3.7.13 \(\int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} (a+c x^2)} \, dx\) [613]

3.7.13.1 Optimal result

Integrand size = 28, antiderivative size = 354 \[ \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \left (a+c x^2\right )} \, dx=-\frac {e \sqrt {f+g x}}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x}}+\frac {e \sqrt {f+g x}}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right ) (e f-d g) \sqrt {d+e x}}+\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {\sqrt {c} f-\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right )^{3/2} \sqrt {\sqrt {c} f-\sqrt {-a} g}}-\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {\sqrt {c} f+\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} \left (\sqrt {c} d+\sqrt {-a} e\right )^{3/2} \sqrt {\sqrt {c} f+\sqrt {-a} g}} \]

output

-e*(g*x+f)^(1/2)/(-d*g+e*f)/(-a)^(1/2)/(-e*(-a)^(1/2)+d*c^(1/2))/(e*x+d)^( 
1/2)+e*(g*x+f)^(1/2)/(-d*g+e*f)/(-a)^(1/2)/(e*(-a)^(1/2)+d*c^(1/2))/(e*x+d 
)^(1/2)+arctanh((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^(1/2)/(-a)^(1/2)/(-e*(-a)^(1/2)+d*c^( 
1/2))^(3/2)/(-g*(-a)^(1/2)+f*c^(1/2))^(1/2)-arctanh((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^(1/ 
2)/(-a)^(1/2)/(e*(-a)^(1/2)+d*c^(1/2))^(3/2)/(g*(-a)^(1/2)+f*c^(1/2))^(1/2 
)
 

3.7.13.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.96 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \left (a+c x^2\right )} \, dx=\frac {2 e^2 \sqrt {f+g x}}{\left (c d^2+a e^2\right ) (-e f+d g) \sqrt {d+e x}}+\frac {i \sqrt {c} \left (\sqrt {c} d+i \sqrt {a} e\right )^2 \arctan \left (\frac {\sqrt {c d^2+a e^2} \sqrt {f+g x}}{\sqrt {-\left (\left (\sqrt {c} d+i \sqrt {a} e\right ) \left (\sqrt {c} f-i \sqrt {a} g\right )\right )} \sqrt {d+e x}}\right )}{\sqrt {a} \left (c d^2+a e^2\right )^{3/2} \sqrt {-\left (\left (\sqrt {c} d+i \sqrt {a} e\right ) \left (\sqrt {c} f-i \sqrt {a} g\right )\right )}}-\frac {i \sqrt {c} \left (\sqrt {c} d-i \sqrt {a} e\right )^2 \arctan \left (\frac {\sqrt {c d^2+a e^2} \sqrt {f+g x}}{\sqrt {-\left (\left (\sqrt {c} d-i \sqrt {a} e\right ) \left (\sqrt {c} f+i \sqrt {a} g\right )\right )} \sqrt {d+e x}}\right )}{\sqrt {a} \left (c d^2+a e^2\right )^{3/2} \sqrt {-\left (\left (\sqrt {c} d-i \sqrt {a} e\right ) \left (\sqrt {c} f+i \sqrt {a} g\right )\right )}} \]

input

Integrate[1/((d + e*x)^(3/2)*Sqrt[f + g*x]*(a + c*x^2)),x]
 

output

(2*e^2*Sqrt[f + g*x])/((c*d^2 + a*e^2)*(-(e*f) + d*g)*Sqrt[d + e*x]) + (I* 
Sqrt[c]*(Sqrt[c]*d + I*Sqrt[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]*(c*d^2 + a*e^2)^(3/2)*Sqrt[-((Sqrt[c]*d + I*Sqrt[a]*e)* 
(Sqrt[c]*f - I*Sqrt[a]*g))]) - (I*Sqrt[c]*(Sqrt[c]*d - I*Sqrt[a]*e)^2*ArcT 
an[(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]*(c*d^2 + a*e^2)^(3/2) 
*Sqrt[-((Sqrt[c]*d - I*Sqrt[a]*e)*(Sqrt[c]*f + I*Sqrt[a]*g))])
 

3.7.13.3 Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 354, 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 = {662, 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.

\(\displaystyle \int \frac {1}{\left (a+c x^2\right ) (d+e x)^{3/2} \sqrt {f+g x}} \, dx\)

\(\Big \downarrow \) 662

\(\displaystyle \int \left (\frac {\sqrt {-a}}{2 a \left (\sqrt {-a}-\sqrt {c} x\right ) (d+e x)^{3/2} \sqrt {f+g x}}+\frac {\sqrt {-a}}{2 a \left (\sqrt {-a}+\sqrt {c} x\right ) (d+e x)^{3/2} \sqrt {f+g x}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {c} f-\sqrt {-a} g}}{\sqrt {f+g x} \sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )}{\sqrt {-a} \left (\sqrt {c} d-\sqrt {-a} e\right )^{3/2} \sqrt {\sqrt {c} f-\sqrt {-a} g}}-\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g+\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{\sqrt {-a} \left (\sqrt {-a} e+\sqrt {c} d\right )^{3/2} \sqrt {\sqrt {-a} g+\sqrt {c} f}}-\frac {e \sqrt {f+g x}}{\sqrt {-a} \sqrt {d+e x} \left (\sqrt {c} d-\sqrt {-a} e\right ) (e f-d g)}+\frac {e \sqrt {f+g x}}{\sqrt {-a} \sqrt {d+e x} \left (\sqrt {-a} e+\sqrt {c} d\right ) (e f-d g)}\)

input

Int[1/((d + e*x)^(3/2)*Sqrt[f + g*x]*(a + c*x^2)),x]
 

output

-((e*Sqrt[f + g*x])/(Sqrt[-a]*(Sqrt[c]*d - Sqrt[-a]*e)*(e*f - d*g)*Sqrt[d 
+ e*x])) + (e*Sqrt[f + g*x])/(Sqrt[-a]*(Sqrt[c]*d + Sqrt[-a]*e)*(e*f - d*g 
)*Sqrt[d + e*x]) + (Sqrt[c]*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]*(Sqrt[c]*d 
 - Sqrt[-a]*e)^(3/2)*Sqrt[Sqrt[c]*f - Sqrt[-a]*g]) - (Sqrt[c]*ArcTanh[(Sqr 
t[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqr 
t[f + g*x])])/(Sqrt[-a]*(Sqrt[c]*d + Sqrt[-a]*e)^(3/2)*Sqrt[Sqrt[c]*f + Sq 
rt[-a]*g])
 

3.7.13.3.1 Defintions of rubi rules used

rule 662

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_) + (c_.)*(x_ 
)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n, 1/(a + c*x^ 
2), x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] &&  !IntegerQ[m] &&  !Inte 
gerQ[n]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

3.7.13.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(10976\) vs. \(2(270)=540\).

Time = 0.47 (sec) , antiderivative size = 10977, normalized size of antiderivative = 31.01

method	result	size
default	\(\text {Expression too large to display}\)	\(10977\)

input

int(1/(e*x+d)^(3/2)/(c*x^2+a)/(g*x+f)^(1/2),x,method=_RETURNVERBOSE)
 

output

result too large to display
 

3.7.13.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 11846 vs. \(2 (270) = 540\).

Time = 56.03 (sec) , antiderivative size = 11846, normalized size of antiderivative = 33.46 \[ \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \left (a+c x^2\right )} \, dx=\text {Too large to display} \]

input

integrate(1/(e*x+d)^(3/2)/(c*x^2+a)/(g*x+f)^(1/2),x, algorithm="fricas")
 

output

Too large to include
 

3.7.13.6 Sympy [F]

\[ \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \left (a+c x^2\right )} \, dx=\int \frac {1}{\left (a + c x^{2}\right ) \left (d + e x\right )^{\frac {3}{2}} \sqrt {f + g x}}\, dx \]

input

integrate(1/(e*x+d)**(3/2)/(c*x**2+a)/(g*x+f)**(1/2),x)
 

output

Integral(1/((a + c*x**2)*(d + e*x)**(3/2)*sqrt(f + g*x)), x)
 

3.7.13.7 Maxima [F]

\[ \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \left (a+c x^2\right )} \, dx=\int { \frac {1}{{\left (c x^{2} + a\right )} {\left (e x + d\right )}^{\frac {3}{2}} \sqrt {g x + f}} \,d x } \]

input

integrate(1/(e*x+d)^(3/2)/(c*x^2+a)/(g*x+f)^(1/2),x, algorithm="maxima")
 

output

integrate(1/((c*x^2 + a)*(e*x + d)^(3/2)*sqrt(g*x + f)), x)
 

3.7.13.8 Giac [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \left (a+c x^2\right )} \, dx=\text {Timed out} \]

input

integrate(1/(e*x+d)^(3/2)/(c*x^2+a)/(g*x+f)^(1/2),x, algorithm="giac")
 

output

Timed out
 

3.7.13.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \left (a+c x^2\right )} \, dx=\int \frac {1}{\sqrt {f+g\,x}\,\left (c\,x^2+a\right )\,{\left (d+e\,x\right )}^{3/2}} \,d x \]

input

int(1/((f + g*x)^(1/2)*(a + c*x^2)*(d + e*x)^(3/2)),x)
 

output

int(1/((f + g*x)^(1/2)*(a + c*x^2)*(d + e*x)^(3/2)), x)