3.17.39 \(\int \frac {(c+d x)^{3/4}}{(a+b x)^{5/2}} \, dx\) [1639]

3.17.39.1 Optimal result

Integrand size = 19, antiderivative size = 221 \[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{5/2}} \, dx=-\frac {2 (c+d x)^{3/4}}{3 b (a+b x)^{3/2}}-\frac {d (c+d x)^{3/4}}{b (b c-a d) \sqrt {a+b x}}+\frac {d \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{7/4} \sqrt [4]{b c-a d} \sqrt {a+b x}}-\frac {d \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{7/4} \sqrt [4]{b c-a d} \sqrt {a+b x}} \]

output

-2/3*(d*x+c)^(3/4)/b/(b*x+a)^(3/2)-d*(d*x+c)^(3/4)/b/(-a*d+b*c)/(b*x+a)^(1 
/2)+d*EllipticE(b^(1/4)*(d*x+c)^(1/4)/(-a*d+b*c)^(1/4),I)*(-d*(b*x+a)/(-a* 
d+b*c))^(1/2)/b^(7/4)/(-a*d+b*c)^(1/4)/(b*x+a)^(1/2)-d*EllipticF(b^(1/4)*( 
d*x+c)^(1/4)/(-a*d+b*c)^(1/4),I)*(-d*(b*x+a)/(-a*d+b*c))^(1/2)/b^(7/4)/(-a 
*d+b*c)^(1/4)/(b*x+a)^(1/2)
 

3.17.39.2 Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.33 \[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{5/2}} \, dx=-\frac {2 (c+d x)^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{4},-\frac {1}{2},\frac {d (a+b x)}{-b c+a d}\right )}{3 b (a+b x)^{3/2} \left (\frac {b (c+d x)}{b c-a d}\right )^{3/4}} \]

input

Integrate[(c + d*x)^(3/4)/(a + b*x)^(5/2),x]
 

output

(-2*(c + d*x)^(3/4)*Hypergeometric2F1[-3/2, -3/4, -1/2, (d*(a + b*x))/(-(b 
*c) + a*d)])/(3*b*(a + b*x)^(3/2)*((b*(c + d*x))/(b*c - a*d))^(3/4))
 

3.17.39.3 Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.22, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {57, 61, 73, 836, 765, 762, 1390, 1388, 327}

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.

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

\(\Big \downarrow \) 57

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

\(\Big \downarrow \) 61

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 836

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

\(\displaystyle \frac {d \left (\frac {2 \left (\frac {\sqrt {b c-a d} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{b c-a d}-\frac {2 (c+d x)^{3/4}}{\sqrt {a+b x} (b c-a d)}\right )}{2 b}-\frac {2 (c+d x)^{3/4}}{3 b (a+b x)^{3/2}}\)

\(\Big \downarrow \) 1390

\(\displaystyle \frac {d \left (\frac {2 \left (\frac {\sqrt {b c-a d} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {1-\frac {b (c+d x)}{b c-a d}}}d\sqrt [4]{c+d x}}{\sqrt {b} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{b c-a d}-\frac {2 (c+d x)^{3/4}}{\sqrt {a+b x} (b c-a d)}\right )}{2 b}-\frac {2 (c+d x)^{3/4}}{3 b (a+b x)^{3/2}}\)

\(\Big \downarrow \) 1388

\(\displaystyle \frac {d \left (\frac {2 \left (\frac {\sqrt {b c-a d} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {\sqrt {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}}{\sqrt {1-\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}}}d\sqrt [4]{c+d x}}{\sqrt {b} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{b c-a d}-\frac {2 (c+d x)^{3/4}}{\sqrt {a+b x} (b c-a d)}\right )}{2 b}-\frac {2 (c+d x)^{3/4}}{3 b (a+b x)^{3/2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {d \left (\frac {2 \left (\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{b c-a d}-\frac {2 (c+d x)^{3/4}}{\sqrt {a+b x} (b c-a d)}\right )}{2 b}-\frac {2 (c+d x)^{3/4}}{3 b (a+b x)^{3/2}}\)

input

Int[(c + d*x)^(3/4)/(a + b*x)^(5/2),x]
 

output

(-2*(c + d*x)^(3/4))/(3*b*(a + b*x)^(3/2)) + (d*((-2*(c + d*x)^(3/4))/((b* 
c - a*d)*Sqrt[a + b*x]) + (2*(((b*c - a*d)^(3/4)*Sqrt[1 - (b*(c + d*x))/(b 
*c - a*d)]*EllipticE[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], 
-1])/(b^(3/4)*Sqrt[a - (b*c)/d + (b*(c + d*x))/d]) - ((b*c - a*d)^(3/4)*Sq 
rt[1 - (b*(c + d*x))/(b*c - a*d)]*EllipticF[ArcSin[(b^(1/4)*(c + d*x)^(1/4 
))/(b*c - a*d)^(1/4)], -1])/(b^(3/4)*Sqrt[a - (b*c)/d + (b*(c + d*x))/d])) 
)/(b*c - a*d)))/(2*b)
 

3.17.39.3.1 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 61

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0 
] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[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 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 765

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt 
[a + b*x^4]   Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ 
[b/a] &&  !GtQ[a, 0]
 

rule 836

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, 
Simp[-q^(-1)   Int[1/Sqrt[a + b*x^4], x], x] + Simp[1/q   Int[(1 + q*x^2)/S 
qrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]
 

rule 1388

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), 
x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, 
 c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer 
Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
 

rule 1390

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt 
[1 + c*(x^4/a)]/Sqrt[a + c*x^4]   Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], 
x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] &&  !GtQ 
[a, 0] &&  !(LtQ[a, 0] && GtQ[c, 0])
 

3.17.39.4 Maple [F]

\[\int \frac {\left (d x +c \right )^{\frac {3}{4}}}{\left (b x +a \right )^{\frac {5}{2}}}d x\]

input

int((d*x+c)^(3/4)/(b*x+a)^(5/2),x)
 

output

int((d*x+c)^(3/4)/(b*x+a)^(5/2),x)
 

3.17.39.5 Fricas [F]

\[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{5/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{4}}}{{\left (b x + a\right )}^{\frac {5}{2}}} \,d x } \]

input

integrate((d*x+c)^(3/4)/(b*x+a)^(5/2),x, algorithm="fricas")
 

output

integral(sqrt(b*x + a)*(d*x + c)^(3/4)/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x 
+ a^3), x)
 

3.17.39.6 Sympy [F]

\[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{5/2}} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{4}}}{\left (a + b x\right )^{\frac {5}{2}}}\, dx \]

input

integrate((d*x+c)**(3/4)/(b*x+a)**(5/2),x)
 

output

Integral((c + d*x)**(3/4)/(a + b*x)**(5/2), x)
 

3.17.39.7 Maxima [F]

\[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{5/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{4}}}{{\left (b x + a\right )}^{\frac {5}{2}}} \,d x } \]

input

integrate((d*x+c)^(3/4)/(b*x+a)^(5/2),x, algorithm="maxima")
 

output

integrate((d*x + c)^(3/4)/(b*x + a)^(5/2), x)
 

3.17.39.8 Giac [F]

\[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{5/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{4}}}{{\left (b x + a\right )}^{\frac {5}{2}}} \,d x } \]

input

integrate((d*x+c)^(3/4)/(b*x+a)^(5/2),x, algorithm="giac")
 

output

integrate((d*x + c)^(3/4)/(b*x + a)^(5/2), x)
 

3.17.39.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{5/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/4}}{{\left (a+b\,x\right )}^{5/2}} \,d x \]

input

int((c + d*x)^(3/4)/(a + b*x)^(5/2),x)
 

output

int((c + d*x)^(3/4)/(a + b*x)^(5/2), x)
 

3.17.39.10 Reduce [F]

\[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{5/2}} \, dx =\text {Too large to display} \]

input

int((c + d*x)**(3/4)/(sqrt(a + b*x)*(a**2 + 2*a*b*x + b**2*x**2)),x)
 

output

(4*(c + d*x)**(3/4)*sqrt(a + b*x)*c + 3*int(((c + d*x)**(3/4)*sqrt(a + b*x 
)*x)/(a**4*c*d + a**4*d**2*x - 2*a**3*b*c**2 + a**3*b*c*d*x + 3*a**3*b*d** 
2*x**2 - 6*a**2*b**2*c**2*x - 3*a**2*b**2*c*d*x**2 + 3*a**2*b**2*d**2*x**3 
 - 6*a*b**3*c**2*x**2 - 5*a*b**3*c*d*x**3 + a*b**3*d**2*x**4 - 2*b**4*c**2 
*x**3 - 2*b**4*c*d*x**4),x)*a**4*d**3 - 9*int(((c + d*x)**(3/4)*sqrt(a + b 
*x)*x)/(a**4*c*d + a**4*d**2*x - 2*a**3*b*c**2 + a**3*b*c*d*x + 3*a**3*b*d 
**2*x**2 - 6*a**2*b**2*c**2*x - 3*a**2*b**2*c*d*x**2 + 3*a**2*b**2*d**2*x* 
*3 - 6*a*b**3*c**2*x**2 - 5*a*b**3*c*d*x**3 + a*b**3*d**2*x**4 - 2*b**4*c* 
*2*x**3 - 2*b**4*c*d*x**4),x)*a**3*b*c*d**2 + 6*int(((c + d*x)**(3/4)*sqrt 
(a + b*x)*x)/(a**4*c*d + a**4*d**2*x - 2*a**3*b*c**2 + a**3*b*c*d*x + 3*a* 
*3*b*d**2*x**2 - 6*a**2*b**2*c**2*x - 3*a**2*b**2*c*d*x**2 + 3*a**2*b**2*d 
**2*x**3 - 6*a*b**3*c**2*x**2 - 5*a*b**3*c*d*x**3 + a*b**3*d**2*x**4 - 2*b 
**4*c**2*x**3 - 2*b**4*c*d*x**4),x)*a**3*b*d**3*x + 6*int(((c + d*x)**(3/4 
)*sqrt(a + b*x)*x)/(a**4*c*d + a**4*d**2*x - 2*a**3*b*c**2 + a**3*b*c*d*x 
+ 3*a**3*b*d**2*x**2 - 6*a**2*b**2*c**2*x - 3*a**2*b**2*c*d*x**2 + 3*a**2* 
b**2*d**2*x**3 - 6*a*b**3*c**2*x**2 - 5*a*b**3*c*d*x**3 + a*b**3*d**2*x**4 
 - 2*b**4*c**2*x**3 - 2*b**4*c*d*x**4),x)*a**2*b**2*c**2*d - 18*int(((c + 
d*x)**(3/4)*sqrt(a + b*x)*x)/(a**4*c*d + a**4*d**2*x - 2*a**3*b*c**2 + a** 
3*b*c*d*x + 3*a**3*b*d**2*x**2 - 6*a**2*b**2*c**2*x - 3*a**2*b**2*c*d*x**2 
 + 3*a**2*b**2*d**2*x**3 - 6*a*b**3*c**2*x**2 - 5*a*b**3*c*d*x**3 + a*b...