[a + x*(b + c*x)]])/(960*c^(
7/2)*Sqrt[-b^2 + 4*a*c]*d)

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Maple [B]  time = 0.012, size = 660, normalized size = 4.4 \[{\frac{1}{10\,cd} \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{5}{2}}}}+{\frac{a}{6\,cd} \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}}{24\,{c}^{2}d} \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}}{4\,cd}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}}-{\frac{a{b}^{2}}{8\,{c}^{2}d}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}}+{\frac{{b}^{4}}{64\,d{c}^{3}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}}-{\frac{{a}^{3}}{cd}\ln \left ({1 \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}}+{\frac{3\,{a}^{2}{b}^{2}}{4\,{c}^{2}d}\ln \left ({1 \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}}-{\frac{3\,a{b}^{4}}{16\,d{c}^{3}}\ln \left ({1 \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}}+{\frac{{b}^{6}}{64\,d{c}^{4}}\ln \left ({1 \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}} \]

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),x)

[Out]

1/10/d/c*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(5/2)+1/6/d/c*((x+1/2*b/c)^2*c+1/4*
(4*a*c-b^2)/c)^(3/2)*a-1/24/d/c^2*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)*b^2+
1/4/d/c*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2)*a^2-1/8/d/c^2*(4*(x+1/2*b/c)^2*c
+(4*a*c-b^2)/c)^(1/2)*a*b^2+1/64/d/c^3*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2)*b
^4-1/d/c/((4*a*c-b^2)/c)^(1/2)*ln((1/2*(4*a*c-b^2)/c+1/2*((4*a*c-b^2)/c)^(1/2)*(
4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2))/(x+1/2*b/c))*a^3+3/4/d/c^2/((4*a*c-b^2)/
c)^(1/2)*ln((1/2*(4*a*c-b^2)/c+1/2*((4*a*c-b^2)/c)^(1/2)*(4*(x+1/2*b/c)^2*c+(4*a
*c-b^2)/c)^(1/2))/(x+1/2*b/c))*a^2*b^2-3/16/d/c^3/((4*a*c-b^2)/c)^(1/2)*ln((1/2*
(4*a*c-b^2)/c+1/2*((4*a*c-b^2)/c)^(1/2)*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2))
/(x+1/2*b/c))*a*b^4+1/64/d/c^4/((4*a*c-b^2)/c)^(1/2)*ln((1/2*(4*a*c-b^2)/c+1/2*(
(4*a*c-b^2)/c)^(1/2)*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2))/(x+1/2*b/c))*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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.332859, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-\frac{b^{2} - 4 \, a c}{c}} \log \left (-\frac{4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c - 4 \, \sqrt{c x^{2} + b x + a} c \sqrt{-\frac{b^{2} - 4 \, a c}{c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) + 4 \,{\left (48 \, c^{4} x^{4} + 96 \, b c^{3} x^{3} + 15 \, b^{4} - 140 \, a b^{2} c + 368 \, a^{2} c^{2} + 4 \,{\left (7 \, b^{2} c^{2} + 44 \, a c^{3}\right )} x^{2} - 4 \,{\left (5 \, b^{3} c - 44 \, a b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{1920 \, c^{3} d}, -\frac{15 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{\frac{b^{2} - 4 \, a c}{c}} \arctan \left (-\frac{b^{2} - 4 \, a c}{2 \, \sqrt{c x^{2} + b x + a} c \sqrt{\frac{b^{2} - 4 \, a c}{c}}}\right ) - 2 \,{\left (48 \, c^{4} x^{4} + 96 \, b c^{3} x^{3} + 15 \, b^{4} - 140 \, a b^{2} c + 368 \, a^{2} c^{2} + 4 \,{\left (7 \, b^{2} c^{2} + 44 \, a c^{3}\right )} x^{2} - 4 \,{\left (5 \, b^{3} c - 44 \, a b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{960 \, c^{3} d}\right ] \]

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),x, algorithm="fricas")

[Out]

[1/1920*(15*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(-(b^2 - 4*a*c)/c)*log(-(4*c^2*x^
2 + 4*b*c*x - b^2 + 8*a*c - 4*sqrt(c*x^2 + b*x + a)*c*sqrt(-(b^2 - 4*a*c)/c))/(4
*c^2*x^2 + 4*b*c*x + b^2)) + 4*(48*c^4*x^4 + 96*b*c^3*x^3 + 15*b^4 - 140*a*b^2*c
+ 368*a^2*c^2 + 4*(7*b^2*c^2 + 44*a*c^3)*x^2 - 4*(5*b^3*c - 44*a*b*c^2)*x)*sqrt
(c*x^2 + b*x + a))/(c^3*d), -1/960*(15*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt((b^2
- 4*a*c)/c)*arctan(-1/2*(b^2 - 4*a*c)/(sqrt(c*x^2 + b*x + a)*c*sqrt((b^2 - 4*a*c
)/c))) - 2*(48*c^4*x^4 + 96*b*c^3*x^3 + 15*b^4 - 140*a*b^2*c + 368*a^2*c^2 + 4*(
7*b^2*c^2 + 44*a*c^3)*x^2 - 4*(5*b^3*c - 44*a*b*c^2)*x)*sqrt(c*x^2 + b*x + a))/(
c^3*d)]

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

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),x)

[Out]

(Integral(a**2*sqrt(a + b*x + c*x**2)/(b + 2*c*x), x) + Integral(b**2*x**2*sqrt(
a + b*x + c*x**2)/(b + 2*c*x), x) + Integral(c**2*x**4*sqrt(a + b*x + c*x**2)/(b
+ 2*c*x), x) + Integral(2*a*b*x*sqrt(a + b*x + c*x**2)/(b + 2*c*x), x) + Integr
al(2*a*c*x**2*sqrt(a + b*x + c*x**2)/(b + 2*c*x), x) + Integral(2*b*c*x**3*sqrt(
a + b*x + c*x**2)/(b + 2*c*x), x))/d

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GIAC/XCAS [A]  time = 0.22542, size = 320, normalized size = 2.15 \[ \frac{1}{480} \, \sqrt{c x^{2} + b x + a}{\left (4 \,{\left ({\left (12 \,{\left (\frac{c x}{d} + \frac{2 \, b}{d}\right )} x + \frac{7 \, b^{2} c^{9} d^{5} + 44 \, a c^{10} d^{5}}{c^{10} d^{6}}\right )} x - \frac{5 \, b^{3} c^{8} d^{5} - 44 \, a b c^{9} d^{5}}{c^{10} d^{6}}\right )} x + \frac{15 \, b^{4} c^{7} d^{5} - 140 \, a b^{2} c^{8} d^{5} + 368 \, a^{2} c^{9} d^{5}}{c^{10} d^{6}}\right )} - \frac{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \arctan \left (-\frac{2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} c + b \sqrt{c}}{\sqrt{b^{2} c - 4 \, a c^{2}}}\right )}{32 \, \sqrt{b^{2} c - 4 \, a c^{2}} c^{3} d} \]

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),x, algorithm="giac")

[Out]

1/480*sqrt(c*x^2 + b*x + a)*(4*((12*(c*x/d + 2*b/d)*x + (7*b^2*c^9*d^5 + 44*a*c^
10*d^5)/(c^10*d^6))*x - (5*b^3*c^8*d^5 - 44*a*b*c^9*d^5)/(c^10*d^6))*x + (15*b^4
*c^7*d^5 - 140*a*b^2*c^8*d^5 + 368*a^2*c^9*d^5)/(c^10*d^6)) - 1/32*(b^6 - 12*a*b
^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*arctan(-(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))*c + b*sqrt(c))/sqrt(b^2*c - 4*a*c^2))/(sqrt(b^2*c - 4*a*c^2)*c^3*d)