\(\int \frac {(d+e x)^{3/2} \sqrt {f+g x}}{a+c x^2} \, dx\) [97]

Optimal result

Integrand size = 28, antiderivative size = 421 \[ \int \frac {(d+e x)^{3/2} \sqrt {f+g x}}{a+c x^2} \, dx=\frac {e \sqrt {d+e x} \sqrt {f+g x}}{c}+\frac {\sqrt {e} (e f+3 d g) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{c \sqrt {g}}+\frac {\left (\sqrt {-a} \left (a e^2 g-c d (2 e f+d g)\right )+\sqrt {c} \left (c d^2 f-a e (e f+2 d g)\right )\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} f-\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} c^{3/2} \sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {\sqrt {c} f-\sqrt {-a} g}}+\frac {\left (\sqrt {-a} \left (a e^2 g-c d (2 e f+d g)\right )-\sqrt {c} \left (c d^2 f-a e (e f+2 d g)\right )\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} f+\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} c^{3/2} \sqrt {\sqrt {c} d+\sqrt {-a} e} \sqrt {\sqrt {c} f+\sqrt {-a} g}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.10 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.04 \[ \int \frac {(d+e x)^{3/2} \sqrt {f+g x}}{a+c x^2} \, dx=\frac {\sqrt {c} e \sqrt {d+e x} \sqrt {f+g x}+\frac {\left (i \sqrt {c} d+\sqrt {a} e\right ) \sqrt {c d^2+a e^2} \left (\sqrt {c} f-i \sqrt {a} g\right ) \arctan \left (\frac {\sqrt {c d^2+a e^2} \sqrt {f+g x}}{\sqrt {-\left (\left (\sqrt {c} d+i \sqrt {a} e\right ) \left (\sqrt {c} f-i \sqrt {a} g\right )\right )} \sqrt {d+e x}}\right )}{\sqrt {a} \sqrt {-\left (\left (\sqrt {c} d+i \sqrt {a} e\right ) \left (\sqrt {c} f-i \sqrt {a} g\right )\right )}}+\frac {\left (-i \sqrt {c} d+\sqrt {a} e\right ) \sqrt {c d^2+a e^2} \left (\sqrt {c} f+i \sqrt {a} g\right ) \arctan \left (\frac {\sqrt {c d^2+a e^2} \sqrt {f+g x}}{\sqrt {-\left (\left (\sqrt {c} d-i \sqrt {a} e\right ) \left (\sqrt {c} f+i \sqrt {a} g\right )\right )} \sqrt {d+e x}}\right )}{\sqrt {a} \sqrt {-\left (\left (\sqrt {c} d-i \sqrt {a} e\right ) \left (\sqrt {c} f+i \sqrt {a} g\right )\right )}}+\frac {\sqrt {c} \sqrt {e} (e f+3 d g) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {g} \sqrt {d+e x}}\right )}{\sqrt {g}}}{c^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 409, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {658, 90, 66, 221, 2348, 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.

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

\(\Big \downarrow \) 658

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 2348

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 658

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_) + (c_.)*(x_ 
)^2), x_Symbol] :> Simp[g/c   Int[Simp[2*e*f + d*g + e*g*x, x]*(d + e*x)^(m 
 - 1)*(f + g*x)^(n - 2), x], x] + Simp[1/c   Int[Simp[c*d*f^2 - 2*a*e*f*g - 
 a*d*g^2 + (c*e*f^2 + 2*c*d*f*g - a*e*g^2)*x, x]*(d + e*x)^(m - 1)*((f + g* 
x)^(n - 2)/(a + c*x^2)), x], x] /; FreeQ[{a, c, d, e, f, g}, x] &&  !Intege 
rQ[m] &&  !IntegerQ[n] && GtQ[m, 0] && GtQ[n, 1]
 

rule 2009

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

rule 2348

Int[(Px_)*((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_. 
)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(c + d*x)^m*(e + f*x)^ 
n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[P 
x, x] && (IntegerQ[p] || (IntegerQ[2*p] && IntegerQ[m] && ILtQ[n, 0])) && 
!(IGtQ[m, 0] && IGtQ[n, 0])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2384\) vs. \(2(333)=666\).

Time = 1.76 (sec) , antiderivative size = 2385, normalized size of antiderivative = 5.67

method	result	size
default	\(\text {Expression too large to display}\)	\(2385\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^{3/2} \sqrt {f+g x}}{a+c x^2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((sqrt(f + g*x)*sqrt(d + e*x)*x)/(a + c*x**2),x)*e + int((sqrt(f + g*x) 
*sqrt(d + e*x))/(a + c*x**2),x)*d