Integrand size = 28, antiderivative size = 337 \[ \int \frac {(d+e x)^{3/2}}{\sqrt {f+g x} \left (a+c x^2\right )} \, dx=\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{c \sqrt {g}}+\frac {\left (c d^2-2 \sqrt {-a} \sqrt {c} d e-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} f-\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} c \sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {\sqrt {c} f-\sqrt {-a} g}}-\frac {\left (c d^2+2 \sqrt {-a} \sqrt {c} d e-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} f+\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} c \sqrt {\sqrt {c} d+\sqrt {-a} e} \sqrt {\sqrt {c} f+\sqrt {-a} g}} \]
2*e^(3/2)*arctanh(g^(1/2)*(e*x+d)^(1/2)/e^(1/2)/(g*x+f)^(1/2))/c/g^(1/2)+a rctanh((e*x+d)^(1/2)*(-g*(-a)^(1/2)+f*c^(1/2))^(1/2)/(g*x+f)^(1/2)/(-e*(-a )^(1/2)+d*c^(1/2))^(1/2))*(c*d^2-a*e^2-2*d*e*(-a)^(1/2)*c^(1/2))/c/(-a)^(1 /2)/(-e*(-a)^(1/2)+d*c^(1/2))^(1/2)/(-g*(-a)^(1/2)+f*c^(1/2))^(1/2)-arctan h((e*x+d)^(1/2)*(g*(-a)^(1/2)+f*c^(1/2))^(1/2)/(g*x+f)^(1/2)/(e*(-a)^(1/2) +d*c^(1/2))^(1/2))*(c*d^2-a*e^2+2*d*e*(-a)^(1/2)*c^(1/2))/c/(-a)^(1/2)/(e* (-a)^(1/2)+d*c^(1/2))^(1/2)/(g*(-a)^(1/2)+f*c^(1/2))^(1/2)
Result contains complex when optimal does not.
Time = 1.02 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.08 \[ \int \frac {(d+e x)^{3/2}}{\sqrt {f+g x} \left (a+c x^2\right )} \, dx=\frac {\frac {\left (i \sqrt {c} d+\sqrt {a} e\right ) \sqrt {c d^2+a e^2} \arctan \left (\frac {\sqrt {c d^2+a e^2} \sqrt {f+g x}}{\sqrt {-\left (\left (\sqrt {c} d+i \sqrt {a} e\right ) \left (\sqrt {c} f-i \sqrt {a} g\right )\right )} \sqrt {d+e x}}\right )}{\sqrt {a} \sqrt {-\left (\left (\sqrt {c} d+i \sqrt {a} e\right ) \left (\sqrt {c} f-i \sqrt {a} g\right )\right )}}+\frac {\left (-i \sqrt {c} d+\sqrt {a} e\right ) \sqrt {c d^2+a e^2} \arctan \left (\frac {\sqrt {c d^2+a e^2} \sqrt {f+g x}}{\sqrt {-\left (\left (\sqrt {c} d-i \sqrt {a} e\right ) \left (\sqrt {c} f+i \sqrt {a} g\right )\right )} \sqrt {d+e x}}\right )}{\sqrt {a} \sqrt {-\left (\left (\sqrt {c} d-i \sqrt {a} e\right ) \left (\sqrt {c} f+i \sqrt {a} g\right )\right )}}+\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {g} \sqrt {d+e x}}\right )}{\sqrt {g}}}{c} \]
(((I*Sqrt[c]*d + Sqrt[a]*e)*Sqrt[c*d^2 + a*e^2]*ArcTan[(Sqrt[c*d^2 + a*e^2 ]*Sqrt[f + g*x])/(Sqrt[-((Sqrt[c]*d + I*Sqrt[a]*e)*(Sqrt[c]*f - I*Sqrt[a]* g))]*Sqrt[d + e*x])])/(Sqrt[a]*Sqrt[-((Sqrt[c]*d + I*Sqrt[a]*e)*(Sqrt[c]*f - I*Sqrt[a]*g))]) + (((-I)*Sqrt[c]*d + Sqrt[a]*e)*Sqrt[c*d^2 + a*e^2]*Arc Tan[(Sqrt[c*d^2 + a*e^2]*Sqrt[f + g*x])/(Sqrt[-((Sqrt[c]*d - I*Sqrt[a]*e)* (Sqrt[c]*f + I*Sqrt[a]*g))]*Sqrt[d + e*x])])/(Sqrt[a]*Sqrt[-((Sqrt[c]*d - I*Sqrt[a]*e)*(Sqrt[c]*f + I*Sqrt[a]*g))]) + (2*e^(3/2)*ArcTanh[(Sqrt[e]*Sq rt[f + g*x])/(Sqrt[g]*Sqrt[d + e*x])])/Sqrt[g])/c
Time = 0.98 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {661, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(d+e x)^{3/2}}{\left (a+c x^2\right ) \sqrt {f+g x}} \, dx\) |
\(\Big \downarrow \) 661 |
\(\displaystyle \int \left (\frac {-a e^2+c d^2+2 c d e x}{c \left (a+c x^2\right ) \sqrt {d+e x} \sqrt {f+g x}}+\frac {e^2}{c \sqrt {d+e x} \sqrt {f+g x}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (-2 \sqrt {-a} \sqrt {c} d e-a e^2+c d^2\right ) \text {arctanh}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {c} f-\sqrt {-a} g}}{\sqrt {f+g x} \sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )}{\sqrt {-a} c \sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {\sqrt {c} f-\sqrt {-a} g}}-\frac {\left (2 \sqrt {-a} \sqrt {c} d e-a e^2+c d^2\right ) \text {arctanh}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g+\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{\sqrt {-a} c \sqrt {\sqrt {-a} e+\sqrt {c} d} \sqrt {\sqrt {-a} g+\sqrt {c} f}}+\frac {2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{c \sqrt {g}}\) |
(2*e^(3/2)*ArcTanh[(Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/(c*Sq rt[g]) + ((c*d^2 - 2*Sqrt[-a]*Sqrt[c]*d*e - a*e^2)*ArcTanh[(Sqrt[Sqrt[c]*f - Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*Sqrt[f + g*x]) ])/(Sqrt[-a]*c*Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*Sqrt[Sqrt[c]*f - Sqrt[-a]*g]) - ((c*d^2 + 2*Sqrt[-a]*Sqrt[c]*d*e - a*e^2)*ArcTanh[(Sqrt[Sqrt[c]*f + Sqrt [-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f + g*x])])/(Sqr t[-a]*c*Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[Sqrt[c]*f + Sqrt[-a]*g])
3.7.10.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_)/(Sqrt[(f_.) + (g_.)*(x_)]*((a_.) + (c_.)*(x_) ^2)), x_Symbol] :> Int[ExpandIntegrand[1/(Sqrt[d + e*x]*Sqrt[f + g*x]), (d + e*x)^(m + 1/2)/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && IGt Q[m + 1/2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(2335\) vs. \(2(257)=514\).
Time = 0.46 (sec) , antiderivative size = 2336, normalized size of antiderivative = 6.93
1/2*(e*x+d)^(1/2)*(g*x+f)^(1/2)*(ln((2*(-a*c)^(1/2)*e*g*x+c*d*g*x+c*e*f*x+ 2*(((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f-a*e*g+c*d*f)/c)^(1/2)*((g*x+f)*(e*x+ d))^(1/2)*c+(-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+2*c*d*f)/(c*x-(-a*c)^(1/2))) *a^2*e^2*g^2*(e*g)^(1/2)*(-((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+a*e*g-c*d*f) /c)^(1/2)-ln((2*(-a*c)^(1/2)*e*g*x+c*d*g*x+c*e*f*x+2*(((-a*c)^(1/2)*d*g+(- a*c)^(1/2)*e*f-a*e*g+c*d*f)/c)^(1/2)*((g*x+f)*(e*x+d))^(1/2)*c+(-a*c)^(1/2 )*d*g+(-a*c)^(1/2)*e*f+2*c*d*f)/(c*x-(-a*c)^(1/2)))*a*c*d^2*g^2*(e*g)^(1/2 )*(-((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+a*e*g-c*d*f)/c)^(1/2)+ln((2*(-a*c)^ (1/2)*e*g*x+c*d*g*x+c*e*f*x+2*(((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f-a*e*g+c* d*f)/c)^(1/2)*((g*x+f)*(e*x+d))^(1/2)*c+(-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+ 2*c*d*f)/(c*x-(-a*c)^(1/2)))*a*c*e^2*f^2*(e*g)^(1/2)*(-((-a*c)^(1/2)*d*g+( -a*c)^(1/2)*e*f+a*e*g-c*d*f)/c)^(1/2)-2*ln((2*(-a*c)^(1/2)*e*g*x+c*d*g*x+c *e*f*x+2*(((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f-a*e*g+c*d*f)/c)^(1/2)*((g*x+f )*(e*x+d))^(1/2)*c+(-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+2*c*d*f)/(c*x-(-a*c)^ (1/2)))*a*d*e*g^2*(e*g)^(1/2)*(-((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+a*e*g-c *d*f)/c)^(1/2)*(-a*c)^(1/2)-ln((2*(-a*c)^(1/2)*e*g*x+c*d*g*x+c*e*f*x+2*((( -a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f-a*e*g+c*d*f)/c)^(1/2)*((g*x+f)*(e*x+d))^( 1/2)*c+(-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+2*c*d*f)/(c*x-(-a*c)^(1/2)))*c^2* d^2*f^2*(e*g)^(1/2)*(-((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+a*e*g-c*d*f)/c)^( 1/2)-2*ln((2*(-a*c)^(1/2)*e*g*x+c*d*g*x+c*e*f*x+2*(((-a*c)^(1/2)*d*g+(-...
Timed out. \[ \int \frac {(d+e x)^{3/2}}{\sqrt {f+g x} \left (a+c x^2\right )} \, dx=\text {Timed out} \]
\[ \int \frac {(d+e x)^{3/2}}{\sqrt {f+g x} \left (a+c x^2\right )} \, dx=\int \frac {\left (d + e x\right )^{\frac {3}{2}}}{\left (a + c x^{2}\right ) \sqrt {f + g x}}\, dx \]
\[ \int \frac {(d+e x)^{3/2}}{\sqrt {f+g x} \left (a+c x^2\right )} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} + a\right )} \sqrt {g x + f}} \,d x } \]
Timed out. \[ \int \frac {(d+e x)^{3/2}}{\sqrt {f+g x} \left (a+c x^2\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(d+e x)^{3/2}}{\sqrt {f+g x} \left (a+c x^2\right )} \, dx=\int \frac {{\left (d+e\,x\right )}^{3/2}}{\sqrt {f+g\,x}\,\left (c\,x^2+a\right )} \,d x \]