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

Optimal. Leaf size=212 \[ \frac {a \log \left (\sqrt {a^2 x^2+b^2}+a x\right )}{c}-\frac {2 a \text {RootSum}\left [\text {$\#$1}^8 c+4 \text {$\#$1}^5 a^2+4 \text {$\#$1}^4 a^2 d-2 \text {$\#$1}^4 b^2 c+b^4 c\& ,\frac {b^2 c \log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-\text {$\#$1}\right )+a^2 (-d) \log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-\text {$\#$1}\right )-\text {$\#$1} a^2 \log \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-\text {$\#$1}\right )}{-2 \text {$\#$1}^4 c-5 \text {$\#$1} a^2-4 a^2 d+2 b^2 c}\& \right ]}{c} \]

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Rubi [F]  time = 26.70, 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 {\sqrt {b^2+a^2 x^2}}{d+c x^2+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

a*d*Defer[Int][(x*Sqrt[b^2 + a^2*x^2])/(b^2 - (d + c*x^2)^2*(d^2 - 2*a*x + 2*c*d*x^2 + c^2*x^4)), x] + a*c*Def
er[Int][(x^3*Sqrt[b^2 + a^2*x^2])/(b^2 - (d + c*x^2)^2*(d^2 - 2*a*x + 2*c*d*x^2 + c^2*x^4)), x] + b^2*Defer[In
t][Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]/(b^2 - (d + c*x^2)^2*(d^2 - 2*a*x + 2*c*d*x^2 + c^2*x^4)), x] + a^2*Defer[I
nt][(x^2*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/(b^2 - (d + c*x^2)^2*(d^2 - 2*a*x + 2*c*d*x^2 + c^2*x^4)), x] + d^2*
Defer[Int][(Sqrt[b^2 + a^2*x^2]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/(b^2 - (d + c*x^2)^2*(d^2 - 2*a*x + 2*c*d*x^2
+ c^2*x^4)), x] + 2*c*d*Defer[Int][(x^2*Sqrt[b^2 + a^2*x^2]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/(b^2 - (d + c*x^
2)^2*(d^2 - 2*a*x + 2*c*d*x^2 + c^2*x^4)), x] + c^2*Defer[Int][(x^4*Sqrt[b^2 + a^2*x^2]*Sqrt[a*x + Sqrt[b^2 +
a^2*x^2]])/(b^2 - (d + c*x^2)^2*(d^2 - 2*a*x + 2*c*d*x^2 + c^2*x^4)), x] + b^2*d*Defer[Int][(-b^2 + (d + c*x^2
)^2*(d^2 - 2*a*x + 2*c*d*x^2 + c^2*x^4))^(-1), x] + b^2*c*Defer[Int][x^2/(-b^2 + (d + c*x^2)^2*(d^2 - 2*a*x +
2*c*d*x^2 + c^2*x^4)), x] + a^2*d*Defer[Int][x^2/(-b^2 + (d + c*x^2)^2*(d^2 - 2*a*x + 2*c*d*x^2 + c^2*x^4)), x
] + a^2*c*Defer[Int][x^4/(-b^2 + (d + c*x^2)^2*(d^2 - 2*a*x + 2*c*d*x^2 + c^2*x^4)), x] + d^3*Defer[Int][Sqrt[
b^2 + a^2*x^2]/(-b^2 + (d + c*x^2)^2*(d^2 - 2*a*x + 2*c*d*x^2 + c^2*x^4)), x] + 3*c*d^2*Defer[Int][(x^2*Sqrt[b
^2 + a^2*x^2])/(-b^2 + (d + c*x^2)^2*(d^2 - 2*a*x + 2*c*d*x^2 + c^2*x^4)), x] + 3*c^2*d*Defer[Int][(x^4*Sqrt[b
^2 + a^2*x^2])/(-b^2 + (d + c*x^2)^2*(d^2 - 2*a*x + 2*c*d*x^2 + c^2*x^4)), x] + c^3*Defer[Int][(x^6*Sqrt[b^2 +
a^2*x^2])/(-b^2 + (d + c*x^2)^2*(d^2 - 2*a*x + 2*c*d*x^2 + c^2*x^4)), x] + a*Defer[Int][(x*Sqrt[b^2 + a^2*x^2
]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/(-b^2 + (d + c*x^2)^2*(d^2 - 2*a*x + 2*c*d*x^2 + c^2*x^4)), x]

Rubi steps

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

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Mathematica [F]  time = 180.01, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

$Aborted

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IntegrateAlgebraic [A]  time = 5.14, size = 212, normalized size = 1.00 \begin {gather*} \frac {a \log \left (a x+\sqrt {b^2+a^2 x^2}\right )}{c}-\frac {2 a \text {RootSum}\left [b^4 c-2 b^2 c \text {$\#$1}^4+4 a^2 d \text {$\#$1}^4+4 a^2 \text {$\#$1}^5+c \text {$\#$1}^8\&,\frac {b^2 c \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-a^2 d \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-a^2 \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}}{2 b^2 c-4 a^2 d-5 a^2 \text {$\#$1}-2 c \text {$\#$1}^4}\&\right ]}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Log[a*x + Sqrt[b^2 + a^2*x^2]])/c - (2*a*RootSum[b^4*c - 2*b^2*c*#1^4 + 4*a^2*d*#1^4 + 4*a^2*#1^5 + c*#1^8
& , (b^2*c*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1] - a^2*d*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1] - a^2*L
og[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]*#1)/(2*b^2*c - 4*a^2*d - 5*a^2*#1 - 2*c*#1^4) & ])/c

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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^2*x^2+b^2)^(1/2)/(d+c*x^2+(a*x+(a^2*x^2+b^2)^(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^{2} x^{2} + b^{2}}}{c x^{2} + d + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.39, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a^{2} x^{2}+b^{2}}}{d +c \,x^{2}+\sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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