3.29.93 \(\int \frac {f+e x}{(h+g x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx\)

Optimal. Leaf size=316 \[ \frac {(f g-e h) \text {RootSum}\left [\text {$\#$1}^8 (-g)+4 \text {$\#$1}^6 d g+2 \text {$\#$1}^4 c g-6 \text {$\#$1}^4 d^2 g-4 \text {$\#$1}^2 c d g+4 \text {$\#$1}^2 d^3 g-a h+b g-c^2 g+2 c d^2 g-d^4 g\& ,\frac {\log \left (\sqrt {\sqrt {\sqrt {a x+b}+c}+d}-\text {$\#$1}\right )}{\text {$\#$1}}\& \right ]}{g^2}+\frac {32 e \left (13 c d-12 d^3\right ) \sqrt {\sqrt {\sqrt {a x+b}+c}+d}}{105 a g}-\frac {32 e \left (5 c-6 d^2\right ) \sqrt {\sqrt {a x+b}+c} \sqrt {\sqrt {\sqrt {a x+b}+c}+d}}{105 a g}+\sqrt {a x+b} \left (\frac {8 e \sqrt {\sqrt {a x+b}+c} \sqrt {\sqrt {\sqrt {a x+b}+c}+d}}{7 a g}-\frac {48 d e \sqrt {\sqrt {\sqrt {a x+b}+c}+d}}{35 a g}\right ) \]

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Rubi [F]  time = 6.15, 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 {f+e x}{(h+g x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(f + e*x)/((h + g*x)*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]]),x]

[Out]

(8*d*(c - d^2)*e*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]])/(a*g) - (8*(c - 3*d^2)*e*(d + Sqrt[c + Sqrt[b + a*x]])^(3/
2))/(3*a*g) - (24*d*e*(d + Sqrt[c + Sqrt[b + a*x]])^(5/2))/(5*a*g) + (8*e*(d + Sqrt[c + Sqrt[b + a*x]])^(7/2))
/(7*a*g) - (8*d*(c - d^2)*(f*g - e*h)*Defer[Subst][Defer[Int][(b*g*(1 - (c^2*g - 2*c*d^2*g + d^4*g + a*h)/(b*g
)) - 4*c*d*(1 - d^2/c)*g*x^2 + 2*c*(1 - (3*d^2)/c)*g*x^4 + 4*d*g*x^6 - g*x^8)^(-1), x], x, Sqrt[d + Sqrt[c + S
qrt[b + a*x]]]])/g + (8*(c - 3*d^2)*(f*g - e*h)*Defer[Subst][Defer[Int][x^2/(b*g*(1 - (c^2*g - 2*c*d^2*g + d^4
*g + a*h)/(b*g)) - 4*c*d*(1 - d^2/c)*g*x^2 + 2*c*(1 - (3*d^2)/c)*g*x^4 + 4*d*g*x^6 - g*x^8), x], x, Sqrt[d + S
qrt[c + Sqrt[b + a*x]]]])/g - (8*(f*g - e*h)*Defer[Subst][Defer[Int][x^6/(b*g*(1 - (c^2*g - 2*c*d^2*g + d^4*g
+ a*h)/(b*g)) - 4*c*d*(1 - d^2/c)*g*x^2 + 2*c*(1 - (3*d^2)/c)*g*x^4 + 4*d*g*x^6 - g*x^8), x], x, Sqrt[d + Sqrt
[c + Sqrt[b + a*x]]]])/g - (24*d*(f*g - e*h)*Defer[Subst][Defer[Int][x^4/(-(b*g*(1 - (c^2*g - 2*c*d^2*g + d^4*
g + a*h)/(b*g))) + 4*c*d*(1 - d^2/c)*g*x^2 - 2*c*(1 - (3*d^2)/c)*g*x^4 - 4*d*g*x^6 + g*x^8), x], x, Sqrt[d + S
qrt[c + Sqrt[b + a*x]]]])/g

Rubi steps

\begin {align*} \int \frac {f+e x}{(h+g x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {x \left (-b e+a f+e x^2\right )}{\left (-b g+a h+g x^2\right ) \sqrt {d+\sqrt {c+x}}} \, dx,x,\sqrt {b+a x}\right )}{a}\\ &=\frac {4 \operatorname {Subst}\left (\int \frac {x \left (-c+x^2\right ) \left (-b e+a f+e \left (c-x^2\right )^2\right )}{\sqrt {d+x} \left (-b g+a h+g \left (c-x^2\right )^2\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=\frac {8 \operatorname {Subst}\left (\int \frac {\left (-d+x^2\right ) \left (-c+\left (d-x^2\right )^2\right ) \left (-b e+a f+e \left (c-\left (d-x^2\right )^2\right )^2\right )}{-b g+a h+g \left (c-\left (d-x^2\right )^2\right )^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{a}\\ &=\frac {8 \operatorname {Subst}\left (\int \frac {\left (d-x^2\right ) \left (c-d^2+2 d x^2-x^4\right ) \left (b e \left (1-\frac {a f}{b e}\right )-e \left (c-\left (d-x^2\right )^2\right )^2\right )}{b g \left (1-\frac {a h}{b g}\right )-g \left (c-\left (d-x^2\right )^2\right )^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{a}\\ &=\frac {8 \operatorname {Subst}\left (\int \left (\frac {d \left (c-d^2\right ) e}{g}-\frac {\left (c-3 d^2\right ) e x^2}{g}-\frac {3 d e x^4}{g}+\frac {e x^6}{g}-\frac {a d \left (c-d^2\right ) (f g-e h)-a \left (c-3 d^2\right ) (f g-e h) x^2-3 a d (f g-e h) x^4+a (f g-e h) x^6}{g \left (b g \left (1-\frac {a h}{b g}\right )-g \left (c-\left (d-x^2\right )^2\right )^2\right )}\right ) \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{a}\\ &=\frac {8 d \left (c-d^2\right ) e \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{a g}-\frac {8 \left (c-3 d^2\right ) e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{3/2}}{3 a g}-\frac {24 d e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{5/2}}{5 a g}+\frac {8 e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{7/2}}{7 a g}-\frac {8 \operatorname {Subst}\left (\int \frac {a d \left (c-d^2\right ) (f g-e h)-a \left (c-3 d^2\right ) (f g-e h) x^2-3 a d (f g-e h) x^4+a (f g-e h) x^6}{b g \left (1-\frac {a h}{b g}\right )-g \left (c-\left (d-x^2\right )^2\right )^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{a g}\\ &=\frac {8 d \left (c-d^2\right ) e \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{a g}-\frac {8 \left (c-3 d^2\right ) e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{3/2}}{3 a g}-\frac {24 d e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{5/2}}{5 a g}+\frac {8 e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{7/2}}{7 a g}-\frac {8 \operatorname {Subst}\left (\int \frac {a (f g-e h) \left (d-x^2\right ) \left (c-d^2+2 d x^2-x^4\right )}{b g \left (1-\frac {a h}{b g}\right )-g \left (c-\left (d-x^2\right )^2\right )^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{a g}\\ &=\frac {8 d \left (c-d^2\right ) e \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{a g}-\frac {8 \left (c-3 d^2\right ) e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{3/2}}{3 a g}-\frac {24 d e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{5/2}}{5 a g}+\frac {8 e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{7/2}}{7 a g}-\frac {(8 (f g-e h)) \operatorname {Subst}\left (\int \frac {\left (d-x^2\right ) \left (c-d^2+2 d x^2-x^4\right )}{b g \left (1-\frac {a h}{b g}\right )-g \left (c-\left (d-x^2\right )^2\right )^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{g}\\ &=\frac {8 d \left (c-d^2\right ) e \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{a g}-\frac {8 \left (c-3 d^2\right ) e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{3/2}}{3 a g}-\frac {24 d e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{5/2}}{5 a g}+\frac {8 e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{7/2}}{7 a g}-\frac {(8 (f g-e h)) \operatorname {Subst}\left (\int \left (\frac {c d \left (1-\frac {d^2}{c}\right )}{b g \left (1-\frac {c^2 g-2 c d^2 g+d^4 g+a h}{b g}\right )-4 c d \left (1-\frac {d^2}{c}\right ) g x^2+2 c \left (1-\frac {3 d^2}{c}\right ) g x^4+4 d g x^6-g x^8}+\frac {3 \left (1-\frac {c}{3 d^2}\right ) d^2 x^2}{b g \left (1-\frac {c^2 g-2 c d^2 g+d^4 g+a h}{b g}\right )-4 c d \left (1-\frac {d^2}{c}\right ) g x^2+2 c \left (1-\frac {3 d^2}{c}\right ) g x^4+4 d g x^6-g x^8}+\frac {x^6}{b g \left (1-\frac {c^2 g-2 c d^2 g+d^4 g+a h}{b g}\right )-4 c d \left (1-\frac {d^2}{c}\right ) g x^2+2 c \left (1-\frac {3 d^2}{c}\right ) g x^4+4 d g x^6-g x^8}+\frac {3 d x^4}{-b g \left (1-\frac {c^2 g-2 c d^2 g+d^4 g+a h}{b g}\right )+4 c d \left (1-\frac {d^2}{c}\right ) g x^2-2 c \left (1-\frac {3 d^2}{c}\right ) g x^4-4 d g x^6+g x^8}\right ) \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{g}\\ &=\frac {8 d \left (c-d^2\right ) e \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{a g}-\frac {8 \left (c-3 d^2\right ) e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{3/2}}{3 a g}-\frac {24 d e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{5/2}}{5 a g}+\frac {8 e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{7/2}}{7 a g}-\frac {(8 (f g-e h)) \operatorname {Subst}\left (\int \frac {x^6}{b g \left (1-\frac {c^2 g-2 c d^2 g+d^4 g+a h}{b g}\right )-4 c d \left (1-\frac {d^2}{c}\right ) g x^2+2 c \left (1-\frac {3 d^2}{c}\right ) g x^4+4 d g x^6-g x^8} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{g}-\frac {(24 d (f g-e h)) \operatorname {Subst}\left (\int \frac {x^4}{-b g \left (1-\frac {c^2 g-2 c d^2 g+d^4 g+a h}{b g}\right )+4 c d \left (1-\frac {d^2}{c}\right ) g x^2-2 c \left (1-\frac {3 d^2}{c}\right ) g x^4-4 d g x^6+g x^8} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{g}+\frac {\left (8 \left (c-3 d^2\right ) (f g-e h)\right ) \operatorname {Subst}\left (\int \frac {x^2}{b g \left (1-\frac {c^2 g-2 c d^2 g+d^4 g+a h}{b g}\right )-4 c d \left (1-\frac {d^2}{c}\right ) g x^2+2 c \left (1-\frac {3 d^2}{c}\right ) g x^4+4 d g x^6-g x^8} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{g}-\frac {\left (8 d \left (c-d^2\right ) (f g-e h)\right ) \operatorname {Subst}\left (\int \frac {1}{b g \left (1-\frac {c^2 g-2 c d^2 g+d^4 g+a h}{b g}\right )-4 c d \left (1-\frac {d^2}{c}\right ) g x^2+2 c \left (1-\frac {3 d^2}{c}\right ) g x^4+4 d g x^6-g x^8} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{g}\\ \end {align*}

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Mathematica [A]  time = 102.55, size = 519, normalized size = 1.64 \begin {gather*} \frac {(f g-e h) \text {RootSum}\left [\text {$\#$1}^8 a h-\text {$\#$1}^8 b g+\text {$\#$1}^8 c^2 g-2 \text {$\#$1}^8 c d^2 g+\text {$\#$1}^8 d^4 g+4 \text {$\#$1}^6 c d g-4 \text {$\#$1}^6 d^3 g-2 \text {$\#$1}^4 c g+6 \text {$\#$1}^4 d^2 g-4 \text {$\#$1}^2 d g+g\&,\frac {\text {$\#$1}^6 d^3 \log \left (\frac {1}{\sqrt {\sqrt {\sqrt {a x+b}+c}+d}}-\text {$\#$1}\right )-\text {$\#$1}^6 c d \log \left (\frac {1}{\sqrt {\sqrt {\sqrt {a x+b}+c}+d}}-\text {$\#$1}\right )-3 \text {$\#$1}^4 d^2 \log \left (\frac {1}{\sqrt {\sqrt {\sqrt {a x+b}+c}+d}}-\text {$\#$1}\right )+\text {$\#$1}^4 c \log \left (\frac {1}{\sqrt {\sqrt {\sqrt {a x+b}+c}+d}}-\text {$\#$1}\right )+3 \text {$\#$1}^2 d \log \left (\frac {1}{\sqrt {\sqrt {\sqrt {a x+b}+c}+d}}-\text {$\#$1}\right )-\log \left (\frac {1}{\sqrt {\sqrt {\sqrt {a x+b}+c}+d}}-\text {$\#$1}\right )}{\text {$\#$1}^7 a h-\text {$\#$1}^7 b g+\text {$\#$1}^7 c^2 g-2 \text {$\#$1}^7 c d^2 g+\text {$\#$1}^7 d^4 g+3 \text {$\#$1}^5 c d g-3 \text {$\#$1}^5 d^3 g-\text {$\#$1}^3 c g+3 \text {$\#$1}^3 d^2 g-\text {$\#$1} d g}\&\right ]}{g}+\frac {8 e \sqrt {\sqrt {\sqrt {a x+b}+c}+d} \left (24 d^2 \sqrt {\sqrt {a x+b}+c}-20 c \sqrt {\sqrt {a x+b}+c}+15 \sqrt {a x+b} \sqrt {\sqrt {a x+b}+c}-18 d \sqrt {a x+b}+52 c d-48 d^3\right )}{105 a g} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(f + e*x)/((h + g*x)*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]]),x]

[Out]

(8*e*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]]*(52*c*d - 48*d^3 - 18*d*Sqrt[b + a*x] - 20*c*Sqrt[c + Sqrt[b + a*x]] +
24*d^2*Sqrt[c + Sqrt[b + a*x]] + 15*Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]]))/(105*a*g) + ((f*g - e*h)*RootSum[g
- 4*d*g*#1^2 - 2*c*g*#1^4 + 6*d^2*g*#1^4 + 4*c*d*g*#1^6 - 4*d^3*g*#1^6 - b*g*#1^8 + c^2*g*#1^8 - 2*c*d^2*g*#1
^8 + d^4*g*#1^8 + a*h*#1^8 & , (-Log[1/Sqrt[d + Sqrt[c + Sqrt[b + a*x]]] - #1] + 3*d*Log[1/Sqrt[d + Sqrt[c + S
qrt[b + a*x]]] - #1]*#1^2 + c*Log[1/Sqrt[d + Sqrt[c + Sqrt[b + a*x]]] - #1]*#1^4 - 3*d^2*Log[1/Sqrt[d + Sqrt[c
+ Sqrt[b + a*x]]] - #1]*#1^4 - c*d*Log[1/Sqrt[d + Sqrt[c + Sqrt[b + a*x]]] - #1]*#1^6 + d^3*Log[1/Sqrt[d + Sq
rt[c + Sqrt[b + a*x]]] - #1]*#1^6)/(-(d*g*#1) - c*g*#1^3 + 3*d^2*g*#1^3 + 3*c*d*g*#1^5 - 3*d^3*g*#1^5 - b*g*#1
^7 + c^2*g*#1^7 - 2*c*d^2*g*#1^7 + d^4*g*#1^7 + a*h*#1^7) & ])/g

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IntegrateAlgebraic [A]  time = 0.55, size = 250, normalized size = 0.79 \begin {gather*} -\frac {8 e \left (20 c-24 d^2-15 \sqrt {b+a x}\right ) \sqrt {c+\sqrt {b+a x}} \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{105 a g}+\frac {16 \left (26 c d e-24 d^3 e-9 d e \sqrt {b+a x}\right ) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{105 a g}+\frac {(f g-e h) \text {RootSum}\left [b g-c^2 g+2 c d^2 g-d^4 g-a h-4 c d g \text {$\#$1}^2+4 d^3 g \text {$\#$1}^2+2 c g \text {$\#$1}^4-6 d^2 g \text {$\#$1}^4+4 d g \text {$\#$1}^6-g \text {$\#$1}^8\&,\frac {\log \left (\sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{g^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(f + e*x)/((h + g*x)*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]]),x]

[Out]

(-8*e*(20*c - 24*d^2 - 15*Sqrt[b + a*x])*Sqrt[c + Sqrt[b + a*x]]*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]])/(105*a*g)
+ (16*(26*c*d*e - 24*d^3*e - 9*d*e*Sqrt[b + a*x])*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]])/(105*a*g) + ((f*g - e*h)*
RootSum[b*g - c^2*g + 2*c*d^2*g - d^4*g - a*h - 4*c*d*g*#1^2 + 4*d^3*g*#1^2 + 2*c*g*#1^4 - 6*d^2*g*#1^4 + 4*d*
g*#1^6 - g*#1^8 & , Log[Sqrt[d + Sqrt[c + Sqrt[b + a*x]]] - #1]/#1 & ])/g^2

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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((e*x+f)/(g*x+h)/(d+(c+(a*x+b)^(1/2))^(1/2))^(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((e*x+f)/(g*x+h)/(d+(c+(a*x+b)^(1/2))^(1/2))^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(t_nostep^2-d)]Unable to divide, perhaps due to rounding error%%%{-1073741824,[2,10,18,5,4,2,0,0]%%%}+%%%{214
7483648,[2,10,18,3,5,2,0,0]%%%}+%%%{1073741824,[2,10,18,3,3,3,0,0]%%%}+%%%{-1073741824,[2,10,18,1,6,2,0,0]%%%}
+%%%{-1073741824,[2,10,18,1,4,3,0,0]%%%}+%%%{2147483648,[2,10,17,5,4,1,1,1]%%%}+%%%{-4294967296,[2,10,17,3,5,1
,1,1]%%%}+%%%{-3221225472,[2,10,17,3,3,2,1,1]%%%}+%%%{2147483648,[2,10,17,1,6,1,1,1]%%%}+%%%{3221225472,[2,10,
17,1,4,2,1,1]%%%}+%%%{-1073741824,[2,10,16,5,4,0,2,2]%%%}+%%%{2147483648,[2,10,16,3,5,0,2,2]%%%}+%%%{322122547
2,[2,10,16,3,3,1,2,2]%%%}+%%%{-1073741824,[2,10,16,1,6,0,2,2]%%%}+%%%{-3221225472,[2,10,16,1,4,1,2,2]%%%}+%%%{
-1073741824,[2,10,15,3,3,0,3,3]%%%}+%%%{1073741824,[2,10,15,1,4,0,3,3]%%%}+%%%{33554432,[2,8,16,4,3,1,0,0]%%%}
+%%%{-50331648,[2,8,16,2,4,1,0,0]%%%}+%%%{16777216,[2,8,16,0,5,1,0,0]%%%}+%%%{-33554432,[2,8,15,4,3,0,1,1]%%%}
+%%%{50331648,[2,8,15,2,4,0,1,1]%%%}+%%%{-16777216,[2,8,15,0,5,0,1,1]%%%}+%%%{-262144,[2,6,14,3,2,0,0,0]%%%}+%
%%{262144,[2,6,14,1,3,0,0,0]%%%}+%%%{-134217728,[1,9,17,4,3,2,0,0]%%%}+%%%{402653184,[1,9,17,2,4,2,0,0]%%%}+%%
%{-268435456,[1,9,17,0,5,2,0,0]%%%}+%%%{268435456,[1,9,16,4,3,1,1,1]%%%}+%%%{-805306368,[1,9,16,2,4,1,1,1]%%%}
+%%%{536870912,[1,9,16,0,5,1,1,1]%%%}+%%%{-134217728,[1,9,15,4,3,0,2,2]%%%}+%%%{402653184,[1,9,15,2,4,0,2,2]%%
%}+%%%{-268435456,[1,9,15,0,5,0,2,2]%%%}+%%%{2097152,[1,7,15,3,2,1,0,0]%%%}+%%%{-2097152,[1,7,15,1,3,1,0,0]%%%
}+%%%{-2097152,[1,7,14,3,2,0,1,1]%%%}+%%%{2097152,[1,7,14,1,3,0,1,1]%%%}+%%%{1073741824,[0,10,18,6,4,2,0,0]%%%
}+%%%{-3221225472,[0,10,18,4,5,2,0,0]%%%}+%%%{3221225472,[0,10,18,2,6,2,0,0]%%%}+%%%{-1073741824,[0,10,18,2,4,
3,0,0]%%%}+%%%{-1073741824,[0,10,18,0,7,2,0,0]%%%}+%%%{1073741824,[0,10,18,0,5,3,0,0]%%%}+%%%{-2147483648,[0,1
0,17,6,4,1,1,1]%%%}+%%%{6442450944,[0,10,17,4,5,1,1,1]%%%}+%%%{-6442450944,[0,10,17,2,6,1,1,1]%%%}+%%%{3221225
472,[0,10,17,2,4,2,1,1]%%%}+%%%{2147483648,[0,10,17,0,7,1,1,1]%%%}+%%%{-3221225472,[0,10,17,0,5,2,1,1]%%%}+%%%
{1073741824,[0,10,16,6,4,0,2,2]%%%}+%%%{-3221225472,[0,10,16,4,5,0,2,2]%%%}+%%%{3221225472,[0,10,16,2,6,0,2,2]
%%%}+%%%{-3221225472,[0,10,16,2,4,1,2,2]%%%}+%%%{-1073741824,[0,10,16,0,7,0,2,2]%%%}+%%%{3221225472,[0,10,16,0
,5,1,2,2]%%%}+%%%{1073741824,[0,10,15,2,4,0,3,3]%%%}+%%%{-1073741824,[0,10,15,0,5,0,3,3]%%%}+%%%{-33554432,[0,
8,16,5,3,1,0,0]%%%}+%%%{67108864,[0,8,16,3,4,1,0,0]%%%}+%%%{-33554432,[0,8,16,1,5,1,0,0]%%%}+%%%{33554432,[0,8
,15,5,3,0,1,1]%%%}+%%%{-67108864,[0,8,15,3,4,0,1,1]%%%}+%%%{33554432,[0,8,15,1,5,0,1,1]%%%}+%%%{262144,[0,6,14
,4,2,0,0,0]%%%}+%%%{-262144,[0,6,14,2,3,0,0,0]%%%} / %%%{4096,[0,4,7,2,2,0,0,0]%%%}+%%%{-4096,[0,4,7,0,3,0,0,0
]%%%} Error: Bad Argument Value

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maple [A]  time = 0.99, size = 338, normalized size = 1.07

method	result	size
derivativedivides	\(\frac {\frac {8 e \left (\frac {\left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {7}{2}}}{7}-\frac {3 \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {5}{2}} d}{5}+d^{2} \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {3}{2}}-\frac {c \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {3}{2}}}{3}-d^{3} \sqrt {d +\sqrt {c +\sqrt {a x +b}}}+c d \sqrt {d +\sqrt {c +\sqrt {a x +b}}}\right )}{g}-\frac {a \left (\munderset {\textit {\_R} =\RootOf \left (g \,\textit {\_Z}^{8}-4 d g \,\textit {\_Z}^{6}+\left (6 d^{2} g -2 c g \right ) \textit {\_Z}^{4}+\left (-4 d^{3} g +4 c d g \right ) \textit {\_Z}^{2}+d^{4} g -2 c \,d^{2} g +c^{2} g +a h -b g \right )}{\sum }\frac {\left (\textit {\_R}^{6} \left (e h -f g \right )+3 d \left (-e h +f g \right ) \textit {\_R}^{4}+\left (3 d^{2} e h -3 d^{2} f g -c e h +c f g \right ) \textit {\_R}^{2}-d^{3} e h +d^{3} f g +c d e h -c d f g \right ) \ln \left (\sqrt {d +\sqrt {c +\sqrt {a x +b}}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5} d +3 \textit {\_R}^{3} d^{2}-\textit {\_R}^{3} c -\textit {\_R} \,d^{3}+\textit {\_R} c d}\right )}{g^{2}}}{a}\)	\(338\)
default	\(\frac {\frac {8 e \left (\frac {\left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {7}{2}}}{7}-\frac {3 \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {5}{2}} d}{5}+d^{2} \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {3}{2}}-\frac {c \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {3}{2}}}{3}-d^{3} \sqrt {d +\sqrt {c +\sqrt {a x +b}}}+c d \sqrt {d +\sqrt {c +\sqrt {a x +b}}}\right )}{g}-\frac {a \left (\munderset {\textit {\_R} =\RootOf \left (g \,\textit {\_Z}^{8}-4 d g \,\textit {\_Z}^{6}+\left (6 d^{2} g -2 c g \right ) \textit {\_Z}^{4}+\left (-4 d^{3} g +4 c d g \right ) \textit {\_Z}^{2}+d^{4} g -2 c \,d^{2} g +c^{2} g +a h -b g \right )}{\sum }\frac {\left (\textit {\_R}^{6} \left (e h -f g \right )+3 d \left (-e h +f g \right ) \textit {\_R}^{4}+\left (3 d^{2} e h -3 d^{2} f g -c e h +c f g \right ) \textit {\_R}^{2}-d^{3} e h +d^{3} f g +c d e h -c d f g \right ) \ln \left (\sqrt {d +\sqrt {c +\sqrt {a x +b}}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5} d +3 \textit {\_R}^{3} d^{2}-\textit {\_R}^{3} c -\textit {\_R} \,d^{3}+\textit {\_R} c d}\right )}{g^{2}}}{a}\)	\(338\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+f)/(g*x+h)/(d+(c+(a*x+b)^(1/2))^(1/2))^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/a*(4*e/g*(1/7*(d+(c+(a*x+b)^(1/2))^(1/2))^(7/2)-3/5*(d+(c+(a*x+b)^(1/2))^(1/2))^(5/2)*d+d^2*(d+(c+(a*x+b)^(1
/2))^(1/2))^(3/2)-1/3*c*(d+(c+(a*x+b)^(1/2))^(1/2))^(3/2)-d^3*(d+(c+(a*x+b)^(1/2))^(1/2))^(1/2)+c*d*(d+(c+(a*x
+b)^(1/2))^(1/2))^(1/2))-1/2*a/g^2*sum((_R^6*(e*h-f*g)+3*d*(-e*h+f*g)*_R^4+(3*d^2*e*h-3*d^2*f*g-c*e*h+c*f*g)*_
R^2-d^3*e*h+d^3*f*g+c*d*e*h-c*d*f*g)/(_R^7-3*_R^5*d+3*_R^3*d^2-_R^3*c-_R*d^3+_R*c*d)*ln((d+(c+(a*x+b)^(1/2))^(
1/2))^(1/2)-_R),_R=RootOf(g*_Z^8-4*d*g*_Z^6+(6*d^2*g-2*c*g)*_Z^4+(-4*d^3*g+4*c*d*g)*_Z^2+d^4*g-2*c*d^2*g+c^2*g
+a*h-b*g)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e x + f}{{\left (g x + h\right )} \sqrt {d + \sqrt {c + \sqrt {a x + b}}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+f)/(g*x+h)/(d+(c+(a*x+b)^(1/2))^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate((e*x + f)/((g*x + h)*sqrt(d + 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 {f+e\,x}{\left (h+g\,x\right )\,\sqrt {d+\sqrt {c+\sqrt {b+a\,x}}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f + e*x)/((h + g*x)*(d + (c + (b + a*x)^(1/2))^(1/2))^(1/2)),x)

[Out]

int((f + e*x)/((h + g*x)*(d + (c + (b + a*x)^(1/2))^(1/2))^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e x + f}{\sqrt {d + \sqrt {c + \sqrt {a x + b}}} \left (g x + h\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+f)/(g*x+h)/(d+(c+(a*x+b)**(1/2))**(1/2))**(1/2),x)

[Out]

Integral((e*x + f)/(sqrt(d + sqrt(c + sqrt(a*x + b)))*(g*x + h)), x)

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