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

Optimal. Leaf size=757 \[ -2 \text {RootSum}\left [\text {$\#$1}^6 a^{5/2} b^2-\text {$\#$1}^5 a^2 b^3-4 \text {$\#$1}^5 a^2 b c-\text {$\#$1}^4 a^{5/2} b^2 c+6 \text {$\#$1}^4 a^{3/2} b^2 c+4 \text {$\#$1}^4 a^{3/2} c^2-8 \text {$\#$1}^3 a^{5/2} b c+2 \text {$\#$1}^3 a^2 b^3 c-2 \text {$\#$1}^3 a b^3 c-8 \text {$\#$1}^3 a b c^2-\text {$\#$1}^2 a^{5/2} b^2 c^2-4 \text {$\#$1}^2 a^{3/2} b^2 c^2+4 \text {$\#$1}^2 a^{3/2} c^3+12 \text {$\#$1}^2 a^2 b^2 c+5 \text {$\#$1}^2 \sqrt {a} b^2 c^2-6 \text {$\#$1} a^{3/2} b^3 c-\text {$\#$1} a^2 b^3 c^2+4 \text {$\#$1} a^2 b c^3+2 \text {$\#$1} a b^3 c^2-4 \text {$\#$1} a b c^3-\text {$\#$1} b^3 c^2+a^{5/2} b^2 c^3-2 a^{3/2} b^2 c^3+a b^4 c+\sqrt {a} b^2 c^3\& ,\frac {2 \text {$\#$1}^3 a \log \left (-\text {$\#$1}+\sqrt {a x^2+b x+c}-\sqrt {a} x\right )-3 \text {$\#$1}^2 \sqrt {a} b \log \left (-\text {$\#$1}+\sqrt {a x^2+b x+c}-\sqrt {a} x\right )+\text {$\#$1} b^2 \log \left (-\text {$\#$1}+\sqrt {a x^2+b x+c}-\sqrt {a} x\right )+2 \text {$\#$1} a c \log \left (-\text {$\#$1}+\sqrt {a x^2+b x+c}-\sqrt {a} x\right )-\sqrt {a} b c \log \left (-\text {$\#$1}+\sqrt {a x^2+b x+c}-\sqrt {a} x\right )}{-6 \text {$\#$1}^5 a^{5/2} b^2+5 \text {$\#$1}^4 a^2 b^3+20 \text {$\#$1}^4 a^2 b c+4 \text {$\#$1}^3 a^{5/2} b^2 c-24 \text {$\#$1}^3 a^{3/2} b^2 c-16 \text {$\#$1}^3 a^{3/2} c^2+24 \text {$\#$1}^2 a^{5/2} b c-6 \text {$\#$1}^2 a^2 b^3 c+6 \text {$\#$1}^2 a b^3 c+24 \text {$\#$1}^2 a b c^2+2 \text {$\#$1} a^{5/2} b^2 c^2+8 \text {$\#$1} a^{3/2} b^2 c^2-8 \text {$\#$1} a^{3/2} c^3-24 \text {$\#$1} a^2 b^2 c-10 \text {$\#$1} \sqrt {a} b^2 c^2+6 a^{3/2} b^3 c+a^2 b^3 c^2-4 a^2 b c^3-2 a b^3 c^2+4 a b c^3+b^3 c^2}\& \right ] \]

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Rubi [F]  time = 6.28, 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 {1}{a b c-(c+a b x)^2 \sqrt {c+b x+a x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

a*b*c*Defer[Int][(a^2*b^2*c^2*(1 - c^3/(a^2*b^2)) - (1 + 4*a)*b*c^4*x - 4*a*b^2*c^3*(1 + (6*a + c/b^2)/4)*x^2
- 6*a^2*b^3*c^2*(1 + (2*(a*b^2 + c))/(3*b^2))*x^3 - 4*a^3*b^4*c*(1 + (a + (6*c)/b^2)/4)*x^4 - a^4*b^5*(1 + (4*
c)/b^2)*x^5 - a^5*b^4*x^6)^(-1), x] + c^2*Defer[Int][Sqrt[c + b*x + a*x^2]/(a^2*b^2*c^2*(1 - c^3/(a^2*b^2)) -
(1 + 4*a)*b*c^4*x - 4*a*b^2*c^3*(1 + (6*a + c/b^2)/4)*x^2 - 6*a^2*b^3*c^2*(1 + (2*(a*b^2 + c))/(3*b^2))*x^3 -
4*a^3*b^4*c*(1 + (a + (6*c)/b^2)/4)*x^4 - a^4*b^5*(1 + (4*c)/b^2)*x^5 - a^5*b^4*x^6), x] + 2*a*b*c*Defer[Int][
(x*Sqrt[c + b*x + a*x^2])/(a^2*b^2*c^2*(1 - c^3/(a^2*b^2)) - (1 + 4*a)*b*c^4*x - 4*a*b^2*c^3*(1 + (6*a + c/b^2
)/4)*x^2 - 6*a^2*b^3*c^2*(1 + (2*(a*b^2 + c))/(3*b^2))*x^3 - 4*a^3*b^4*c*(1 + (a + (6*c)/b^2)/4)*x^4 - a^4*b^5
*(1 + (4*c)/b^2)*x^5 - a^5*b^4*x^6), x] + a^2*b^2*Defer[Int][(x^2*Sqrt[c + b*x + a*x^2])/(a^2*b^2*c^2*(1 - c^3
/(a^2*b^2)) - (1 + 4*a)*b*c^4*x - 4*a*b^2*c^3*(1 + (6*a + c/b^2)/4)*x^2 - 6*a^2*b^3*c^2*(1 + (2*(a*b^2 + c))/(
3*b^2))*x^3 - 4*a^3*b^4*c*(1 + (a + (6*c)/b^2)/4)*x^4 - a^4*b^5*(1 + (4*c)/b^2)*x^5 - a^5*b^4*x^6), x]

Rubi steps

\begin {align*} \int \frac {1}{a b c-(c+a b x)^2 \sqrt {c+b x+a x^2}} \, dx &=\int \left (\frac {a b c}{a^2 b^2 c^2 \left (1-\frac {c^3}{a^2 b^2}\right )-(1+4 a) b c^4 x-4 a b^2 c^3 \left (1+\frac {1}{4} \left (6 a+\frac {c}{b^2}\right )\right ) x^2-6 a^2 b^3 c^2 \left (1+\frac {2 \left (a b^2+c\right )}{3 b^2}\right ) x^3-4 a^3 b^4 c \left (1+\frac {1}{4} \left (a+\frac {6 c}{b^2}\right )\right ) x^4-a^4 b^5 \left (1+\frac {4 c}{b^2}\right ) x^5-a^5 b^4 x^6}+\frac {c^2 \sqrt {c+b x+a x^2}}{a^2 b^2 c^2 \left (1-\frac {c^3}{a^2 b^2}\right )-(1+4 a) b c^4 x-4 a b^2 c^3 \left (1+\frac {1}{4} \left (6 a+\frac {c}{b^2}\right )\right ) x^2-6 a^2 b^3 c^2 \left (1+\frac {2 \left (a b^2+c\right )}{3 b^2}\right ) x^3-4 a^3 b^4 c \left (1+\frac {1}{4} \left (a+\frac {6 c}{b^2}\right )\right ) x^4-a^4 b^5 \left (1+\frac {4 c}{b^2}\right ) x^5-a^5 b^4 x^6}+\frac {2 a b c x \sqrt {c+b x+a x^2}}{a^2 b^2 c^2 \left (1-\frac {c^3}{a^2 b^2}\right )-(1+4 a) b c^4 x-4 a b^2 c^3 \left (1+\frac {1}{4} \left (6 a+\frac {c}{b^2}\right )\right ) x^2-6 a^2 b^3 c^2 \left (1+\frac {2 \left (a b^2+c\right )}{3 b^2}\right ) x^3-4 a^3 b^4 c \left (1+\frac {1}{4} \left (a+\frac {6 c}{b^2}\right )\right ) x^4-a^4 b^5 \left (1+\frac {4 c}{b^2}\right ) x^5-a^5 b^4 x^6}+\frac {a^2 b^2 x^2 \sqrt {c+b x+a x^2}}{a^2 b^2 c^2 \left (1-\frac {c^3}{a^2 b^2}\right )-(1+4 a) b c^4 x-4 a b^2 c^3 \left (1+\frac {1}{4} \left (6 a+\frac {c}{b^2}\right )\right ) x^2-6 a^2 b^3 c^2 \left (1+\frac {2 \left (a b^2+c\right )}{3 b^2}\right ) x^3-4 a^3 b^4 c \left (1+\frac {1}{4} \left (a+\frac {6 c}{b^2}\right )\right ) x^4-a^4 b^5 \left (1+\frac {4 c}{b^2}\right ) x^5-a^5 b^4 x^6}\right ) \, dx\\ &=\left (a^2 b^2\right ) \int \frac {x^2 \sqrt {c+b x+a x^2}}{a^2 b^2 c^2 \left (1-\frac {c^3}{a^2 b^2}\right )-(1+4 a) b c^4 x-4 a b^2 c^3 \left (1+\frac {1}{4} \left (6 a+\frac {c}{b^2}\right )\right ) x^2-6 a^2 b^3 c^2 \left (1+\frac {2 \left (a b^2+c\right )}{3 b^2}\right ) x^3-4 a^3 b^4 c \left (1+\frac {1}{4} \left (a+\frac {6 c}{b^2}\right )\right ) x^4-a^4 b^5 \left (1+\frac {4 c}{b^2}\right ) x^5-a^5 b^4 x^6} \, dx+(a b c) \int \frac {1}{a^2 b^2 c^2 \left (1-\frac {c^3}{a^2 b^2}\right )-(1+4 a) b c^4 x-4 a b^2 c^3 \left (1+\frac {1}{4} \left (6 a+\frac {c}{b^2}\right )\right ) x^2-6 a^2 b^3 c^2 \left (1+\frac {2 \left (a b^2+c\right )}{3 b^2}\right ) x^3-4 a^3 b^4 c \left (1+\frac {1}{4} \left (a+\frac {6 c}{b^2}\right )\right ) x^4-a^4 b^5 \left (1+\frac {4 c}{b^2}\right ) x^5-a^5 b^4 x^6} \, dx+(2 a b c) \int \frac {x \sqrt {c+b x+a x^2}}{a^2 b^2 c^2 \left (1-\frac {c^3}{a^2 b^2}\right )-(1+4 a) b c^4 x-4 a b^2 c^3 \left (1+\frac {1}{4} \left (6 a+\frac {c}{b^2}\right )\right ) x^2-6 a^2 b^3 c^2 \left (1+\frac {2 \left (a b^2+c\right )}{3 b^2}\right ) x^3-4 a^3 b^4 c \left (1+\frac {1}{4} \left (a+\frac {6 c}{b^2}\right )\right ) x^4-a^4 b^5 \left (1+\frac {4 c}{b^2}\right ) x^5-a^5 b^4 x^6} \, dx+c^2 \int \frac {\sqrt {c+b x+a x^2}}{a^2 b^2 c^2 \left (1-\frac {c^3}{a^2 b^2}\right )-(1+4 a) b c^4 x-4 a b^2 c^3 \left (1+\frac {1}{4} \left (6 a+\frac {c}{b^2}\right )\right ) x^2-6 a^2 b^3 c^2 \left (1+\frac {2 \left (a b^2+c\right )}{3 b^2}\right ) x^3-4 a^3 b^4 c \left (1+\frac {1}{4} \left (a+\frac {6 c}{b^2}\right )\right ) x^4-a^4 b^5 \left (1+\frac {4 c}{b^2}\right ) x^5-a^5 b^4 x^6} \, dx\\ \end {align*}

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Mathematica [F]  time = 2.06, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{a b c-(c+a b x)^2 \sqrt {c+b x+a x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 3.31, size = 757, normalized size = 1.00 \begin {gather*} -2 \text {RootSum}\left [a b^4 c+\sqrt {a} b^2 c^3-2 a^{3/2} b^2 c^3+a^{5/2} b^2 c^3-6 a^{3/2} b^3 c \text {$\#$1}-b^3 c^2 \text {$\#$1}+2 a b^3 c^2 \text {$\#$1}-a^2 b^3 c^2 \text {$\#$1}-4 a b c^3 \text {$\#$1}+4 a^2 b c^3 \text {$\#$1}+12 a^2 b^2 c \text {$\#$1}^2+5 \sqrt {a} b^2 c^2 \text {$\#$1}^2-4 a^{3/2} b^2 c^2 \text {$\#$1}^2-a^{5/2} b^2 c^2 \text {$\#$1}^2+4 a^{3/2} c^3 \text {$\#$1}^2-8 a^{5/2} b c \text {$\#$1}^3-2 a b^3 c \text {$\#$1}^3+2 a^2 b^3 c \text {$\#$1}^3-8 a b c^2 \text {$\#$1}^3+6 a^{3/2} b^2 c \text {$\#$1}^4-a^{5/2} b^2 c \text {$\#$1}^4+4 a^{3/2} c^2 \text {$\#$1}^4-a^2 b^3 \text {$\#$1}^5-4 a^2 b c \text {$\#$1}^5+a^{5/2} b^2 \text {$\#$1}^6\&,\frac {-\sqrt {a} b c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+b^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 a c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}-3 \sqrt {a} b \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+2 a \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^3}{6 a^{3/2} b^3 c+b^3 c^2-2 a b^3 c^2+a^2 b^3 c^2+4 a b c^3-4 a^2 b c^3-24 a^2 b^2 c \text {$\#$1}-10 \sqrt {a} b^2 c^2 \text {$\#$1}+8 a^{3/2} b^2 c^2 \text {$\#$1}+2 a^{5/2} b^2 c^2 \text {$\#$1}-8 a^{3/2} c^3 \text {$\#$1}+24 a^{5/2} b c \text {$\#$1}^2+6 a b^3 c \text {$\#$1}^2-6 a^2 b^3 c \text {$\#$1}^2+24 a b c^2 \text {$\#$1}^2-24 a^{3/2} b^2 c \text {$\#$1}^3+4 a^{5/2} b^2 c \text {$\#$1}^3-16 a^{3/2} c^2 \text {$\#$1}^3+5 a^2 b^3 \text {$\#$1}^4+20 a^2 b c \text {$\#$1}^4-6 a^{5/2} b^2 \text {$\#$1}^5}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2*RootSum[a*b^4*c + Sqrt[a]*b^2*c^3 - 2*a^(3/2)*b^2*c^3 + a^(5/2)*b^2*c^3 - 6*a^(3/2)*b^3*c*#1 - b^3*c^2*#1 +
2*a*b^3*c^2*#1 - a^2*b^3*c^2*#1 - 4*a*b*c^3*#1 + 4*a^2*b*c^3*#1 + 12*a^2*b^2*c*#1^2 + 5*Sqrt[a]*b^2*c^2*#1^2
- 4*a^(3/2)*b^2*c^2*#1^2 - a^(5/2)*b^2*c^2*#1^2 + 4*a^(3/2)*c^3*#1^2 - 8*a^(5/2)*b*c*#1^3 - 2*a*b^3*c*#1^3 + 2
*a^2*b^3*c*#1^3 - 8*a*b*c^2*#1^3 + 6*a^(3/2)*b^2*c*#1^4 - a^(5/2)*b^2*c*#1^4 + 4*a^(3/2)*c^2*#1^4 - a^2*b^3*#1
^5 - 4*a^2*b*c*#1^5 + a^(5/2)*b^2*#1^6 & , (-(Sqrt[a]*b*c*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #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
- 3*Sqrt[a]*b*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^2] - #1]*#1^2 + 2*a*Log[-(Sqrt[a]*x) + Sqrt[c + b*x + a*x^
2] - #1]*#1^3)/(6*a^(3/2)*b^3*c + b^3*c^2 - 2*a*b^3*c^2 + a^2*b^3*c^2 + 4*a*b*c^3 - 4*a^2*b*c^3 - 24*a^2*b^2*c
*#1 - 10*Sqrt[a]*b^2*c^2*#1 + 8*a^(3/2)*b^2*c^2*#1 + 2*a^(5/2)*b^2*c^2*#1 - 8*a^(3/2)*c^3*#1 + 24*a^(5/2)*b*c*
#1^2 + 6*a*b^3*c*#1^2 - 6*a^2*b^3*c*#1^2 + 24*a*b*c^2*#1^2 - 24*a^(3/2)*b^2*c*#1^3 + 4*a^(5/2)*b^2*c*#1^3 - 16
*a^(3/2)*c^2*#1^3 + 5*a^2*b^3*#1^4 + 20*a^2*b*c*#1^4 - 6*a^(5/2)*b^2*#1^5) & ]

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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(1/(a*b*c-(a*b*x+c)^2*(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(1/(a*b*c-(a*b*x+c)^2*(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:Eval
uation time: 5.57sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [B]  time = 0.74, size = 8156, normalized size = 10.77

method	result	size
default	\(\text {Expression too large to display}\)	\(8156\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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