\(\int \frac {(a+b x)^{7/2}}{(c+d x)^{3/2}} \, dx\) [352]

Optimal result

Integrand size = 19, antiderivative size = 174 \[ \int \frac {(a+b x)^{7/2}}{(c+d x)^{3/2}} \, dx=-\frac {2 (a+b x)^{7/2}}{d \sqrt {c+d x}}+\frac {35 b (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 d^4}-\frac {35 b (b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 d^3}+\frac {7 b (a+b x)^{5/2} \sqrt {c+d x}}{3 d^2}-\frac {35 \sqrt {b} (b c-a d)^3 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 d^{9/2}} \] Output:

-2*(b*x+a)^(7/2)/d/(d*x+c)^(1/2)+35/8*b*(-a*d+b*c)^2*(b*x+a)^(1/2)*(d*x+c) 
^(1/2)/d^4-35/12*b*(-a*d+b*c)*(b*x+a)^(3/2)*(d*x+c)^(1/2)/d^3+7/3*b*(b*x+a 
)^(5/2)*(d*x+c)^(1/2)/d^2-35/8*b^(1/2)*(-a*d+b*c)^3*arctanh(d^(1/2)*(b*x+a 
)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/d^(9/2)
 

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x)^{7/2}}{(c+d x)^{3/2}} \, dx=\frac {\sqrt {a+b x} \left (-48 a^3 d^3+3 a^2 b d^2 (77 c+29 d x)+2 a b^2 d \left (-140 c^2-49 c d x+19 d^2 x^2\right )+b^3 \left (105 c^3+35 c^2 d x-14 c d^2 x^2+8 d^3 x^3\right )\right )}{24 d^4 \sqrt {c+d x}}-\frac {35 \sqrt {b} (b c-a d)^3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{8 d^{9/2}} \] Input:

Integrate[(a + b*x)^(7/2)/(c + d*x)^(3/2),x]
 

Output:

(Sqrt[a + b*x]*(-48*a^3*d^3 + 3*a^2*b*d^2*(77*c + 29*d*x) + 2*a*b^2*d*(-14 
0*c^2 - 49*c*d*x + 19*d^2*x^2) + b^3*(105*c^3 + 35*c^2*d*x - 14*c*d^2*x^2 
+ 8*d^3*x^3)))/(24*d^4*Sqrt[c + d*x]) - (35*Sqrt[b]*(b*c - a*d)^3*ArcTanh[ 
(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(8*d^(9/2))
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {57, 60, 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.

\(\displaystyle \int \frac {(a+b x)^{7/2}}{(c+d x)^{3/2}} \, dx\)

\(\Big \downarrow \) 57

\(\displaystyle \frac {7 b \int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}}dx}{d}-\frac {2 (a+b x)^{7/2}}{d \sqrt {c+d x}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {7 b \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}}dx}{6 d}\right )}{d}-\frac {2 (a+b x)^{7/2}}{d \sqrt {c+d x}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {7 b \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}}dx}{4 d}\right )}{6 d}\right )}{d}-\frac {2 (a+b x)^{7/2}}{d \sqrt {c+d x}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {7 b \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 d}\right )}{4 d}\right )}{6 d}\right )}{d}-\frac {2 (a+b x)^{7/2}}{d \sqrt {c+d x}}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {7 b \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{d}\right )}{4 d}\right )}{6 d}\right )}{d}-\frac {2 (a+b x)^{7/2}}{d \sqrt {c+d x}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {7 b \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{3/2}}\right )}{4 d}\right )}{6 d}\right )}{d}-\frac {2 (a+b x)^{7/2}}{d \sqrt {c+d x}}\)

Input:

Int[(a + b*x)^(7/2)/(c + d*x)^(3/2),x]
 

Output:

(-2*(a + b*x)^(7/2))/(d*Sqrt[c + d*x]) + (7*b*(((a + b*x)^(5/2)*Sqrt[c + d 
*x])/(3*d) - (5*(b*c - a*d)*(((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)))/(6*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]

\[\int \frac {\left (b x +a \right )^{\frac {7}{2}}}{\left (x d +c \right )^{\frac {3}{2}}}d x\]

Input:

int((b*x+a)^(7/2)/(d*x+c)^(3/2),x)
 

Output:

int((b*x+a)^(7/2)/(d*x+c)^(3/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (138) = 276\).

Time = 0.25 (sec) , antiderivative size = 603, normalized size of antiderivative = 3.47 \[ \int \frac {(a+b x)^{7/2}}{(c+d x)^{3/2}} \, dx=\left [-\frac {105 \, {\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3} + {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} x\right )} \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (8 \, b^{3} d^{3} x^{3} + 105 \, b^{3} c^{3} - 280 \, a b^{2} c^{2} d + 231 \, a^{2} b c d^{2} - 48 \, a^{3} d^{3} - 2 \, {\left (7 \, b^{3} c d^{2} - 19 \, a b^{2} d^{3}\right )} x^{2} + {\left (35 \, b^{3} c^{2} d - 98 \, a b^{2} c d^{2} + 87 \, a^{2} b d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, {\left (d^{5} x + c d^{4}\right )}}, \frac {105 \, {\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3} + {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} x\right )} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right ) + 2 \, {\left (8 \, b^{3} d^{3} x^{3} + 105 \, b^{3} c^{3} - 280 \, a b^{2} c^{2} d + 231 \, a^{2} b c d^{2} - 48 \, a^{3} d^{3} - 2 \, {\left (7 \, b^{3} c d^{2} - 19 \, a b^{2} d^{3}\right )} x^{2} + {\left (35 \, b^{3} c^{2} d - 98 \, a b^{2} c d^{2} + 87 \, a^{2} b d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (d^{5} x + c d^{4}\right )}}\right ] \] Input:

integrate((b*x+a)^(7/2)/(d*x+c)^(3/2),x, algorithm="fricas")
 

Output:

[-1/96*(105*(b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3 + (b^3* 
c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*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)*sq 
rt(b*x + a)*sqrt(d*x + c)*sqrt(b/d) + 8*(b^2*c*d + a*b*d^2)*x) - 4*(8*b^3* 
d^3*x^3 + 105*b^3*c^3 - 280*a*b^2*c^2*d + 231*a^2*b*c*d^2 - 48*a^3*d^3 - 2 
*(7*b^3*c*d^2 - 19*a*b^2*d^3)*x^2 + (35*b^3*c^2*d - 98*a*b^2*c*d^2 + 87*a^ 
2*b*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(d^5*x + c*d^4), 1/48*(105*(b^3*c 
^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3 + (b^3*c^3*d - 3*a*b^2*c^ 
2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*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*(8*b^3*d^3*x^3 + 105*b^3*c^3 - 280*a*b^2*c^2*d + 231*a^2*b 
*c*d^2 - 48*a^3*d^3 - 2*(7*b^3*c*d^2 - 19*a*b^2*d^3)*x^2 + (35*b^3*c^2*d - 
 98*a*b^2*c*d^2 + 87*a^2*b*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(d^5*x + c 
*d^4)]
 

Sympy [F]

\[ \int \frac {(a+b x)^{7/2}}{(c+d x)^{3/2}} \, dx=\int \frac {\left (a + b x\right )^{\frac {7}{2}}}{\left (c + d x\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((b*x+a)**(7/2)/(d*x+c)**(3/2),x)
 

Output:

Integral((a + b*x)**(7/2)/(c + d*x)**(3/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^{7/2}}{(c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x+a)^(7/2)/(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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (138) = 276\).

Time = 0.19 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.60 \[ \int \frac {(a+b x)^{7/2}}{(c+d x)^{3/2}} \, dx=\frac {{\left ({\left (b x + a\right )} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )} b^{2}}{d {\left | b \right |}} - \frac {7 \, {\left (b^{3} c d^{5} - a b^{2} d^{6}\right )}}{d^{7} {\left | b \right |}}\right )} + \frac {35 \, {\left (b^{4} c^{2} d^{4} - 2 \, a b^{3} c d^{5} + a^{2} b^{2} d^{6}\right )}}{d^{7} {\left | b \right |}}\right )} + \frac {105 \, {\left (b^{5} c^{3} d^{3} - 3 \, a b^{4} c^{2} d^{4} + 3 \, a^{2} b^{3} c d^{5} - a^{3} b^{2} d^{6}\right )}}{d^{7} {\left | b \right |}}\right )} \sqrt {b x + a}}{24 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} + \frac {35 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{8 \, \sqrt {b d} d^{4} {\left | b \right |}} \] Input:

integrate((b*x+a)^(7/2)/(d*x+c)^(3/2),x, algorithm="giac")
 

Output:

1/24*((b*x + a)*(2*(b*x + a)*(4*(b*x + a)*b^2/(d*abs(b)) - 7*(b^3*c*d^5 - 
a*b^2*d^6)/(d^7*abs(b))) + 35*(b^4*c^2*d^4 - 2*a*b^3*c*d^5 + a^2*b^2*d^6)/ 
(d^7*abs(b))) + 105*(b^5*c^3*d^3 - 3*a*b^4*c^2*d^4 + 3*a^2*b^3*c*d^5 - a^3 
*b^2*d^6)/(d^7*abs(b)))*sqrt(b*x + a)/sqrt(b^2*c + (b*x + a)*b*d - a*b*d) 
+ 35/8*(b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*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^4*abs(b))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{7/2}}{(c+d x)^{3/2}} \, dx=\int \frac {{\left (a+b\,x\right )}^{7/2}}{{\left (c+d\,x\right )}^{3/2}} \,d x \] Input:

int((a + b*x)^(7/2)/(c + d*x)^(3/2),x)
 

Output:

int((a + b*x)^(7/2)/(c + d*x)^(3/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 724, normalized size of antiderivative = 4.16 \[ \int \frac {(a+b x)^{7/2}}{(c+d x)^{3/2}} \, dx =\text {Too large to display} \] Input:

int((b*x+a)^(7/2)/(d*x+c)^(3/2),x)
 

Output:

( - 384*sqrt(c + d*x)*sqrt(a + b*x)*a**3*d**4 + 1848*sqrt(c + d*x)*sqrt(a 
+ b*x)*a**2*b*c*d**3 + 696*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b*d**4*x - 224 
0*sqrt(c + d*x)*sqrt(a + b*x)*a*b**2*c**2*d**2 - 784*sqrt(c + d*x)*sqrt(a 
+ b*x)*a*b**2*c*d**3*x + 304*sqrt(c + d*x)*sqrt(a + b*x)*a*b**2*d**4*x**2 
+ 840*sqrt(c + d*x)*sqrt(a + b*x)*b**3*c**3*d + 280*sqrt(c + d*x)*sqrt(a + 
 b*x)*b**3*c**2*d**2*x - 112*sqrt(c + d*x)*sqrt(a + b*x)*b**3*c*d**3*x**2 
+ 64*sqrt(c + d*x)*sqrt(a + b*x)*b**3*d**4*x**3 + 840*sqrt(d)*sqrt(b)*log( 
(sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**3*c*d* 
*3 + 840*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x 
))/sqrt(a*d - b*c))*a**3*d**4*x - 2520*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a 
 + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**2*b*c**2*d**2 - 2520* 
sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a 
*d - b*c))*a**2*b*c*d**3*x + 2520*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b* 
x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a*b**2*c**3*d + 2520*sqrt(d)* 
sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c 
))*a*b**2*c**2*d**2*x - 840*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + s 
qrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*b**3*c**4 - 840*sqrt(d)*sqrt(b)*log 
((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*b**3*c** 
3*d*x - 525*sqrt(d)*sqrt(b)*a**3*c*d**3 - 525*sqrt(d)*sqrt(b)*a**3*d**4*x 
+ 1575*sqrt(d)*sqrt(b)*a**2*b*c**2*d**2 + 1575*sqrt(d)*sqrt(b)*a**2*b*c...