3.17.38 \(\int \frac {(c+d x)^{3/4}}{(a+b x)^{3/2}} \, dx\) [1638]

3.17.38.1 Optimal result

Integrand size = 19, antiderivative size = 184 \[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{3/2}} \, dx=-\frac {2 (c+d x)^{3/4}}{b \sqrt {a+b x}}+\frac {6 (b c-a d)^{3/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{7/4} \sqrt {a+b x}}-\frac {6 (b c-a d)^{3/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{7/4} \sqrt {a+b x}} \]

output

-2*(d*x+c)^(3/4)/b/(b*x+a)^(1/2)+6*(-a*d+b*c)^(3/4)*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)/(b*x+a 
)^(1/2)-6*(-a*d+b*c)^(3/4)*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)/(b*x+a)^(1/2)
 

3.17.38.2 Mathematica [C] (verified)

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

Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.39 \[ \int \frac {(c+d x)^{3/4}}{(a+b x)^{3/2}} \, dx=-\frac {2 (c+d x)^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},-\frac {1}{2},\frac {1}{2},\frac {d (a+b x)}{-b c+a d}\right )}{b \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{3/4}} \]

input

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

output

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

3.17.38.3 Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.20, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {57, 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.

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

\(\Big \downarrow \) 57

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

\(\Big \downarrow \) 73

\(\displaystyle \frac {6 \int \frac {\sqrt {c+d x}}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{b}-\frac {2 (c+d x)^{3/4}}{b \sqrt {a+b x}}\)

\(\Big \downarrow \) 836

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

\(\displaystyle \frac {6 \left (\frac {\sqrt {b c-a d} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{b}-\frac {2 (c+d x)^{3/4}}{b \sqrt {a+b x}}\)

\(\Big \downarrow \) 1390

\(\displaystyle \frac {6 \left (\frac {\sqrt {b c-a d} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {1-\frac {b (c+d x)}{b c-a d}}}d\sqrt [4]{c+d x}}{\sqrt {b} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{b}-\frac {2 (c+d x)^{3/4}}{b \sqrt {a+b x}}\)

\(\Big \downarrow \) 1388

\(\displaystyle \frac {6 \left (\frac {\sqrt {b c-a d} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {\sqrt {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}}{\sqrt {1-\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}}}d\sqrt [4]{c+d x}}{\sqrt {b} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{b}-\frac {2 (c+d x)^{3/4}}{b \sqrt {a+b x}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {6 \left (\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{b}-\frac {2 (c+d x)^{3/4}}{b \sqrt {a+b x}}\)

input

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

output

(-2*(c + d*x)^(3/4))/(b*Sqrt[a + b*x]) + (6*(((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)*Sqrt[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
 

3.17.38.3.1 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 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])
 

3.17.38.4 Maple [F]

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

input

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

output

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

3.17.38.5 Fricas [F]

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

input

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

output

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

3.17.38.6 Sympy [F]

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

input

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

output

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

3.17.38.7 Maxima [F]

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

input

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

output

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

3.17.38.8 Giac [F]

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

input

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

output

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

3.17.38.9 Mupad [F(-1)]

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

input

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

output

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