\(\int \frac {(c+d x)^{3/4}}{\sqrt {a+b x}} \, dx\) [439]

Optimal result

Integrand size = 19, antiderivative size = 147 \[ \int \frac {(c+d x)^{3/4}}{\sqrt {a+b x}} \, dx=\frac {12 (b c-a d) \sqrt {a+b x}}{5 b^2 \sqrt [4]{c+d x}}+\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 b}-\frac {12 (b c-a d)^{3/2} \sqrt [4]{\frac {b (c+d x)}{b c-a d}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )\right |2\right )}{5 b^2 \sqrt {d} \sqrt [4]{c+d x}} \] Output:

12/5*(-a*d+b*c)*(b*x+a)^(1/2)/b^2/(d*x+c)^(1/4)+4/5*(b*x+a)^(1/2)*(d*x+c)^ 
(3/4)/b-12/5*(-a*d+b*c)^(3/2)*(b*(d*x+c)/(-a*d+b*c))^(1/4)*EllipticE(sin(1 
/2*arctan(d^(1/2)*(b*x+a)^(1/2)/(-a*d+b*c)^(1/2))),2^(1/2))/b^2/d^(1/2)/(d 
*x+c)^(1/4)
 

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 = 0.48 \[ \int \frac {(c+d x)^{3/4}}{\sqrt {a+b x}} \, dx=\frac {2 \sqrt {a+b x} (c+d x)^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {1}{2},\frac {3}{2},\frac {d (a+b x)}{-b c+a d}\right )}{b \left (\frac {b (c+d x)}{b c-a d}\right )^{3/4}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.61, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {60, 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.

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 73

\(\displaystyle \frac {12 (b c-a d) \int \frac {\sqrt {c+d x}}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{5 b d}+\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 b}\)

\(\Big \downarrow \) 836

\(\displaystyle \frac {12 (b c-a d) \left (\frac {\sqrt {b c-a d} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}-\frac {\sqrt {b c-a d} \int \frac {1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}\right )}{5 b d}+\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 b}\)

\(\Big \downarrow \) 765

\(\displaystyle \frac {12 (b c-a d) \left (\frac {\sqrt {b c-a d} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}-\frac {\sqrt {b c-a d} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {1}{\sqrt {1-\frac {b (c+d x)}{b c-a d}}}d\sqrt [4]{c+d x}}{\sqrt {b} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{5 b d}+\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 b}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {12 (b c-a d) \left (\frac {\sqrt {b c-a d} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{5 b d}+\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 b}\)

\(\Big \downarrow \) 1390

\(\displaystyle \frac {12 (b c-a d) \left (\frac {\sqrt {b c-a d} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {1-\frac {b (c+d x)}{b c-a d}}}d\sqrt [4]{c+d x}}{\sqrt {b} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{5 b d}+\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 b}\)

\(\Big \downarrow \) 1388

\(\displaystyle \frac {12 (b c-a d) \left (\frac {\sqrt {b c-a d} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {\sqrt {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}}{\sqrt {1-\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}}}d\sqrt [4]{c+d x}}{\sqrt {b} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{5 b d}+\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 b}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {12 (b c-a d) \left (\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{5 b d}+\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 b}\)

Input:

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

Output:

(4*Sqrt[a + b*x]*(c + d*x)^(3/4))/(5*b) + (12*(b*c - a*d)*(((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[A 
rcSin[(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])))/(5*b*d)
 

Defintions of rubi rules used

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 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])
 

Maple [F]

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

Input:

int((d*x+c)^(3/4)/(b*x+a)^(1/2),x)
 

Output:

int((d*x+c)^(3/4)/(b*x+a)^(1/2),x)
 

Fricas [F]

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

integrate((d*x+c)^(3/4)/(b*x+a)^(1/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate((d*x+c)**(3/4)/(b*x+a)**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate((d*x+c)^(3/4)/(b*x+a)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((d*x+c)^(3/4)/(b*x+a)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((c + d*x)^(3/4)/(a + b*x)^(1/2),x)
 

Output:

int((c + d*x)^(3/4)/(a + b*x)^(1/2), x)
 

Reduce [F]

\[ \int \frac {(c+d x)^{3/4}}{\sqrt {a+b x}} \, dx=\frac {4 \left (d x +c \right )^{\frac {3}{4}} \sqrt {b x +a}\, c +9 \left (\int \frac {\left (d x +c \right )^{\frac {3}{4}} \sqrt {b x +a}\, x}{3 a b \,d^{2} x^{2}+2 b^{2} c d \,x^{2}+3 a^{2} d^{2} x +5 a b c d x +2 b^{2} c^{2} x +3 a^{2} c d +2 a b \,c^{2}}d x \right ) a^{2} d^{3}-3 \left (\int \frac {\left (d x +c \right )^{\frac {3}{4}} \sqrt {b x +a}\, x}{3 a b \,d^{2} x^{2}+2 b^{2} c d \,x^{2}+3 a^{2} d^{2} x +5 a b c d x +2 b^{2} c^{2} x +3 a^{2} c d +2 a b \,c^{2}}d x \right ) a b c \,d^{2}-6 \left (\int \frac {\left (d x +c \right )^{\frac {3}{4}} \sqrt {b x +a}\, x}{3 a b \,d^{2} x^{2}+2 b^{2} c d \,x^{2}+3 a^{2} d^{2} x +5 a b c d x +2 b^{2} c^{2} x +3 a^{2} c d +2 a b \,c^{2}}d x \right ) b^{2} c^{2} d}{3 a d +2 b c} \] Input:

int((d*x+c)^(3/4)/(b*x+a)^(1/2),x)
 

Output:

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