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

Optimal. Leaf size=455 \[ -4 a \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6 c-2 \text {$\#$1}^4 b+6 \text {$\#$1}^4 c^2-\text {$\#$1}^3 a^2+4 \text {$\#$1}^2 b c-4 \text {$\#$1}^2 c^3+\text {$\#$1} a^2 c+b^2-2 b c^2+c^4\& ,\frac {\text {$\#$1}^6 (-c) \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )+3 \text {$\#$1}^4 c^2 \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )-2 \text {$\#$1}^4 b \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )-\text {$\#$1}^3 a^2 \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )-3 \text {$\#$1}^2 c^3 \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )+4 \text {$\#$1}^2 b c \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )+\text {$\#$1} a^2 c \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )+b^2 \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )+c^4 \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )-2 b c^2 \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )}{8 \text {$\#$1}^7-24 \text {$\#$1}^5 c-8 \text {$\#$1}^3 b+24 \text {$\#$1}^3 c^2-3 \text {$\#$1}^2 a^2+8 \text {$\#$1} b c-8 \text {$\#$1} c^3+a^2 c}\& \right ]+4 a \sqrt {\sqrt {a x+b}+c}+\frac {2 (a x+b)^{3/2}}{3 a} \]

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Rubi [F]  time = 4.24, 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 {x^2 \sqrt {b+a x}}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {x^2 \sqrt {b+a x}}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {\left (-b x+x^3\right )^2}{b^2-2 b x^2+x^4-a^2 x \sqrt {c+x}} \, dx,x,\sqrt {b+a x}\right )}{a}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {x^2 \left (-b+x^2\right )^2}{b^2-2 b x^2+x^4-a^2 x \sqrt {c+x}} \, dx,x,\sqrt {b+a x}\right )}{a}\\ &=\frac {4 \operatorname {Subst}\left (\int \frac {x \left (c-x^2\right )^2 \left (b-\left (c-x^2\right )^2\right )^2}{b^2-2 b \left (c-x^2\right )^2+\left (c-x^2\right )^4-a^2 x \left (-c+x^2\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=\frac {4 \operatorname {Subst}\left (\int \frac {x \left (c-x^2\right )^2 \left (b-c^2+2 c x^2-x^4\right )^2}{b^2-2 b \left (c-x^2\right )^2+\left (c-x^2\right )^4-a^2 x \left (-c+x^2\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=\frac {4 \operatorname {Subst}\left (\int \left (a^2+c^2 x-2 c x^3+x^5-\frac {a^2 \left (b-c^2\right )^2+a^4 c x+a^2 c \left (4 b-3 c^2\right ) x^2-a^4 x^3-a^2 \left (2 b-3 c^2\right ) x^4-a^2 c x^6}{b^2-2 b \left (c-x^2\right )^2+\left (c-x^2\right )^4-a^2 x \left (-c+x^2\right )}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=\frac {2 c^2 \sqrt {b+a x}}{a}+4 a \sqrt {c+\sqrt {b+a x}}-\frac {2 c \left (c+\sqrt {b+a x}\right )^2}{a}+\frac {2 \left (c+\sqrt {b+a x}\right )^3}{3 a}-\frac {4 \operatorname {Subst}\left (\int \frac {a^2 \left (b-c^2\right )^2+a^4 c x+a^2 c \left (4 b-3 c^2\right ) x^2-a^4 x^3-a^2 \left (2 b-3 c^2\right ) x^4-a^2 c x^6}{b^2-2 b \left (c-x^2\right )^2+\left (c-x^2\right )^4-a^2 x \left (-c+x^2\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=\frac {2 c^2 \sqrt {b+a x}}{a}+4 a \sqrt {c+\sqrt {b+a x}}-\frac {2 c \left (c+\sqrt {b+a x}\right )^2}{a}+\frac {2 \left (c+\sqrt {b+a x}\right )^3}{3 a}-\frac {4 \operatorname {Subst}\left (\int \frac {a^2 \left (b^2 \left (1+\frac {-2 b c^2+c^4}{b^2}\right )+a^2 c x+4 b c \left (1-\frac {3 c^2}{4 b}\right ) x^2-a^2 x^3-2 b \left (1-\frac {3 c^2}{2 b}\right ) x^4-c x^6\right )}{b^2-2 b \left (c-x^2\right )^2+\left (c-x^2\right )^4-a^2 x \left (-c+x^2\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=\frac {2 c^2 \sqrt {b+a x}}{a}+4 a \sqrt {c+\sqrt {b+a x}}-\frac {2 c \left (c+\sqrt {b+a x}\right )^2}{a}+\frac {2 \left (c+\sqrt {b+a x}\right )^3}{3 a}-(4 a) \operatorname {Subst}\left (\int \frac {b^2 \left (1+\frac {-2 b c^2+c^4}{b^2}\right )+a^2 c x+4 b c \left (1-\frac {3 c^2}{4 b}\right ) x^2-a^2 x^3-2 b \left (1-\frac {3 c^2}{2 b}\right ) x^4-c x^6}{b^2-2 b \left (c-x^2\right )^2+\left (c-x^2\right )^4-a^2 x \left (-c+x^2\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )\\ &=\frac {2 c^2 \sqrt {b+a x}}{a}+4 a \sqrt {c+\sqrt {b+a x}}-\frac {2 c \left (c+\sqrt {b+a x}\right )^2}{a}+\frac {2 \left (c+\sqrt {b+a x}\right )^3}{3 a}-(4 a) \operatorname {Subst}\left (\int \left (\frac {a^2 x^3}{-b^2 \left (1+\frac {-2 b c^2+c^4}{b^2}\right )-a^2 c x-4 b c \left (1-\frac {c^2}{b}\right ) x^2+a^2 x^3+2 b \left (1-\frac {3 c^2}{b}\right ) x^4+4 c x^6-x^8}+\frac {c x^6}{-b^2 \left (1+\frac {-2 b c^2+c^4}{b^2}\right )-a^2 c x-4 b c \left (1-\frac {c^2}{b}\right ) x^2+a^2 x^3+2 b \left (1-\frac {3 c^2}{b}\right ) x^4+4 c x^6-x^8}+\frac {\left (b-c^2\right )^2}{b^2 \left (1+\frac {-2 b c^2+c^4}{b^2}\right )+a^2 c x+4 b c \left (1-\frac {c^2}{b}\right ) x^2-a^2 x^3-2 b \left (1-\frac {3 c^2}{b}\right ) x^4-4 c x^6+x^8}+\frac {a^2 c x}{b^2 \left (1+\frac {-2 b c^2+c^4}{b^2}\right )+a^2 c x+4 b c \left (1-\frac {c^2}{b}\right ) x^2-a^2 x^3-2 b \left (1-\frac {3 c^2}{b}\right ) x^4-4 c x^6+x^8}+\frac {c \left (4 b-3 c^2\right ) x^2}{b^2 \left (1+\frac {-2 b c^2+c^4}{b^2}\right )+a^2 c x+4 b c \left (1-\frac {c^2}{b}\right ) x^2-a^2 x^3-2 b \left (1-\frac {3 c^2}{b}\right ) x^4-4 c x^6+x^8}+\frac {\left (-2 b+3 c^2\right ) x^4}{b^2 \left (1+\frac {-2 b c^2+c^4}{b^2}\right )+a^2 c x+4 b c \left (1-\frac {c^2}{b}\right ) x^2-a^2 x^3-2 b \left (1-\frac {3 c^2}{b}\right ) x^4-4 c x^6+x^8}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )\\ &=\frac {2 c^2 \sqrt {b+a x}}{a}+4 a \sqrt {c+\sqrt {b+a x}}-\frac {2 c \left (c+\sqrt {b+a x}\right )^2}{a}+\frac {2 \left (c+\sqrt {b+a x}\right )^3}{3 a}-\left (4 a^3\right ) \operatorname {Subst}\left (\int \frac {x^3}{-b^2 \left (1+\frac {-2 b c^2+c^4}{b^2}\right )-a^2 c x-4 b c \left (1-\frac {c^2}{b}\right ) x^2+a^2 x^3+2 b \left (1-\frac {3 c^2}{b}\right ) x^4+4 c x^6-x^8} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )-(4 a c) \operatorname {Subst}\left (\int \frac {x^6}{-b^2 \left (1+\frac {-2 b c^2+c^4}{b^2}\right )-a^2 c x-4 b c \left (1-\frac {c^2}{b}\right ) x^2+a^2 x^3+2 b \left (1-\frac {3 c^2}{b}\right ) x^4+4 c x^6-x^8} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )-\left (4 a^3 c\right ) \operatorname {Subst}\left (\int \frac {x}{b^2 \left (1+\frac {-2 b c^2+c^4}{b^2}\right )+a^2 c x+4 b c \left (1-\frac {c^2}{b}\right ) x^2-a^2 x^3-2 b \left (1-\frac {3 c^2}{b}\right ) x^4-4 c x^6+x^8} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )+\left (4 a \left (2 b-3 c^2\right )\right ) \operatorname {Subst}\left (\int \frac {x^4}{b^2 \left (1+\frac {-2 b c^2+c^4}{b^2}\right )+a^2 c x+4 b c \left (1-\frac {c^2}{b}\right ) x^2-a^2 x^3-2 b \left (1-\frac {3 c^2}{b}\right ) x^4-4 c x^6+x^8} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )-\left (4 a c \left (4 b-3 c^2\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{b^2 \left (1+\frac {-2 b c^2+c^4}{b^2}\right )+a^2 c x+4 b c \left (1-\frac {c^2}{b}\right ) x^2-a^2 x^3-2 b \left (1-\frac {3 c^2}{b}\right ) x^4-4 c x^6+x^8} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )-\left (4 a \left (b-c^2\right )^2\right ) \operatorname {Subst}\left (\int \frac {1}{b^2 \left (1+\frac {-2 b c^2+c^4}{b^2}\right )+a^2 c x+4 b c \left (1-\frac {c^2}{b}\right ) x^2-a^2 x^3-2 b \left (1-\frac {3 c^2}{b}\right ) x^4-4 c x^6+x^8} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )\\ &=\frac {2 c^2 \sqrt {b+a x}}{a}+4 a \sqrt {c+\sqrt {b+a x}}-\frac {2 c \left (c+\sqrt {b+a x}\right )^2}{a}+\frac {2 \left (c+\sqrt {b+a x}\right )^3}{3 a}-\left (4 a^3\right ) \operatorname {Subst}\left (\int \frac {x^3}{-b^2+2 b \left (c-x^2\right )^2-\left (c-x^2\right ) \left (c^3+a^2 x-3 c^2 x^2+3 c x^4-x^6\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )-(4 a c) \operatorname {Subst}\left (\int \frac {x^6}{-b^2+2 b \left (c-x^2\right )^2-\left (c-x^2\right ) \left (c^3+a^2 x-3 c^2 x^2+3 c x^4-x^6\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )-\left (4 a^3 c\right ) \operatorname {Subst}\left (\int \frac {x}{b^2-2 b \left (c-x^2\right )^2+\left (c-x^2\right ) \left (c^3+a^2 x-3 c^2 x^2+3 c x^4-x^6\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )+\left (4 a \left (2 b-3 c^2\right )\right ) \operatorname {Subst}\left (\int \frac {x^4}{b^2-2 b \left (c-x^2\right )^2+\left (c-x^2\right ) \left (c^3+a^2 x-3 c^2 x^2+3 c x^4-x^6\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )-\left (4 a c \left (4 b-3 c^2\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{b^2-2 b \left (c-x^2\right )^2+\left (c-x^2\right ) \left (c^3+a^2 x-3 c^2 x^2+3 c x^4-x^6\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )-\left (4 a \left (b-c^2\right )^2\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-2 b \left (c-x^2\right )^2+\left (c-x^2\right ) \left (c^3+a^2 x-3 c^2 x^2+3 c x^4-x^6\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )\\ \end {align*}

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Mathematica [F]  time = 2.80, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 \sqrt {b+a x}}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.00, size = 459, normalized size = 1.01 \begin {gather*} 4 a \sqrt {c+\sqrt {b+a x}}+\frac {2 \left (c^3+(b+a x)^{3/2}\right )}{3 a}-4 a \text {RootSum}\left [b^2-2 b c^2+c^4+a^2 c \text {$\#$1}+4 b c \text {$\#$1}^2-4 c^3 \text {$\#$1}^2-a^2 \text {$\#$1}^3-2 b \text {$\#$1}^4+6 c^2 \text {$\#$1}^4-4 c \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {b^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )-2 b c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )+c^4 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )+a^2 c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+4 b c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2-3 c^3 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2-a^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3-2 b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^4+3 c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^4-c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^6}{a^2 c+8 b c \text {$\#$1}-8 c^3 \text {$\#$1}-3 a^2 \text {$\#$1}^2-8 b \text {$\#$1}^3+24 c^2 \text {$\#$1}^3-24 c \text {$\#$1}^5+8 \text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

4*a*Sqrt[c + Sqrt[b + a*x]] + (2*(c^3 + (b + a*x)^(3/2)))/(3*a) - 4*a*RootSum[b^2 - 2*b*c^2 + c^4 + a^2*c*#1 +
4*b*c*#1^2 - 4*c^3*#1^2 - a^2*#1^3 - 2*b*#1^4 + 6*c^2*#1^4 - 4*c*#1^6 + #1^8 & , (b^2*Log[Sqrt[c + Sqrt[b + a
*x]] - #1] - 2*b*c^2*Log[Sqrt[c + Sqrt[b + a*x]] - #1] + c^4*Log[Sqrt[c + Sqrt[b + a*x]] - #1] + a^2*c*Log[Sqr
t[c + Sqrt[b + a*x]] - #1]*#1 + 4*b*c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^2 - 3*c^3*Log[Sqrt[c + Sqrt[b + a*x
]] - #1]*#1^2 - a^2*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^3 - 2*b*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^4 + 3*c^
2*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^4 - c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^6)/(a^2*c + 8*b*c*#1 - 8*c^3
*#1 - 3*a^2*#1^2 - 8*b*#1^3 + 24*c^2*#1^3 - 24*c*#1^5 + 8*#1^7) & ]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.20, size = 264, normalized size = 0.58

method	result	size
derivativedivides	\(-\frac {2 \left (-\frac {\left (c +\sqrt {a x +b}\right )^{3}}{3}+c \left (c +\sqrt {a x +b}\right )^{2}-c^{2} \left (c +\sqrt {a x +b}\right )-2 a^{2} \sqrt {c +\sqrt {a x +b}}+2 a^{2} \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 c \,\textit {\_Z}^{6}+\left (6 c^{2}-2 b \right ) \textit {\_Z}^{4}-a^{2} \textit {\_Z}^{3}+\left (-4 c^{3}+4 b c \right ) \textit {\_Z}^{2}+a^{2} c \textit {\_Z} +c^{4}-2 b \,c^{2}+b^{2}\right )}{\sum }\frac {\left (-\textit {\_R}^{6} c +\left (3 c^{2}-2 b \right ) \textit {\_R}^{4}-\textit {\_R}^{3} a^{2}+c \left (-3 c^{2}+4 b \right ) \textit {\_R}^{2}+a^{2} c \textit {\_R} +c^{4}-2 b \,c^{2}+b^{2}\right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{8 \textit {\_R}^{7}-24 c \,\textit {\_R}^{5}+24 \textit {\_R}^{3} c^{2}-8 \textit {\_R}^{3} b -3 \textit {\_R}^{2} a^{2}-8 \textit {\_R} \,c^{3}+8 \textit {\_R} b c +a^{2} c}\right )\right )}{a}\)	\(264\)
default	\(-\frac {2 \left (-\frac {\left (c +\sqrt {a x +b}\right )^{3}}{3}+c \left (c +\sqrt {a x +b}\right )^{2}-c^{2} \left (c +\sqrt {a x +b}\right )-2 a^{2} \sqrt {c +\sqrt {a x +b}}+2 a^{2} \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 c \,\textit {\_Z}^{6}+\left (6 c^{2}-2 b \right ) \textit {\_Z}^{4}-a^{2} \textit {\_Z}^{3}+\left (-4 c^{3}+4 b c \right ) \textit {\_Z}^{2}+a^{2} c \textit {\_Z} +c^{4}-2 b \,c^{2}+b^{2}\right )}{\sum }\frac {\left (-\textit {\_R}^{6} c +\left (3 c^{2}-2 b \right ) \textit {\_R}^{4}-\textit {\_R}^{3} a^{2}+c \left (-3 c^{2}+4 b \right ) \textit {\_R}^{2}+a^{2} c \textit {\_R} +c^{4}-2 b \,c^{2}+b^{2}\right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{8 \textit {\_R}^{7}-24 c \,\textit {\_R}^{5}+24 \textit {\_R}^{3} c^{2}-8 \textit {\_R}^{3} b -3 \textit {\_R}^{2} a^{2}-8 \textit {\_R} \,c^{3}+8 \textit {\_R} b c +a^{2} c}\right )\right )}{a}\)	\(264\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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