\(\int \frac {(a+b x)^{5/2}}{(c+d x)^{3/2}} \, dx\) [353]

Optimal result

Integrand size = 19, antiderivative size = 138 \[ \int \frac {(a+b x)^{5/2}}{(c+d x)^{3/2}} \, dx=-\frac {2 (a+b x)^{5/2}}{d \sqrt {c+d x}}-\frac {15 b (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^3}+\frac {5 b (a+b x)^{3/2} \sqrt {c+d x}}{2 d^2}+\frac {15 \sqrt {b} (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 d^{7/2}} \] Output:

-2*(b*x+a)^(5/2)/d/(d*x+c)^(1/2)-15/4*b*(-a*d+b*c)*(b*x+a)^(1/2)*(d*x+c)^( 
1/2)/d^3+5/2*b*(b*x+a)^(3/2)*(d*x+c)^(1/2)/d^2+15/4*b^(1/2)*(-a*d+b*c)^2*a 
rctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/d^(7/2)
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x)^{5/2}}{(c+d x)^{3/2}} \, dx=\frac {\sqrt {a+b x} \left (-8 a^2 d^2+a b d (25 c+9 d x)+b^2 \left (-15 c^2-5 c d x+2 d^2 x^2\right )\right )}{4 d^3 \sqrt {c+d x}}+\frac {15 \sqrt {b} (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{4 d^{7/2}} \] Input:

Integrate[(a + b*x)^(5/2)/(c + d*x)^(3/2),x]
 

Output:

(Sqrt[a + b*x]*(-8*a^2*d^2 + a*b*d*(25*c + 9*d*x) + b^2*(-15*c^2 - 5*c*d*x 
 + 2*d^2*x^2)))/(4*d^3*Sqrt[c + d*x]) + (15*Sqrt[b]*(b*c - a*d)^2*ArcTanh[ 
(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(4*d^(7/2))
 

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {57, 60, 60, 66, 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.

\(\displaystyle \int \frac {(a+b x)^{5/2}}{(c+d x)^{3/2}} \, dx\)

\(\Big \downarrow \) 57

\(\displaystyle \frac {5 b \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}}dx}{d}-\frac {2 (a+b x)^{5/2}}{d \sqrt {c+d x}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {5 b \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}}dx}{4 d}\right )}{d}-\frac {2 (a+b x)^{5/2}}{d \sqrt {c+d x}}\)

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 66

\(\displaystyle \frac {5 b \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{d}\right )}{4 d}\right )}{d}-\frac {2 (a+b x)^{5/2}}{d \sqrt {c+d x}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {5 b \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{3/2}}\right )}{4 d}\right )}{d}-\frac {2 (a+b x)^{5/2}}{d \sqrt {c+d x}}\)

Input:

Int[(a + b*x)^(5/2)/(c + d*x)^(3/2),x]
 

Output:

(-2*(a + b*x)^(5/2))/(d*Sqrt[c + d*x]) + (5*b*(((a + b*x)^(3/2)*Sqrt[c + d 
*x])/(2*d) - (3*(b*c - a*d)*((Sqrt[a + b*x]*Sqrt[c + d*x])/d - ((b*c - a*d 
)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(Sqrt[b]*d^(3/ 
2))))/(4*d)))/d
 

Defintions of rubi rules used

rule 57

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}, x] & 
& GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m 
+ n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c 
, d, m, n, x]
 

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 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 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 [F]

\[\int \frac {\left (b x +a \right )^{\frac {5}{2}}}{\left (x d +c \right )^{\frac {3}{2}}}d x\]

Input:

int((b*x+a)^(5/2)/(d*x+c)^(3/2),x)
 

Output:

int((b*x+a)^(5/2)/(d*x+c)^(3/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 441, normalized size of antiderivative = 3.20 \[ \int \frac {(a+b x)^{5/2}}{(c+d x)^{3/2}} \, dx=\left [\frac {15 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x\right )} \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (2 \, b^{2} d^{2} x^{2} - 15 \, b^{2} c^{2} + 25 \, a b c d - 8 \, a^{2} d^{2} - {\left (5 \, b^{2} c d - 9 \, a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, {\left (d^{4} x + c d^{3}\right )}}, -\frac {15 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x\right )} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right ) - 2 \, {\left (2 \, b^{2} d^{2} x^{2} - 15 \, b^{2} c^{2} + 25 \, a b c d - 8 \, a^{2} d^{2} - {\left (5 \, b^{2} c d - 9 \, a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, {\left (d^{4} x + c d^{3}\right )}}\right ] \] Input:

integrate((b*x+a)^(5/2)/(d*x+c)^(3/2),x, algorithm="fricas")
 

Output:

[1/16*(15*(b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2 + (b^2*c^2*d - 2*a*b*c*d^2 + 
a^2*d^3)*x)*sqrt(b/d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 
4*(2*b*d^2*x + b*c*d + a*d^2)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b/d) + 8*(b 
^2*c*d + a*b*d^2)*x) + 4*(2*b^2*d^2*x^2 - 15*b^2*c^2 + 25*a*b*c*d - 8*a^2* 
d^2 - (5*b^2*c*d - 9*a*b*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(d^4*x + c*d 
^3), -1/8*(15*(b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2 + (b^2*c^2*d - 2*a*b*c*d^ 
2 + a^2*d^3)*x)*sqrt(-b/d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(b*x + a)* 
sqrt(d*x + c)*sqrt(-b/d)/(b^2*d*x^2 + a*b*c + (b^2*c + a*b*d)*x)) - 2*(2*b 
^2*d^2*x^2 - 15*b^2*c^2 + 25*a*b*c*d - 8*a^2*d^2 - (5*b^2*c*d - 9*a*b*d^2) 
*x)*sqrt(b*x + a)*sqrt(d*x + c))/(d^4*x + c*d^3)]
 

Sympy [F]

\[ \int \frac {(a+b x)^{5/2}}{(c+d x)^{3/2}} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}}}{\left (c + d x\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((b*x+a)**(5/2)/(d*x+c)**(3/2),x)
 

Output:

Integral((a + b*x)**(5/2)/(c + d*x)**(3/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^{5/2}}{(c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x+a)^(5/2)/(d*x+c)^(3/2),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(a*d-b*c>0)', see `assume?` for m 
ore detail
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.46 \[ \int \frac {(a+b x)^{5/2}}{(c+d x)^{3/2}} \, dx=\frac {\sqrt {b x + a} {\left ({\left (b x + a\right )} {\left (\frac {2 \, {\left (b x + a\right )} b^{2}}{d {\left | b \right |}} - \frac {5 \, {\left (b^{3} c d^{3} - a b^{2} d^{4}\right )}}{d^{5} {\left | b \right |}}\right )} - \frac {15 \, {\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )}}{d^{5} {\left | b \right |}}\right )}}{4 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} - \frac {15 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{4 \, \sqrt {b d} d^{3} {\left | b \right |}} \] Input:

integrate((b*x+a)^(5/2)/(d*x+c)^(3/2),x, algorithm="giac")
 

Output:

1/4*sqrt(b*x + a)*((b*x + a)*(2*(b*x + a)*b^2/(d*abs(b)) - 5*(b^3*c*d^3 - 
a*b^2*d^4)/(d^5*abs(b))) - 15*(b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)/ 
(d^5*abs(b)))/sqrt(b^2*c + (b*x + a)*b*d - a*b*d) - 15/4*(b^4*c^2 - 2*a*b^ 
3*c*d + a^2*b^2*d^2)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x 
+ a)*b*d - a*b*d)))/(sqrt(b*d)*d^3*abs(b))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{5/2}}{(c+d x)^{3/2}} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}}{{\left (c+d\,x\right )}^{3/2}} \,d x \] Input:

int((a + b*x)^(5/2)/(c + d*x)^(3/2),x)
 

Output:

int((a + b*x)^(5/2)/(c + d*x)^(3/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 485, normalized size of antiderivative = 3.51 \[ \int \frac {(a+b x)^{5/2}}{(c+d x)^{3/2}} \, dx=\frac {-8 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{2} d^{3}+25 \sqrt {d x +c}\, \sqrt {b x +a}\, a b c \,d^{2}+9 \sqrt {d x +c}\, \sqrt {b x +a}\, a b \,d^{3} x -15 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{2} c^{2} d -5 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{2} c \,d^{2} x +2 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{2} d^{3} x^{2}+15 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{2} c \,d^{2}+15 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{2} d^{3} x -30 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a b \,c^{2} d -30 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a b c \,d^{2} x +15 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b^{2} c^{3}+15 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b^{2} c^{2} d x -10 \sqrt {d}\, \sqrt {b}\, a^{2} c \,d^{2}-10 \sqrt {d}\, \sqrt {b}\, a^{2} d^{3} x +20 \sqrt {d}\, \sqrt {b}\, a b \,c^{2} d +20 \sqrt {d}\, \sqrt {b}\, a b c \,d^{2} x -10 \sqrt {d}\, \sqrt {b}\, b^{2} c^{3}-10 \sqrt {d}\, \sqrt {b}\, b^{2} c^{2} d x}{4 d^{4} \left (d x +c \right )} \] Input:

int((b*x+a)^(5/2)/(d*x+c)^(3/2),x)
 

Output:

( - 8*sqrt(c + d*x)*sqrt(a + b*x)*a**2*d**3 + 25*sqrt(c + d*x)*sqrt(a + b* 
x)*a*b*c*d**2 + 9*sqrt(c + d*x)*sqrt(a + b*x)*a*b*d**3*x - 15*sqrt(c + d*x 
)*sqrt(a + b*x)*b**2*c**2*d - 5*sqrt(c + d*x)*sqrt(a + b*x)*b**2*c*d**2*x 
+ 2*sqrt(c + d*x)*sqrt(a + b*x)*b**2*d**3*x**2 + 15*sqrt(d)*sqrt(b)*log((s 
qrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**2*c*d**2 
 + 15*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/ 
sqrt(a*d - b*c))*a**2*d**3*x - 30*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b* 
x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a*b*c**2*d - 30*sqrt(d)*sqrt( 
b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a* 
b*c*d**2*x + 15*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt( 
c + d*x))/sqrt(a*d - b*c))*b**2*c**3 + 15*sqrt(d)*sqrt(b)*log((sqrt(d)*sqr 
t(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*b**2*c**2*d*x - 10*sq 
rt(d)*sqrt(b)*a**2*c*d**2 - 10*sqrt(d)*sqrt(b)*a**2*d**3*x + 20*sqrt(d)*sq 
rt(b)*a*b*c**2*d + 20*sqrt(d)*sqrt(b)*a*b*c*d**2*x - 10*sqrt(d)*sqrt(b)*b* 
*2*c**3 - 10*sqrt(d)*sqrt(b)*b**2*c**2*d*x)/(4*d**4*(c + d*x))