3.31.31 \(\int \frac {(c+b x+a x^2)^{3/2}}{1-x \sqrt {c+b x+a x^2}} \, dx\)

Optimal. Leaf size=437 \[ -2 \text {RootSum}\left [\text {$\#$1}^4 \sqrt {a}-\text {$\#$1}^3 b+4 \text {$\#$1}^2 a-4 \text {$\#$1} \sqrt {a} b+\text {$\#$1} b c-\sqrt {a} c^2+b^2\& ,\frac {\text {$\#$1}^3 \sqrt {a} c \log \left (-\text {$\#$1}+\sqrt {a x^2+b x+c}-\sqrt {a} x\right )+\text {$\#$1}^2 \sqrt {a} b \log \left (-\text {$\#$1}+\sqrt {a x^2+b x+c}-\sqrt {a} x\right )-\text {$\#$1}^2 b c \log \left (-\text {$\#$1}+\sqrt {a x^2+b x+c}-\sqrt {a} x\right )+2 \text {$\#$1} a^{3/2} \log \left (-\text {$\#$1}+\sqrt {a x^2+b x+c}-\sqrt {a} x\right )-\text {$\#$1} b^2 \log \left (-\text {$\#$1}+\sqrt {a x^2+b x+c}-\sqrt {a} x\right )+\text {$\#$1} \sqrt {a} c^2 \log \left (-\text {$\#$1}+\sqrt {a x^2+b x+c}-\sqrt {a} x\right )+2 \text {$\#$1} a c \log \left (-\text {$\#$1}+\sqrt {a x^2+b x+c}-\sqrt {a} x\right )-a b \log \left (-\text {$\#$1}+\sqrt {a x^2+b x+c}-\sqrt {a} x\right )}{4 \text {$\#$1}^3 \sqrt {a}-3 \text {$\#$1}^2 b+8 \text {$\#$1} a-4 \sqrt {a} b+b c}\& \right ]+\left (\sqrt {a}+c\right ) \log \left (-2 \sqrt {a} \sqrt {a x^2+b x+c}+2 a x+b\right )-\frac {1}{2} x (a x+2 b) \]

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Rubi [F]  time = 0.99, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (c+b x+a x^2\right )^{3/2}}{1-x \sqrt {c+b x+a x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(c + b*x + a*x^2)^(3/2)/(1 - x*Sqrt[c + b*x + a*x^2]),x]

[Out]

-(b*x) - (a*x^2)/2 - (c*Log[-1 + c*x^2 + b*x^3 + a*x^4])/4 + Defer[Int][(c + b*x + a*x^2)^(3/2)/(1 - c*x^2 - b
*x^3 - a*x^4), x] - b*Defer[Int][(-1 + c*x^2 + b*x^3 + a*x^4)^(-1), x] - ((2*a + c^2)*Defer[Int][x/(-1 + c*x^2
+ b*x^3 + a*x^4), x])/2 - (b*c*Defer[Int][x^2/(-1 + c*x^2 + b*x^3 + a*x^4), x])/4

Rubi steps

\begin {align*} \int \frac {\left (c+b x+a x^2\right )^{3/2}}{1-x \sqrt {c+b x+a x^2}} \, dx &=\int \left (\frac {\left (c+b x+a x^2\right )^{3/2}}{1-c x^2-b x^3-a x^4}-\frac {x \left (c+b x+a x^2\right )^2}{-1+c x^2+b x^3+a x^4}\right ) \, dx\\ &=\int \frac {\left (c+b x+a x^2\right )^{3/2}}{1-c x^2-b x^3-a x^4} \, dx-\int \frac {x \left (c+b x+a x^2\right )^2}{-1+c x^2+b x^3+a x^4} \, dx\\ &=\int \frac {\left (c+b x+a x^2\right )^{3/2}}{1-c x^2-b x^3-a x^4} \, dx-\int \left (b+a x+\frac {b+\left (a+c^2\right ) x+b c x^2+a c x^3}{-1+c x^2+b x^3+a x^4}\right ) \, dx\\ &=-b x-\frac {a x^2}{2}+\int \frac {\left (c+b x+a x^2\right )^{3/2}}{1-c x^2-b x^3-a x^4} \, dx-\int \frac {b+\left (a+c^2\right ) x+b c x^2+a c x^3}{-1+c x^2+b x^3+a x^4} \, dx\\ &=-b x-\frac {a x^2}{2}-\frac {1}{4} c \log \left (-1+c x^2+b x^3+a x^4\right )-\frac {\int \frac {4 a b+2 a \left (2 a+c^2\right ) x+a b c x^2}{-1+c x^2+b x^3+a x^4} \, dx}{4 a}+\int \frac {\left (c+b x+a x^2\right )^{3/2}}{1-c x^2-b x^3-a x^4} \, dx\\ &=-b x-\frac {a x^2}{2}-\frac {1}{4} c \log \left (-1+c x^2+b x^3+a x^4\right )-\frac {\int \left (\frac {4 a b}{-1+c x^2+b x^3+a x^4}+\frac {2 a \left (2 a+c^2\right ) x}{-1+c x^2+b x^3+a x^4}+\frac {a b c x^2}{-1+c x^2+b x^3+a x^4}\right ) \, dx}{4 a}+\int \frac {\left (c+b x+a x^2\right )^{3/2}}{1-c x^2-b x^3-a x^4} \, dx\\ &=-b x-\frac {a x^2}{2}-\frac {1}{4} c \log \left (-1+c x^2+b x^3+a x^4\right )-b \int \frac {1}{-1+c x^2+b x^3+a x^4} \, dx-\frac {1}{4} (b c) \int \frac {x^2}{-1+c x^2+b x^3+a x^4} \, dx-\frac {1}{2} \left (2 a+c^2\right ) \int \frac {x}{-1+c x^2+b x^3+a x^4} \, dx+\int \frac {\left (c+b x+a x^2\right )^{3/2}}{1-c x^2-b x^3-a x^4} \, dx\\ \end {align*}

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Mathematica [B]  time = 5.65, size = 9145, normalized size = 20.93 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(c + b*x + a*x^2)^(3/2)/(1 - x*Sqrt[c + b*x + a*x^2]),x]

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 0.72, size = 437, normalized size = 1.00 \begin {gather*} -\frac {1}{2} x (2 b+a x)+\left (\sqrt {a}+c\right ) \log \left (b+2 a x-2 \sqrt {a} \sqrt {c+b x+a x^2}\right )-2 \text {RootSum}\left [b^2-\sqrt {a} c^2-4 \sqrt {a} b \text {$\#$1}+b c \text {$\#$1}+4 a \text {$\#$1}^2-b \text {$\#$1}^3+\sqrt {a} \text {$\#$1}^4\&,\frac {-a b \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+2 a^{3/2} \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}-b^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 a c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}+\sqrt {a} c^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}+\sqrt {a} b \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-b c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+\sqrt {a} c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^3}{-4 \sqrt {a} b+b c+8 a \text {$\#$1}-3 b \text {$\#$1}^2+4 \sqrt {a} \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(c + b*x + a*x^2)^(3/2)/(1 - x*Sqrt[c + b*x + a*x^2]),x]

[Out]

-1/2*(x*(2*b + a*x)) + (Sqrt[a] + c)*Log[b + 2*a*x - 2*Sqrt[a]*Sqrt[c + b*x + a*x^2]] - 2*RootSum[b^2 - Sqrt[a
]*c^2 - 4*Sqrt[a]*b*#1 + b*c*#1 + 4*a*#1^2 - b*#1^3 + Sqrt[a]*#1^4 & , (-(a*b*Log[-(Sqrt[a]*x) + Sqrt[c + b*x
+ a*x^2] - #1]) + 2*a^(3/2)*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 - b^2*Log[-(Sqrt[a]*x) + Sqrt[c
+ b*x + a*x^2] - #1]*#1 + 2*a*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1 + Sqrt[a]*c^2*Log[-(Sqrt[a]*
x) + Sqrt[c + b*x + a*x^2] - #1]*#1 + Sqrt[a]*b*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1^2 - b*c*Log[
-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1^2 + Sqrt[a]*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1^
3)/(-4*Sqrt[a]*b + b*c + 8*a*#1 - 3*b*#1^2 + 4*Sqrt[a]*#1^3) & ]

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^2+b*x+c)^(3/2)/(1-x*(a*x^2+b*x+c)^(1/2)),x, algorithm="fricas")

[Out]

Timed out

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^2+b*x+c)^(3/2)/(1-x*(a*x^2+b*x+c)^(1/2)),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [B]  time = 0.41, size = 487, normalized size = 1.11

method	result	size
default	\(\frac {\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8} a -2 \sqrt {a}\, \textit {\_Z}^{7} b +\textit {\_Z}^{6} b^{2}+2 \sqrt {a}\, \textit {\_Z}^{5} b c +\left (-2 a \,c^{2}-2 b^{2} c -16 a^{2}\right ) \textit {\_Z}^{4}+\left (32 a^{\frac {3}{2}} b +2 \sqrt {a}\, c^{2} b \right ) \textit {\_Z}^{3}+\left (b^{2} c^{2}-24 a \,b^{2}\right ) \textit {\_Z}^{2}+\left (-2 \sqrt {a}\, c^{3} b +8 \sqrt {a}\, b^{3}\right ) \textit {\_Z} +a \,c^{4}-b^{4}\right )}{\sum }\frac {\left (2 a \left (\left (a c +b^{2}\right ) \textit {\_R}^{5}+2 \left (a \,c^{2}+b^{2} c +2 a^{2}\right ) \textit {\_R}^{3}+a \left (c^{3}+3 b^{2}\right ) \textit {\_R} \right )+b \left (-a^{\frac {3}{2}} \textit {\_R}^{6}+\textit {\_R}^{4} \left (-6 a^{\frac {3}{2}} c -b^{2} \sqrt {a}\right )+\textit {\_R}^{2} \left (-12 a^{\frac {5}{2}}-5 a^{\frac {3}{2}} c^{2}\right )-a^{\frac {3}{2}} b^{2}\right )\right ) \ln \left (\sqrt {a \,x^{2}+b x +c}-\sqrt {a}\, x -\textit {\_R} \right )}{4 \textit {\_R}^{7} a +3 \textit {\_R}^{5} b^{2}-4 \textit {\_R}^{3} a \,c^{2}-4 \textit {\_R}^{3} b^{2} c -32 \textit {\_R}^{3} a^{2}+\textit {\_R} \,b^{2} c^{2}-24 a \,b^{2} \textit {\_R} +b \left (-7 \textit {\_R}^{6} \sqrt {a}+5 c \,\textit {\_R}^{4} \sqrt {a}+48 \textit {\_R}^{2} a^{\frac {3}{2}}+3 c^{2} \textit {\_R}^{2} \sqrt {a}-\sqrt {a}\, c^{3}+4 b^{2} \sqrt {a}\right )}}{\sqrt {a}}+\sqrt {a}\, \ln \left (2 \left (-\sqrt {a}\, x +\sqrt {a \,x^{2}+b x +c}\right ) \sqrt {a}-b \right )-\frac {a \,x^{2}}{2}-b x -\left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{4}+b \,\textit {\_Z}^{3}+c \,\textit {\_Z}^{2}-1\right )}{\sum }\frac {\left (\textit {\_R}^{3} a c +\textit {\_R}^{2} b c +\left (c^{2}+a \right ) \textit {\_R} +b \right ) \ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3} a +3 \textit {\_R}^{2} b +2 \textit {\_R} c}\right )\)	\(487\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^2+b*x+c)^(3/2)/(1-x*(a*x^2+b*x+c)^(1/2)),x,method=_RETURNVERBOSE)

[Out]

1/a^(1/2)*sum((2*a*((a*c+b^2)*_R^5+2*(a*c^2+b^2*c+2*a^2)*_R^3+a*(c^3+3*b^2)*_R)+b*(-a^(3/2)*_R^6+_R^4*(-6*a^(3
/2)*c-b^2*a^(1/2))+_R^2*(-12*a^(5/2)-5*a^(3/2)*c^2)-a^(3/2)*b^2))/(4*_R^7*a+3*_R^5*b^2-4*_R^3*a*c^2-4*_R^3*b^2
*c-32*_R^3*a^2+_R*b^2*c^2-24*a*b^2*_R+b*(-7*_R^6*a^(1/2)+5*c*_R^4*a^(1/2)+48*_R^2*a^(3/2)+3*c^2*_R^2*a^(1/2)-a
^(1/2)*c^3+4*b^2*a^(1/2)))*ln((a*x^2+b*x+c)^(1/2)-a^(1/2)*x-_R),_R=RootOf(_Z^8*a-2*a^(1/2)*_Z^7*b+_Z^6*b^2+2*a
^(1/2)*_Z^5*b*c+(-2*a*c^2-2*b^2*c-16*a^2)*_Z^4+(32*a^(3/2)*b+2*a^(1/2)*c^2*b)*_Z^3+(b^2*c^2-24*a*b^2)*_Z^2+(-2
*a^(1/2)*c^3*b+8*a^(1/2)*b^3)*_Z+a*c^4-b^4))+a^(1/2)*ln(2*(-a^(1/2)*x+(a*x^2+b*x+c)^(1/2))*a^(1/2)-b)-1/2*a*x^
2-b*x-sum((_R^3*a*c+_R^2*b*c+(c^2+a)*_R+b)/(4*_R^3*a+3*_R^2*b+2*_R*c)*ln(x-_R),_R=RootOf(_Z^4*a+_Z^3*b+_Z^2*c-
1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {1}{2} \, a x^{2} - b x - c \log \relax (x) - \int \frac {a x^{2} + b x + c}{\sqrt {a x^{2} + b x + c} x^{2} - x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^2+b*x+c)^(3/2)/(1-x*(a*x^2+b*x+c)^(1/2)),x, algorithm="maxima")

[Out]

-1/2*a*x^2 - b*x - c*log(x) - integrate((a*x^2 + b*x + c)/(sqrt(a*x^2 + b*x + c)*x^2 - x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {{\left (a\,x^2+b\,x+c\right )}^{3/2}}{x\,\sqrt {a\,x^2+b\,x+c}-1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(c + b*x + a*x^2)^(3/2)/(x*(c + b*x + a*x^2)^(1/2) - 1),x)

[Out]

int(-(c + b*x + a*x^2)^(3/2)/(x*(c + b*x + a*x^2)^(1/2) - 1), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {c \sqrt {a x^{2} + b x + c}}{x \sqrt {a x^{2} + b x + c} - 1}\, dx - \int \frac {a x^{2} \sqrt {a x^{2} + b x + c}}{x \sqrt {a x^{2} + b x + c} - 1}\, dx - \int \frac {b x \sqrt {a x^{2} + b x + c}}{x \sqrt {a x^{2} + b x + c} - 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**2+b*x+c)**(3/2)/(1-x*(a*x**2+b*x+c)**(1/2)),x)

[Out]

-Integral(c*sqrt(a*x**2 + b*x + c)/(x*sqrt(a*x**2 + b*x + c) - 1), x) - Integral(a*x**2*sqrt(a*x**2 + b*x + c)
/(x*sqrt(a*x**2 + b*x + c) - 1), x) - Integral(b*x*sqrt(a*x**2 + b*x + c)/(x*sqrt(a*x**2 + b*x + c) - 1), x)

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