\(\int x^{3/2} (a+b x)^{3/4} \, dx\) [648]

Optimal result

Integrand size = 15, antiderivative size = 145 \[ \int x^{3/2} (a+b x)^{3/4} \, dx=\frac {8 a^3 \sqrt {x}}{65 b^2 \sqrt [4]{a+b x}}-\frac {4 a^2 x^{3/2}}{195 b \sqrt [4]{a+b x}}+\frac {4 a x^{5/2}}{39 \sqrt [4]{a+b x}}+\frac {4}{13} x^{5/2} (a+b x)^{3/4}-\frac {16 a^{7/2} \sqrt [4]{\frac {a+b x}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right |2\right )}{65 b^{5/2} \sqrt [4]{a+b x}} \] Output:

8/65*a^3*x^(1/2)/b^2/(b*x+a)^(1/4)-4/195*a^2*x^(3/2)/b/(b*x+a)^(1/4)+4/39* 
a*x^(5/2)/(b*x+a)^(1/4)+4/13*x^(5/2)*(b*x+a)^(3/4)-16/65*a^(7/2)*((b*x+a)/ 
a)^(1/4)*EllipticE(sin(1/2*arctan(b^(1/2)*x^(1/2)/a^(1/2))),2^(1/2))/b^(5/ 
2)/(b*x+a)^(1/4)
 

Mathematica [C] (verified)

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

Time = 10.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.32 \[ \int x^{3/2} (a+b x)^{3/4} \, dx=\frac {2 x^{5/2} (a+b x)^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {5}{2},\frac {7}{2},-\frac {b x}{a}\right )}{5 \left (1+\frac {b x}{a}\right )^{3/4}} \] Input:

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

Output:

(2*x^(5/2)*(a + b*x)^(3/4)*Hypergeometric2F1[-3/4, 5/2, 7/2, -((b*x)/a)])/ 
(5*(1 + (b*x)/a)^(3/4))
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.42, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.733, Rules used = {60, 60, 60, 73, 836, 27, 765, 762, 1390, 1389, 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.

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

\(\Big \downarrow \) 60

\(\displaystyle \frac {3}{13} a \int \frac {x^{3/2}}{\sqrt [4]{a+b x}}dx+\frac {4}{13} x^{5/2} (a+b x)^{3/4}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {3}{13} a \left (\frac {4 x^{3/2} (a+b x)^{3/4}}{9 b}-\frac {2 a \int \frac {\sqrt {x}}{\sqrt [4]{a+b x}}dx}{3 b}\right )+\frac {4}{13} x^{5/2} (a+b x)^{3/4}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {3}{13} a \left (\frac {4 x^{3/2} (a+b x)^{3/4}}{9 b}-\frac {2 a \left (\frac {4 \sqrt {x} (a+b x)^{3/4}}{5 b}-\frac {2 a \int \frac {1}{\sqrt {x} \sqrt [4]{a+b x}}dx}{5 b}\right )}{3 b}\right )+\frac {4}{13} x^{5/2} (a+b x)^{3/4}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {3}{13} a \left (\frac {4 x^{3/2} (a+b x)^{3/4}}{9 b}-\frac {2 a \left (\frac {4 \sqrt {x} (a+b x)^{3/4}}{5 b}-\frac {8 a \int \frac {\sqrt {a+b x}}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}d\sqrt [4]{a+b x}}{5 b^2}\right )}{3 b}\right )+\frac {4}{13} x^{5/2} (a+b x)^{3/4}\)

\(\Big \downarrow \) 836

\(\displaystyle \frac {3}{13} a \left (\frac {4 x^{3/2} (a+b x)^{3/4}}{9 b}-\frac {2 a \left (\frac {4 \sqrt {x} (a+b x)^{3/4}}{5 b}-\frac {8 a \left (\sqrt {a} \int \frac {\sqrt {a}+\sqrt {a+b x}}{\sqrt {a} \sqrt {\frac {a+b x}{b}-\frac {a}{b}}}d\sqrt [4]{a+b x}-\sqrt {a} \int \frac {1}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}d\sqrt [4]{a+b x}\right )}{5 b^2}\right )}{3 b}\right )+\frac {4}{13} x^{5/2} (a+b x)^{3/4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{13} a \left (\frac {4 x^{3/2} (a+b x)^{3/4}}{9 b}-\frac {2 a \left (\frac {4 \sqrt {x} (a+b x)^{3/4}}{5 b}-\frac {8 a \left (\int \frac {\sqrt {a}+\sqrt {a+b x}}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}d\sqrt [4]{a+b x}-\sqrt {a} \int \frac {1}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}d\sqrt [4]{a+b x}\right )}{5 b^2}\right )}{3 b}\right )+\frac {4}{13} x^{5/2} (a+b x)^{3/4}\)

\(\Big \downarrow \) 765

\(\displaystyle \frac {3}{13} a \left (\frac {4 x^{3/2} (a+b x)^{3/4}}{9 b}-\frac {2 a \left (\frac {4 \sqrt {x} (a+b x)^{3/4}}{5 b}-\frac {8 a \left (\int \frac {\sqrt {a}+\sqrt {a+b x}}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}d\sqrt [4]{a+b x}-\frac {\sqrt {a} \sqrt {1-\frac {a+b x}{a}} \int \frac {1}{\sqrt {1-\frac {a+b x}{a}}}d\sqrt [4]{a+b x}}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}\right )}{5 b^2}\right )}{3 b}\right )+\frac {4}{13} x^{5/2} (a+b x)^{3/4}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {3}{13} a \left (\frac {4 x^{3/2} (a+b x)^{3/4}}{9 b}-\frac {2 a \left (\frac {4 \sqrt {x} (a+b x)^{3/4}}{5 b}-\frac {8 a \left (\int \frac {\sqrt {a}+\sqrt {a+b x}}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}d\sqrt [4]{a+b x}-\frac {a^{3/4} \sqrt {1-\frac {a+b x}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a+b x}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}\right )}{5 b^2}\right )}{3 b}\right )+\frac {4}{13} x^{5/2} (a+b x)^{3/4}\)

\(\Big \downarrow \) 1390

\(\displaystyle \frac {3}{13} a \left (\frac {4 x^{3/2} (a+b x)^{3/4}}{9 b}-\frac {2 a \left (\frac {4 \sqrt {x} (a+b x)^{3/4}}{5 b}-\frac {8 a \left (\frac {\sqrt {1-\frac {a+b x}{a}} \int \frac {\sqrt {a}+\sqrt {a+b x}}{\sqrt {1-\frac {a+b x}{a}}}d\sqrt [4]{a+b x}}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}-\frac {a^{3/4} \sqrt {1-\frac {a+b x}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a+b x}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}\right )}{5 b^2}\right )}{3 b}\right )+\frac {4}{13} x^{5/2} (a+b x)^{3/4}\)

\(\Big \downarrow \) 1389

\(\displaystyle \frac {3}{13} a \left (\frac {4 x^{3/2} (a+b x)^{3/4}}{9 b}-\frac {2 a \left (\frac {4 \sqrt {x} (a+b x)^{3/4}}{5 b}-\frac {8 a \left (\frac {\sqrt {a} \sqrt {1-\frac {a+b x}{a}} \int \frac {\sqrt {\frac {\sqrt {a+b x}}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {a+b x}}{\sqrt {a}}}}d\sqrt [4]{a+b x}}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}-\frac {a^{3/4} \sqrt {1-\frac {a+b x}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a+b x}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}\right )}{5 b^2}\right )}{3 b}\right )+\frac {4}{13} x^{5/2} (a+b x)^{3/4}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {3}{13} a \left (\frac {4 x^{3/2} (a+b x)^{3/4}}{9 b}-\frac {2 a \left (\frac {4 \sqrt {x} (a+b x)^{3/4}}{5 b}-\frac {8 a \left (\frac {a^{3/4} \sqrt {1-\frac {a+b x}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{a+b x}}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}-\frac {a^{3/4} \sqrt {1-\frac {a+b x}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a+b x}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}\right )}{5 b^2}\right )}{3 b}\right )+\frac {4}{13} x^{5/2} (a+b x)^{3/4}\)

Input:

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

Output:

(4*x^(5/2)*(a + b*x)^(3/4))/13 + (3*a*((4*x^(3/2)*(a + b*x)^(3/4))/(9*b) - 
 (2*a*((4*Sqrt[x]*(a + b*x)^(3/4))/(5*b) - (8*a*((a^(3/4)*Sqrt[1 - (a + b* 
x)/a]*EllipticE[ArcSin[(a + b*x)^(1/4)/a^(1/4)], -1])/Sqrt[-(a/b) + (a + b 
*x)/b] - (a^(3/4)*Sqrt[1 - (a + b*x)/a]*EllipticF[ArcSin[(a + b*x)^(1/4)/a 
^(1/4)], -1])/Sqrt[-(a/b) + (a + b*x)/b]))/(5*b^2)))/(3*b)))/13
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 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 1389

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.55 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.21 \[ \int x^{3/2} (a+b x)^{3/4} \, dx=\frac {2 a^{\frac {3}{4}} x^{\frac {5}{2}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{5} \] Input:

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

Output:

2*a**(3/4)*x**(5/2)*hyper((-3/4, 5/2), (7/2,), b*x*exp_polar(I*pi)/a)/5
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^{3/2} (a+b x)^{3/4} \, dx=\frac {-\frac {8 \sqrt {x}\, \left (b x +a \right )^{\frac {3}{4}} a^{2}}{65}+\frac {4 \sqrt {x}\, \left (b x +a \right )^{\frac {3}{4}} a b x}{39}+\frac {4 \sqrt {x}\, \left (b x +a \right )^{\frac {3}{4}} b^{2} x^{2}}{13}+\frac {4 \left (\int \frac {\sqrt {x}\, \left (b x +a \right )^{\frac {3}{4}}}{b \,x^{2}+a x}d x \right ) a^{3}}{65}}{b^{2}} \] Input:

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

Output:

(4*( - 6*sqrt(x)*(a + b*x)**(3/4)*a**2 + 5*sqrt(x)*(a + b*x)**(3/4)*a*b*x 
+ 15*sqrt(x)*(a + b*x)**(3/4)*b**2*x**2 + 3*int((sqrt(x)*(a + b*x)**(3/4)) 
/(a*x + b*x**2),x)*a**3))/(195*b**2)