\(\int \frac {(e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx\) [99]

Optimal result

Integrand size = 22, antiderivative size = 152 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx=\frac {2 f h \sqrt {e+f x}}{d^3}-\frac {(3 d f g+4 d e h-7 c f h) \sqrt {e+f x}}{4 d^3 (c+d x)}-\frac {(d g-c h) (e+f x)^{3/2}}{2 d^2 (c+d x)^2}-\frac {3 f (d f g+4 d e h-5 c f h) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{4 d^{7/2} \sqrt {d e-c f}} \] Output:

2*f*h*(f*x+e)^(1/2)/d^3-1/4*(-7*c*f*h+4*d*e*h+3*d*f*g)*(f*x+e)^(1/2)/d^3/( 
d*x+c)-1/2*(-c*h+d*g)*(f*x+e)^(3/2)/d^2/(d*x+c)^2-3/4*f*(-5*c*f*h+4*d*e*h+ 
d*f*g)*arctanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^(1/2))/d^(7/2)/(-c*f+d*e)^ 
(1/2)
 

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.91 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx=-\frac {\sqrt {e+f x} \left (-15 c^2 f h+c d (3 f g+2 e h-25 f h x)+d^2 (f x (5 g-8 h x)+2 e (g+2 h x))\right )}{4 d^3 (c+d x)^2}+\frac {3 f (d f g+4 d e h-5 c f h) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{4 d^{7/2} \sqrt {-d e+c f}} \] Input:

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

Output:

-1/4*(Sqrt[e + f*x]*(-15*c^2*f*h + c*d*(3*f*g + 2*e*h - 25*f*h*x) + d^2*(f 
*x*(5*g - 8*h*x) + 2*e*(g + 2*h*x))))/(d^3*(c + d*x)^2) + (3*f*(d*f*g + 4* 
d*e*h - 5*c*f*h)*ArcTan[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[-(d*e) + c*f]])/(4*d^ 
(7/2)*Sqrt[-(d*e) + c*f])
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {87, 51, 60, 73, 221}

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.

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

\(\Big \downarrow \) 87

\(\displaystyle \frac {(-5 c f h+4 d e h+d f g) \int \frac {(e+f x)^{3/2}}{(c+d x)^2}dx}{4 d (d e-c f)}-\frac {(e+f x)^{5/2} (d g-c h)}{2 d (c+d x)^2 (d e-c f)}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {(-5 c f h+4 d e h+d f g) \left (\frac {3 f \int \frac {\sqrt {e+f x}}{c+d x}dx}{2 d}-\frac {(e+f x)^{3/2}}{d (c+d x)}\right )}{4 d (d e-c f)}-\frac {(e+f x)^{5/2} (d g-c h)}{2 d (c+d x)^2 (d e-c f)}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(-5 c f h+4 d e h+d f g) \left (\frac {3 f \left (\frac {(d e-c f) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{d}+\frac {2 \sqrt {e+f x}}{d}\right )}{2 d}-\frac {(e+f x)^{3/2}}{d (c+d x)}\right )}{4 d (d e-c f)}-\frac {(e+f x)^{5/2} (d g-c h)}{2 d (c+d x)^2 (d e-c f)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {(-5 c f h+4 d e h+d f g) \left (\frac {3 f \left (\frac {2 (d e-c f) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x}}{d f}+\frac {2 \sqrt {e+f x}}{d}\right )}{2 d}-\frac {(e+f x)^{3/2}}{d (c+d x)}\right )}{4 d (d e-c f)}-\frac {(e+f x)^{5/2} (d g-c h)}{2 d (c+d x)^2 (d e-c f)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\left (\frac {3 f \left (\frac {2 \sqrt {e+f x}}{d}-\frac {2 \sqrt {d e-c f} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{3/2}}\right )}{2 d}-\frac {(e+f x)^{3/2}}{d (c+d x)}\right ) (-5 c f h+4 d e h+d f g)}{4 d (d e-c f)}-\frac {(e+f x)^{5/2} (d g-c h)}{2 d (c+d x)^2 (d e-c f)}\)

Input:

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

Output:

-1/2*((d*g - c*h)*(e + f*x)^(5/2))/(d*(d*e - c*f)*(c + d*x)^2) + ((d*f*g + 
 4*d*e*h - 5*c*f*h)*(-((e + f*x)^(3/2)/(d*(c + d*x))) + (3*f*((2*Sqrt[e + 
f*x])/d - (2*Sqrt[d*e - c*f]*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c* 
f]])/d^(3/2)))/(2*d)))/(4*d*(d*e - c*f))
 

Defintions of rubi rules used

rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
 

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 87

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

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]
 

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.96

method	result	size
pseudoelliptic	\(-\frac {15 \left (\left (x d +c \right )^{2} f \left (\frac {\left (-4 e h -f g \right ) d}{5}+c f h \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )-\sqrt {\left (c f -d e \right ) d}\, \left (\frac {\left (-\left (-\frac {8 h x}{5}+g \right ) x f -\frac {2 e \left (2 h x +g \right )}{5}\right ) d^{2}}{3}-\frac {2 c \left (\frac {\left (-25 h x +3 g \right ) f}{2}+e h \right ) d}{15}+c^{2} f h \right ) \sqrt {f x +e}\right )}{4 \sqrt {\left (c f -d e \right ) d}\, d^{3} \left (x d +c \right )^{2}}\)	\(146\)
derivativedivides	\(2 f \left (\frac {h \sqrt {f x +e}}{d^{3}}-\frac {\frac {\left (-\frac {9}{8} c d f h +\frac {1}{2} d^{2} e h +\frac {5}{8} d^{2} f g \right ) \left (f x +e \right )^{\frac {3}{2}}+\left (-\frac {7}{8} c^{2} f^{2} h +\frac {11}{8} c d e h f +\frac {3}{8} c d \,f^{2} g -\frac {1}{2} d^{2} e^{2} h -\frac {3}{8} d^{2} e g f \right ) \sqrt {f x +e}}{\left (\left (f x +e \right ) d +c f -d e \right )^{2}}+\frac {3 \left (5 c f h -4 d e h -d f g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{8 \sqrt {\left (c f -d e \right ) d}}}{d^{3}}\right )\)	\(173\)
default	\(2 f \left (\frac {h \sqrt {f x +e}}{d^{3}}-\frac {\frac {\left (-\frac {9}{8} c d f h +\frac {1}{2} d^{2} e h +\frac {5}{8} d^{2} f g \right ) \left (f x +e \right )^{\frac {3}{2}}+\left (-\frac {7}{8} c^{2} f^{2} h +\frac {11}{8} c d e h f +\frac {3}{8} c d \,f^{2} g -\frac {1}{2} d^{2} e^{2} h -\frac {3}{8} d^{2} e g f \right ) \sqrt {f x +e}}{\left (\left (f x +e \right ) d +c f -d e \right )^{2}}+\frac {3 \left (5 c f h -4 d e h -d f g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{8 \sqrt {\left (c f -d e \right ) d}}}{d^{3}}\right )\)	\(173\)
risch	\(\frac {2 f h \sqrt {f x +e}}{d^{3}}-\frac {2 f \left (\frac {\left (-\frac {9}{8} c d f h +\frac {1}{2} d^{2} e h +\frac {5}{8} d^{2} f g \right ) \left (f x +e \right )^{\frac {3}{2}}+\left (-\frac {7}{8} c^{2} f^{2} h +\frac {11}{8} c d e h f +\frac {3}{8} c d \,f^{2} g -\frac {1}{2} d^{2} e^{2} h -\frac {3}{8} d^{2} e g f \right ) \sqrt {f x +e}}{\left (\left (f x +e \right ) d +c f -d e \right )^{2}}+\frac {3 \left (5 c f h -4 d e h -d f g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{8 \sqrt {\left (c f -d e \right ) d}}\right )}{d^{3}}\)	\(173\)

Input:

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

Output:

-15/4*((d*x+c)^2*f*(1/5*(-4*e*h-f*g)*d+c*f*h)*arctan(d*(f*x+e)^(1/2)/((c*f 
-d*e)*d)^(1/2))-((c*f-d*e)*d)^(1/2)*(1/3*(-(-8/5*h*x+g)*x*f-2/5*e*(2*h*x+g 
))*d^2-2/15*c*(1/2*(-25*h*x+3*g)*f+e*h)*d+c^2*f*h)*(f*x+e)^(1/2))/((c*f-d* 
e)*d)^(1/2)/d^3/(d*x+c)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 346 vs. \(2 (130) = 260\).

Time = 0.18 (sec) , antiderivative size = 706, normalized size of antiderivative = 4.64 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx=\left [-\frac {3 \, {\left (c^{2} d f^{2} g + {\left (d^{3} f^{2} g + {\left (4 \, d^{3} e f - 5 \, c d^{2} f^{2}\right )} h\right )} x^{2} + {\left (4 \, c^{2} d e f - 5 \, c^{3} f^{2}\right )} h + 2 \, {\left (c d^{2} f^{2} g + {\left (4 \, c d^{2} e f - 5 \, c^{2} d f^{2}\right )} h\right )} x\right )} \sqrt {d^{2} e - c d f} \log \left (\frac {d f x + 2 \, d e - c f + 2 \, \sqrt {d^{2} e - c d f} \sqrt {f x + e}}{d x + c}\right ) - 2 \, {\left (8 \, {\left (d^{4} e f - c d^{3} f^{2}\right )} h x^{2} - {\left (2 \, d^{4} e^{2} + c d^{3} e f - 3 \, c^{2} d^{2} f^{2}\right )} g - {\left (2 \, c d^{3} e^{2} - 17 \, c^{2} d^{2} e f + 15 \, c^{3} d f^{2}\right )} h - {\left (5 \, {\left (d^{4} e f - c d^{3} f^{2}\right )} g + {\left (4 \, d^{4} e^{2} - 29 \, c d^{3} e f + 25 \, c^{2} d^{2} f^{2}\right )} h\right )} x\right )} \sqrt {f x + e}}{8 \, {\left (c^{2} d^{5} e - c^{3} d^{4} f + {\left (d^{7} e - c d^{6} f\right )} x^{2} + 2 \, {\left (c d^{6} e - c^{2} d^{5} f\right )} x\right )}}, \frac {3 \, {\left (c^{2} d f^{2} g + {\left (d^{3} f^{2} g + {\left (4 \, d^{3} e f - 5 \, c d^{2} f^{2}\right )} h\right )} x^{2} + {\left (4 \, c^{2} d e f - 5 \, c^{3} f^{2}\right )} h + 2 \, {\left (c d^{2} f^{2} g + {\left (4 \, c d^{2} e f - 5 \, c^{2} d f^{2}\right )} h\right )} x\right )} \sqrt {-d^{2} e + c d f} \arctan \left (\frac {\sqrt {-d^{2} e + c d f} \sqrt {f x + e}}{d f x + d e}\right ) + {\left (8 \, {\left (d^{4} e f - c d^{3} f^{2}\right )} h x^{2} - {\left (2 \, d^{4} e^{2} + c d^{3} e f - 3 \, c^{2} d^{2} f^{2}\right )} g - {\left (2 \, c d^{3} e^{2} - 17 \, c^{2} d^{2} e f + 15 \, c^{3} d f^{2}\right )} h - {\left (5 \, {\left (d^{4} e f - c d^{3} f^{2}\right )} g + {\left (4 \, d^{4} e^{2} - 29 \, c d^{3} e f + 25 \, c^{2} d^{2} f^{2}\right )} h\right )} x\right )} \sqrt {f x + e}}{4 \, {\left (c^{2} d^{5} e - c^{3} d^{4} f + {\left (d^{7} e - c d^{6} f\right )} x^{2} + 2 \, {\left (c d^{6} e - c^{2} d^{5} f\right )} x\right )}}\right ] \] Input:

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

Output:

[-1/8*(3*(c^2*d*f^2*g + (d^3*f^2*g + (4*d^3*e*f - 5*c*d^2*f^2)*h)*x^2 + (4 
*c^2*d*e*f - 5*c^3*f^2)*h + 2*(c*d^2*f^2*g + (4*c*d^2*e*f - 5*c^2*d*f^2)*h 
)*x)*sqrt(d^2*e - c*d*f)*log((d*f*x + 2*d*e - c*f + 2*sqrt(d^2*e - c*d*f)* 
sqrt(f*x + e))/(d*x + c)) - 2*(8*(d^4*e*f - c*d^3*f^2)*h*x^2 - (2*d^4*e^2 
+ c*d^3*e*f - 3*c^2*d^2*f^2)*g - (2*c*d^3*e^2 - 17*c^2*d^2*e*f + 15*c^3*d* 
f^2)*h - (5*(d^4*e*f - c*d^3*f^2)*g + (4*d^4*e^2 - 29*c*d^3*e*f + 25*c^2*d 
^2*f^2)*h)*x)*sqrt(f*x + e))/(c^2*d^5*e - c^3*d^4*f + (d^7*e - c*d^6*f)*x^ 
2 + 2*(c*d^6*e - c^2*d^5*f)*x), 1/4*(3*(c^2*d*f^2*g + (d^3*f^2*g + (4*d^3* 
e*f - 5*c*d^2*f^2)*h)*x^2 + (4*c^2*d*e*f - 5*c^3*f^2)*h + 2*(c*d^2*f^2*g + 
 (4*c*d^2*e*f - 5*c^2*d*f^2)*h)*x)*sqrt(-d^2*e + c*d*f)*arctan(sqrt(-d^2*e 
 + c*d*f)*sqrt(f*x + e)/(d*f*x + d*e)) + (8*(d^4*e*f - c*d^3*f^2)*h*x^2 - 
(2*d^4*e^2 + c*d^3*e*f - 3*c^2*d^2*f^2)*g - (2*c*d^3*e^2 - 17*c^2*d^2*e*f 
+ 15*c^3*d*f^2)*h - (5*(d^4*e*f - c*d^3*f^2)*g + (4*d^4*e^2 - 29*c*d^3*e*f 
 + 25*c^2*d^2*f^2)*h)*x)*sqrt(f*x + e))/(c^2*d^5*e - c^3*d^4*f + (d^7*e - 
c*d^6*f)*x^2 + 2*(c*d^6*e - c^2*d^5*f)*x)]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(c*f-d*e>0)', see `assume?` for m 
ore detail
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.49 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx=\frac {2 \, \sqrt {f x + e} f h}{d^{3}} + \frac {3 \, {\left (d f^{2} g + 4 \, d e f h - 5 \, c f^{2} h\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{4 \, \sqrt {-d^{2} e + c d f} d^{3}} - \frac {5 \, {\left (f x + e\right )}^{\frac {3}{2}} d^{2} f^{2} g - 3 \, \sqrt {f x + e} d^{2} e f^{2} g + 3 \, \sqrt {f x + e} c d f^{3} g + 4 \, {\left (f x + e\right )}^{\frac {3}{2}} d^{2} e f h - 4 \, \sqrt {f x + e} d^{2} e^{2} f h - 9 \, {\left (f x + e\right )}^{\frac {3}{2}} c d f^{2} h + 11 \, \sqrt {f x + e} c d e f^{2} h - 7 \, \sqrt {f x + e} c^{2} f^{3} h}{4 \, {\left ({\left (f x + e\right )} d - d e + c f\right )}^{2} d^{3}} \] Input:

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

Output:

2*sqrt(f*x + e)*f*h/d^3 + 3/4*(d*f^2*g + 4*d*e*f*h - 5*c*f^2*h)*arctan(sqr 
t(f*x + e)*d/sqrt(-d^2*e + c*d*f))/(sqrt(-d^2*e + c*d*f)*d^3) - 1/4*(5*(f* 
x + e)^(3/2)*d^2*f^2*g - 3*sqrt(f*x + e)*d^2*e*f^2*g + 3*sqrt(f*x + e)*c*d 
*f^3*g + 4*(f*x + e)^(3/2)*d^2*e*f*h - 4*sqrt(f*x + e)*d^2*e^2*f*h - 9*(f* 
x + e)^(3/2)*c*d*f^2*h + 11*sqrt(f*x + e)*c*d*e*f^2*h - 7*sqrt(f*x + e)*c^ 
2*f^3*h)/(((f*x + e)*d - d*e + c*f)^2*d^3)
 

Mupad [B] (verification not implemented)

Time = 2.46 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.69 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx=\frac {2\,f\,h\,\sqrt {e+f\,x}}{d^3}-\frac {{\left (e+f\,x\right )}^{3/2}\,\left (\frac {5\,g\,d^2\,f^2}{4}+e\,h\,d^2\,f-\frac {9\,c\,h\,d\,f^2}{4}\right )-\sqrt {e+f\,x}\,\left (\frac {7\,h\,c^2\,f^3}{4}-\frac {11\,h\,c\,d\,e\,f^2}{4}-\frac {3\,g\,c\,d\,f^3}{4}+h\,d^2\,e^2\,f+\frac {3\,g\,d^2\,e\,f^2}{4}\right )}{d^5\,{\left (e+f\,x\right )}^2-\left (e+f\,x\right )\,\left (2\,d^5\,e-2\,c\,d^4\,f\right )+d^5\,e^2+c^2\,d^3\,f^2-2\,c\,d^4\,e\,f}+\frac {3\,f\,\mathrm {atan}\left (\frac {\sqrt {d}\,f\,\sqrt {e+f\,x}\,\left (4\,d\,e\,h-5\,c\,f\,h+d\,f\,g\right )}{\sqrt {c\,f-d\,e}\,\left (d\,f^2\,g-5\,c\,f^2\,h+4\,d\,e\,f\,h\right )}\right )\,\left (4\,d\,e\,h-5\,c\,f\,h+d\,f\,g\right )}{4\,d^{7/2}\,\sqrt {c\,f-d\,e}} \] Input:

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

Output:

(2*f*h*(e + f*x)^(1/2))/d^3 - ((e + f*x)^(3/2)*((5*d^2*f^2*g)/4 - (9*c*d*f 
^2*h)/4 + d^2*e*f*h) - (e + f*x)^(1/2)*((7*c^2*f^3*h)/4 - (3*c*d*f^3*g)/4 
+ (3*d^2*e*f^2*g)/4 + d^2*e^2*f*h - (11*c*d*e*f^2*h)/4))/(d^5*(e + f*x)^2 
- (e + f*x)*(2*d^5*e - 2*c*d^4*f) + d^5*e^2 + c^2*d^3*f^2 - 2*c*d^4*e*f) + 
 (3*f*atan((d^(1/2)*f*(e + f*x)^(1/2)*(4*d*e*h - 5*c*f*h + d*f*g))/((c*f - 
 d*e)^(1/2)*(d*f^2*g - 5*c*f^2*h + 4*d*e*f*h)))*(4*d*e*h - 5*c*f*h + d*f*g 
))/(4*d^(7/2)*(c*f - d*e)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 681, normalized size of antiderivative = 4.48 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx=\frac {-15 \sqrt {d}\, \sqrt {c f -d e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, d}{\sqrt {d}\, \sqrt {c f -d e}}\right ) c^{3} f^{2} h +12 \sqrt {d}\, \sqrt {c f -d e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, d}{\sqrt {d}\, \sqrt {c f -d e}}\right ) c^{2} d e f h +3 \sqrt {d}\, \sqrt {c f -d e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, d}{\sqrt {d}\, \sqrt {c f -d e}}\right ) c^{2} d \,f^{2} g -30 \sqrt {d}\, \sqrt {c f -d e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, d}{\sqrt {d}\, \sqrt {c f -d e}}\right ) c^{2} d \,f^{2} h x +24 \sqrt {d}\, \sqrt {c f -d e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, d}{\sqrt {d}\, \sqrt {c f -d e}}\right ) c \,d^{2} e f h x +6 \sqrt {d}\, \sqrt {c f -d e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, d}{\sqrt {d}\, \sqrt {c f -d e}}\right ) c \,d^{2} f^{2} g x -15 \sqrt {d}\, \sqrt {c f -d e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, d}{\sqrt {d}\, \sqrt {c f -d e}}\right ) c \,d^{2} f^{2} h \,x^{2}+12 \sqrt {d}\, \sqrt {c f -d e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, d}{\sqrt {d}\, \sqrt {c f -d e}}\right ) d^{3} e f h \,x^{2}+3 \sqrt {d}\, \sqrt {c f -d e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, d}{\sqrt {d}\, \sqrt {c f -d e}}\right ) d^{3} f^{2} g \,x^{2}+15 \sqrt {f x +e}\, c^{3} d \,f^{2} h -17 \sqrt {f x +e}\, c^{2} d^{2} e f h -3 \sqrt {f x +e}\, c^{2} d^{2} f^{2} g +25 \sqrt {f x +e}\, c^{2} d^{2} f^{2} h x +2 \sqrt {f x +e}\, c \,d^{3} e^{2} h +\sqrt {f x +e}\, c \,d^{3} e f g -29 \sqrt {f x +e}\, c \,d^{3} e f h x -5 \sqrt {f x +e}\, c \,d^{3} f^{2} g x +8 \sqrt {f x +e}\, c \,d^{3} f^{2} h \,x^{2}+2 \sqrt {f x +e}\, d^{4} e^{2} g +4 \sqrt {f x +e}\, d^{4} e^{2} h x +5 \sqrt {f x +e}\, d^{4} e f g x -8 \sqrt {f x +e}\, d^{4} e f h \,x^{2}}{4 d^{4} \left (c \,d^{2} f \,x^{2}-d^{3} e \,x^{2}+2 c^{2} d f x -2 c \,d^{2} e x +c^{3} f -c^{2} d e \right )} \] Input:

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

Output:

( - 15*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - 
d*e)))*c**3*f**2*h + 12*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sq 
rt(d)*sqrt(c*f - d*e)))*c**2*d*e*f*h + 3*sqrt(d)*sqrt(c*f - d*e)*atan((sqr 
t(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*c**2*d*f**2*g - 30*sqrt(d)*sqrt(c 
*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*c**2*d*f**2*h* 
x + 24*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - 
d*e)))*c*d**2*e*f*h*x + 6*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/( 
sqrt(d)*sqrt(c*f - d*e)))*c*d**2*f**2*g*x - 15*sqrt(d)*sqrt(c*f - d*e)*ata 
n((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*c*d**2*f**2*h*x**2 + 12*sqr 
t(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*d** 
3*e*f*h*x**2 + 3*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*s 
qrt(c*f - d*e)))*d**3*f**2*g*x**2 + 15*sqrt(e + f*x)*c**3*d*f**2*h - 17*sq 
rt(e + f*x)*c**2*d**2*e*f*h - 3*sqrt(e + f*x)*c**2*d**2*f**2*g + 25*sqrt(e 
 + f*x)*c**2*d**2*f**2*h*x + 2*sqrt(e + f*x)*c*d**3*e**2*h + sqrt(e + f*x) 
*c*d**3*e*f*g - 29*sqrt(e + f*x)*c*d**3*e*f*h*x - 5*sqrt(e + f*x)*c*d**3*f 
**2*g*x + 8*sqrt(e + f*x)*c*d**3*f**2*h*x**2 + 2*sqrt(e + f*x)*d**4*e**2*g 
 + 4*sqrt(e + f*x)*d**4*e**2*h*x + 5*sqrt(e + f*x)*d**4*e*f*g*x - 8*sqrt(e 
 + f*x)*d**4*e*f*h*x**2)/(4*d**4*(c**3*f - c**2*d*e + 2*c**2*d*f*x - 2*c*d 
**2*e*x + c*d**2*f*x**2 - d**3*e*x**2))