3.23.23 \(\int \frac {x \sqrt {b+a x}}{x+\sqrt {c+\sqrt {b+a x}}} \, dx\)

Optimal. Leaf size=165 \[ 4 \text {RootSum}\left [-\text {$\#$1}^4+2 \text {$\#$1}^2 c-\text {$\#$1} a+b-c^2\& ,\frac {\text {$\#$1}^3 a \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )-\text {$\#$1}^2 b \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )}{4 \text {$\#$1}^3-4 \text {$\#$1} c+a}\& \right ]-\frac {4}{3} c \sqrt {\sqrt {a x+b}+c}+\sqrt {a x+b} \left (\frac {2 (a x+b)}{3 a}-\frac {4}{3} \sqrt {\sqrt {a x+b}+c}\right ) \]

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Rubi [F]  time = 1.66, 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 \sqrt {b+a x}}{x+\sqrt {c+\sqrt {b+a x}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(2*c^2*Sqrt[b + a*x])/a - (4*(c + Sqrt[b + a*x])^(3/2))/3 - (2*c*(c + Sqrt[b + a*x])^2)/a + (2*(c + Sqrt[b + a
*x])^3)/(3*a) + a*Log[b - c^2 - a*Sqrt[c + Sqrt[b + a*x]] + 2*c*(c + Sqrt[b + a*x]) - (c + Sqrt[b + a*x])^2] -
a^2*Defer[Subst][Defer[Int][(-b + c^2 + a*x - 2*c*x^2 + x^4)^(-1), x], x, Sqrt[c + Sqrt[b + a*x]]] + 4*a*c*De
fer[Subst][Defer[Int][x/(-b + c^2 + a*x - 2*c*x^2 + x^4), x], x, Sqrt[c + Sqrt[b + a*x]]] - 4*b*Defer[Subst][D
efer[Int][x^2/(-b + c^2 + a*x - 2*c*x^2 + x^4), x], x, Sqrt[c + Sqrt[b + a*x]]]

Rubi steps

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

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Mathematica [F]  time = 4.56, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \sqrt {b+a x}}{x+\sqrt {c+\sqrt {b+a x}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.00, size = 142, normalized size = 0.86 \begin {gather*} -\frac {4}{3} \left (c+\sqrt {b+a x}\right )^{3/2}+\frac {2 \left (c^3+(b+a x)^{3/2}\right )}{3 a}+4 \text {RootSum}\left [b-c^2-a \text {$\#$1}+2 c \text {$\#$1}^2-\text {$\#$1}^4\&,\frac {-b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2+a \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{a-4 c \text {$\#$1}+4 \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(c + Sqrt[b + a*x])^(3/2))/3 + (2*(c^3 + (b + a*x)^(3/2)))/(3*a) + 4*RootSum[b - c^2 - a*#1 + 2*c*#1^2 - #
1^4 & , (-(b*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^2) + a*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^3)/(a - 4*c*#1 +
4*#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(x*(a*x+b)^(1/2)/(x+(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}{x + \sqrt {c + \sqrt {a x + b}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.11, size = 129, normalized size = 0.78

method	result	size
derivativedivides	\(\frac {\frac {2 \left (c +\sqrt {a x +b}\right )^{3}}{3}-2 c \left (c +\sqrt {a x +b}\right )^{2}-\frac {4 a \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}+2 c^{2} \left (c +\sqrt {a x +b}\right )+4 a \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-2 c \,\textit {\_Z}^{2}+a \textit {\_Z} +c^{2}-b \right )}{\sum }\frac {\left (\textit {\_R}^{3} a -\textit {\_R}^{2} b \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-4 \textit {\_R} c +a}\right )}{a}\)	\(129\)
default	\(\frac {\frac {2 \left (c +\sqrt {a x +b}\right )^{3}}{3}-2 c \left (c +\sqrt {a x +b}\right )^{2}-\frac {4 a \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}+2 c^{2} \left (c +\sqrt {a x +b}\right )+4 a \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-2 c \,\textit {\_Z}^{2}+a \textit {\_Z} +c^{2}-b \right )}{\sum }\frac {\left (\textit {\_R}^{3} a -\textit {\_R}^{2} b \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-4 \textit {\_R} c +a}\right )}{a}\)	\(129\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a*x+b)^(1/2)/(x+(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))^2-2/3*a*(c+(a*x+b)^(1/2))^(3/2)+c^2*(c+(a*x+b)^(1/2))+2*a*sum
((_R^3*a-_R^2*b)/(4*_R^3-4*_R*c+a)*ln((c+(a*x+b)^(1/2))^(1/2)-_R),_R=RootOf(_Z^4-2*_Z^2*c+_Z*a+c^2-b)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x\,\sqrt {b+a\,x}}{x+\sqrt {c+\sqrt {b+a\,x}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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