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

Optimal. Leaf size=179 \[ -\frac{3 \left (A \left (4 a c+b^2\right )+4 a b B\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{8 \sqrt{a}}+\frac{3 \left (4 a B c+4 A b c+b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 \sqrt{c}}-\frac{(A-B x) \left (a+b x+c x^2\right )^{3/2}}{2 x^2}-\frac{3 \sqrt{a+b x+c x^2} (2 a B-x (2 A c+b B)+A b)}{4 x} \]

[Out]

(-3*(A*b + 2*a*B - (b*B + 2*A*c)*x)*Sqrt[a + b*x + c*x^2])/(4*x) - ((A - B*x)*(a
+ b*x + c*x^2)^(3/2))/(2*x^2) - (3*(4*a*b*B + A*(b^2 + 4*a*c))*ArcTanh[(2*a + b
*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(8*Sqrt[a]) + (3*(b^2*B + 4*A*b*c + 4*a*
B*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*Sqrt[c])

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Rubi [A]  time = 0.44945, antiderivative size = 179, 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 \[ -\frac{3 \left (A \left (4 a c+b^2\right )+4 a b B\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{8 \sqrt{a}}+\frac{3 \left (4 a B c+4 A b c+b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 \sqrt{c}}-\frac{(A-B x) \left (a+b x+c x^2\right )^{3/2}}{2 x^2}-\frac{3 \sqrt{a+b x+c x^2} (2 a B-x (2 A c+b B)+A b)}{4 x} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(A*b + 2*a*B - (b*B + 2*A*c)*x)*Sqrt[a + b*x + c*x^2])/(4*x) - ((A - B*x)*(a
+ b*x + c*x^2)^(3/2))/(2*x^2) - (3*(4*a*b*B + A*(b^2 + 4*a*c))*ArcTanh[(2*a + b
*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(8*Sqrt[a]) + (3*(b^2*B + 4*A*b*c + 4*a*
B*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*Sqrt[c])

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Rubi in Sympy [A]  time = 54.2547, size = 178, normalized size = 0.99 \[ - \frac{3 \sqrt{a + b x + c x^{2}} \left (2 A b + 4 B a - x \left (4 A c + 2 B b\right )\right )}{8 x} - \frac{\left (2 A - 2 B x\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{4 x^{2}} + \frac{3 \left (4 A b c + 4 B a c + B b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{8 \sqrt{c}} - \frac{3 \left (4 A a c + A b^{2} + 4 B a b\right ) \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{8 \sqrt{a}} \]

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**3,x)

[Out]

-3*sqrt(a + b*x + c*x**2)*(2*A*b + 4*B*a - x*(4*A*c + 2*B*b))/(8*x) - (2*A - 2*B
*x)*(a + b*x + c*x**2)**(3/2)/(4*x**2) + 3*(4*A*b*c + 4*B*a*c + B*b**2)*atanh((b
+ 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(8*sqrt(c)) - 3*(4*A*a*c + A*b**2
+ 4*B*a*b)*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x**2)))/(8*sqrt(a))

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Mathematica [A]  time = 0.770672, size = 183, normalized size = 1.02 \[ \frac{1}{8} \left (\frac{3 \log (x) \left (A \left (4 a c+b^2\right )+4 a b B\right )}{\sqrt{a}}-\frac{3 \left (A \left (4 a c+b^2\right )+4 a b B\right ) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )}{\sqrt{a}}+\frac{3 \left (4 a B c+4 A b c+b^2 B\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{\sqrt{c}}+\frac{2 \sqrt{a+x (b+c x)} (x (A (4 c x-5 b)+B x (5 b+2 c x))-2 a (A+2 B x))}{x^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[a + x*(b + c*x)]*(-2*a*(A + 2*B*x) + x*(B*x*(5*b + 2*c*x) + A*(-5*b + 4
*c*x))))/x^2 + (3*(4*a*b*B + A*(b^2 + 4*a*c))*Log[x])/Sqrt[a] - (3*(4*a*b*B + A*
(b^2 + 4*a*c))*Log[2*a + b*x + 2*Sqrt[a]*Sqrt[a + x*(b + c*x)]])/Sqrt[a] + (3*(b
^2*B + 4*A*b*c + 4*a*B*c)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/Sqrt
[c])/8

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Maple [B]  time = 0.016, size = 463, normalized size = 2.6 \[ -{\frac{A}{2\,a{x}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{2}}}}-{\frac{Ab}{4\,{a}^{2}x} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{2}}}}+{\frac{{b}^{2}A}{4\,{a}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{b}^{2}A}{4\,a}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,{b}^{2}A}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{Abcx}{4\,{a}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,Abcx}{4\,a}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,Ab}{2}\sqrt{c}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ) }+{\frac{Ac}{2\,a} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,Ac}{2}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,Ac}{2}\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ) }-{\frac{B}{ax} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{2}}}}+{\frac{Bb}{a} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{b}^{2}B}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{9\,Bb}{4}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,Bb}{2}\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ) }+{\frac{Bcx}{a} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,Bcx}{2}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,Ba}{2}\sqrt{c}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*A/a/x^2*(c*x^2+b*x+a)^(5/2)-1/4*A*b/a^2/x*(c*x^2+b*x+a)^(5/2)+1/4*A*b^2/a^2
*(c*x^2+b*x+a)^(3/2)+3/4*A*b^2/a*(c*x^2+b*x+a)^(1/2)-3/8*A*b^2/a^(1/2)*ln((2*a+b
*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+1/4*A*b/a^2*c*(c*x^2+b*x+a)^(3/2)*x+3/4*A*b
/a*c*(c*x^2+b*x+a)^(1/2)*x+3/2*A*b*c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^
(1/2))+1/2*A/a*c*(c*x^2+b*x+a)^(3/2)+3/2*A*c*(c*x^2+b*x+a)^(1/2)-3/2*A*a^(1/2)*c
*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-B/a/x*(c*x^2+b*x+a)^(5/2)+B*b/a*(
c*x^2+b*x+a)^(3/2)+3/8*B*b^2/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))
+9/4*B*b*(c*x^2+b*x+a)^(1/2)-3/2*B*b*a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)
^(1/2))/x)+B/a*c*(c*x^2+b*x+a)^(3/2)*x+3/2*B*c*(c*x^2+b*x+a)^(1/2)*x+3/2*B*c^(1/
2)*a*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))

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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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.34766, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(3*(B*b^2 + 4*(B*a + A*b)*c)*sqrt(a)*x^2*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)) + 3*(4*B*a*b + A*b^2
+ 4*A*a*c)*sqrt(c)*x^2*log((4*(a*b*x + 2*a^2)*sqrt(c*x^2 + b*x + a) - (8*a*b*x +
(b^2 + 4*a*c)*x^2 + 8*a^2)*sqrt(a))/x^2) + 4*(2*B*c*x^3 + (5*B*b + 4*A*c)*x^2 -
2*A*a - (4*B*a + 5*A*b)*x)*sqrt(c*x^2 + b*x + a)*sqrt(a)*sqrt(c))/(sqrt(a)*sqrt
(c)*x^2), 1/16*(6*(B*b^2 + 4*(B*a + A*b)*c)*sqrt(a)*x^2*arctan(1/2*(2*c*x + b)*s
qrt(-c)/(sqrt(c*x^2 + b*x + a)*c)) + 3*(4*B*a*b + A*b^2 + 4*A*a*c)*sqrt(-c)*x^2*
log((4*(a*b*x + 2*a^2)*sqrt(c*x^2 + b*x + a) - (8*a*b*x + (b^2 + 4*a*c)*x^2 + 8*
a^2)*sqrt(a))/x^2) + 4*(2*B*c*x^3 + (5*B*b + 4*A*c)*x^2 - 2*A*a - (4*B*a + 5*A*b
)*x)*sqrt(c*x^2 + b*x + a)*sqrt(a)*sqrt(-c))/(sqrt(a)*sqrt(-c)*x^2), -1/16*(6*(4
*B*a*b + A*b^2 + 4*A*a*c)*sqrt(c)*x^2*arctan(1/2*(b*x + 2*a)*sqrt(-a)/(sqrt(c*x^
2 + b*x + a)*a)) - 3*(B*b^2 + 4*(B*a + A*b)*c)*sqrt(-a)*x^2*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)) - 4*(2*B
*c*x^3 + (5*B*b + 4*A*c)*x^2 - 2*A*a - (4*B*a + 5*A*b)*x)*sqrt(c*x^2 + b*x + a)*
sqrt(-a)*sqrt(c))/(sqrt(-a)*sqrt(c)*x^2), -1/8*(3*(4*B*a*b + A*b^2 + 4*A*a*c)*sq
rt(-c)*x^2*arctan(1/2*(b*x + 2*a)*sqrt(-a)/(sqrt(c*x^2 + b*x + a)*a)) - 3*(B*b^2
+ 4*(B*a + A*b)*c)*sqrt(-a)*x^2*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b
*x + a)*c)) - 2*(2*B*c*x^3 + (5*B*b + 4*A*c)*x^2 - 2*A*a - (4*B*a + 5*A*b)*x)*sq
rt(c*x^2 + b*x + a)*sqrt(-a)*sqrt(-c))/(sqrt(-a)*sqrt(-c)*x^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.92563, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x