3.927 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x} \, dx\)

Optimal. Leaf size=218 \[ a^{3/2} (-A) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )+\frac{\left (\left (b^2-4 a c\right ) \left (-12 a B c-8 A b c+3 b^2 B\right )+64 a A b c^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{5/2}}-\frac{\sqrt{a+b x+c x^2} \left (2 c x \left (-12 a B c-8 A b c+3 b^2 B\right )-64 a A c^2-12 a b B c-8 A b^2 c+3 b^3 B\right )}{64 c^2}+\frac{\left (a+b x+c x^2\right )^{3/2} (8 A c+3 b B+6 B c x)}{24 c} \]

[Out]

-((3*b^3*B - 8*A*b^2*c - 12*a*b*B*c - 64*a*A*c^2 + 2*c*(3*b^2*B - 8*A*b*c - 12*a
*B*c)*x)*Sqrt[a + b*x + c*x^2])/(64*c^2) + ((3*b*B + 8*A*c + 6*B*c*x)*(a + b*x +
c*x^2)^(3/2))/(24*c) - a^(3/2)*A*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x +
c*x^2])] + ((64*a*A*b*c^2 + (b^2 - 4*a*c)*(3*b^2*B - 8*A*b*c - 12*a*B*c))*ArcTan
h[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(128*c^(5/2))

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Rubi [A]  time = 0.59845, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ a^{3/2} (-A) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )+\frac{\left (\left (b^2-4 a c\right ) \left (-12 a B c-8 A b c+3 b^2 B\right )+64 a A b c^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{5/2}}-\frac{\sqrt{a+b x+c x^2} \left (2 c x \left (-12 a B c-8 A b c+3 b^2 B\right )-64 a A c^2-12 a b B c-8 A b^2 c+3 b^3 B\right )}{64 c^2}+\frac{\left (a+b x+c x^2\right )^{3/2} (8 A c+3 b B+6 B c x)}{24 c} \]

Antiderivative was successfully verified.

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

[Out]

-((3*b^3*B - 8*A*b^2*c - 12*a*b*B*c - 64*a*A*c^2 + 2*c*(3*b^2*B - 8*A*b*c - 12*a
*B*c)*x)*Sqrt[a + b*x + c*x^2])/(64*c^2) + ((3*b*B + 8*A*c + 6*B*c*x)*(a + b*x +
c*x^2)^(3/2))/(24*c) - a^(3/2)*A*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x +
c*x^2])] + ((64*a*A*b*c^2 + (b^2 - 4*a*c)*(3*b^2*B - 8*A*b*c - 12*a*B*c))*ArcTan
h[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(128*c^(5/2))

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Rubi in Sympy [A]  time = 58.3662, size = 221, normalized size = 1.01 \[ - A a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )} + \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (4 A c + \frac{3 B b}{2} + 3 B c x\right )}{12 c} - \frac{\sqrt{a + b x + c x^{2}} \left (- 16 A a c^{2} + \frac{b \left (- 8 A b c - 12 B a c + 3 B b^{2}\right )}{4} + \frac{c x \left (- 8 A b c - 12 B a c + 3 B b^{2}\right )}{2}\right )}{16 c^{2}} + \frac{\left (64 A a b c^{2} + \left (- 4 a c + b^{2}\right ) \left (- 8 A b c - 12 B a c + 3 B b^{2}\right )\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{128 c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a**(3/2)*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x**2))) + (a + b*x + c
*x**2)**(3/2)*(4*A*c + 3*B*b/2 + 3*B*c*x)/(12*c) - sqrt(a + b*x + c*x**2)*(-16*A
*a*c**2 + b*(-8*A*b*c - 12*B*a*c + 3*B*b**2)/4 + c*x*(-8*A*b*c - 12*B*a*c + 3*B*
b**2)/2)/(16*c**2) + (64*A*a*b*c**2 + (-4*a*c + b**2)*(-8*A*b*c - 12*B*a*c + 3*B
*b**2))*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(128*c**(5/2))

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Mathematica [A]  time = 0.505093, size = 211, normalized size = 0.97 \[ -a^{3/2} A \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )+a^{3/2} A \log (x)+\frac{\left (48 a^2 B c^2+96 a A b c^2-24 a b^2 B c-8 A b^3 c+3 b^4 B\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{128 c^{5/2}}+\frac{\sqrt{a+x (b+c x)} \left (4 b c (15 a B+2 c x (14 A+9 B x))+8 c^2 \left (32 a A+15 a B x+8 A c x^2+6 B c x^3\right )+6 b^2 c (4 A+B x)-9 b^3 B\right )}{192 c^2} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + x*(b + c*x)]*(-9*b^3*B + 6*b^2*c*(4*A + B*x) + 8*c^2*(32*a*A + 15*a*B*
x + 8*A*c*x^2 + 6*B*c*x^3) + 4*b*c*(15*a*B + 2*c*x*(14*A + 9*B*x))))/(192*c^2) +
a^(3/2)*A*Log[x] - a^(3/2)*A*Log[2*a + b*x + 2*Sqrt[a]*Sqrt[a + x*(b + c*x)]] +
((3*b^4*B - 8*A*b^3*c - 24*a*b^2*B*c + 96*a*A*b*c^2 + 48*a^2*B*c^2)*Log[b + 2*c
*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(128*c^(5/2))

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Maple [B]  time = 0.012, size = 390, normalized size = 1.8 \[{\frac{Bx}{4} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{Bb}{8\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,aBx}{8}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,{b}^{2}Bx}{32\,c}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,abB}{16\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,B{b}^{3}}{64\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,{a}^{2}B}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}-{\frac{3\,a{b}^{2}B}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{3\,{b}^{4}B}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{A}{3} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{Abx}{4}\sqrt{c{x}^{2}+bx+a}}+{\frac{{b}^{2}A}{8\,c}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,abA}{4}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}-{\frac{A{b}^{3}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+Aa\sqrt{c{x}^{2}+bx+a}-A{a}^{{\frac{3}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*B*x*(c*x^2+b*x+a)^(3/2)+1/8*B/c*(c*x^2+b*x+a)^(3/2)*b+3/8*B*(c*x^2+b*x+a)^(1
/2)*x*a-3/32*B/c*(c*x^2+b*x+a)^(1/2)*x*b^2+3/16*B/c*(c*x^2+b*x+a)^(1/2)*b*a-3/64
*B/c^2*(c*x^2+b*x+a)^(1/2)*b^3+3/8*B/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a
)^(1/2))*a^2-3/16*B/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^2*a+3/
128*B/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^4+1/3*A*(c*x^2+b*x+a
)^(3/2)+1/4*A*b*(c*x^2+b*x+a)^(1/2)*x+1/8*A/c*(c*x^2+b*x+a)^(1/2)*b^2+3/4*A*b/c^
(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-1/16*A/c^(3/2)*ln((1/2*b+c*x
)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^3+A*a*(c*x^2+b*x+a)^(1/2)-A*a^(3/2)*ln((2*a+b*x
+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)

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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)*(B*x + A)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 4.80402, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/768*(384*A*a^(3/2)*c^(5/2)*log(-(8*a*b*x + (b^2 + 4*a*c)*x^2 - 4*sqrt(c*x^2 +
b*x + a)*(b*x + 2*a)*sqrt(a) + 8*a^2)/x^2) + 4*(48*B*c^3*x^3 - 9*B*b^3 + 256*A*
a*c^2 + 8*(9*B*b*c^2 + 8*A*c^3)*x^2 + 12*(5*B*a*b + 2*A*b^2)*c + 2*(3*B*b^2*c +
4*(15*B*a + 14*A*b)*c^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) + 3*(3*B*b^4 + 48*(B*a
^2 + 2*A*a*b)*c^2 - 8*(3*B*a*b^2 + A*b^3)*c)*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^(5/2), 1/384*(192*A*
a^(3/2)*sqrt(-c)*c^2*log(-(8*a*b*x + (b^2 + 4*a*c)*x^2 - 4*sqrt(c*x^2 + b*x + a)
*(b*x + 2*a)*sqrt(a) + 8*a^2)/x^2) + 2*(48*B*c^3*x^3 - 9*B*b^3 + 256*A*a*c^2 + 8
*(9*B*b*c^2 + 8*A*c^3)*x^2 + 12*(5*B*a*b + 2*A*b^2)*c + 2*(3*B*b^2*c + 4*(15*B*a
+ 14*A*b)*c^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) + 3*(3*B*b^4 + 48*(B*a^2 + 2*A
*a*b)*c^2 - 8*(3*B*a*b^2 + A*b^3)*c)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2
+ b*x + a)*c)))/(sqrt(-c)*c^2), -1/768*(768*A*sqrt(-a)*a*c^(5/2)*arctan(1/2*(b*
x + 2*a)/(sqrt(c*x^2 + b*x + a)*sqrt(-a))) - 4*(48*B*c^3*x^3 - 9*B*b^3 + 256*A*a
*c^2 + 8*(9*B*b*c^2 + 8*A*c^3)*x^2 + 12*(5*B*a*b + 2*A*b^2)*c + 2*(3*B*b^2*c + 4
*(15*B*a + 14*A*b)*c^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) - 3*(3*B*b^4 + 48*(B*a^
2 + 2*A*a*b)*c^2 - 8*(3*B*a*b^2 + A*b^3)*c)*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^(5/2), -1/384*(384*A*
sqrt(-a)*a*sqrt(-c)*c^2*arctan(1/2*(b*x + 2*a)/(sqrt(c*x^2 + b*x + a)*sqrt(-a)))
- 2*(48*B*c^3*x^3 - 9*B*b^3 + 256*A*a*c^2 + 8*(9*B*b*c^2 + 8*A*c^3)*x^2 + 12*(5
*B*a*b + 2*A*b^2)*c + 2*(3*B*b^2*c + 4*(15*B*a + 14*A*b)*c^2)*x)*sqrt(c*x^2 + b*
x + a)*sqrt(-c) - 3*(3*B*b^4 + 48*(B*a^2 + 2*A*a*b)*c^2 - 8*(3*B*a*b^2 + A*b^3)*
c)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/(sqrt(-c)*c^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((A + B*x)*(a + b*x + c*x**2)**(3/2)/x, x)

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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)*(B*x + A)/x,x, algorithm="giac")

[Out]

Exception raised: TypeError