\(\int \frac {x^{7/4}}{(a+b x)^{7/4}} \, dx\) [742]

Optimal result

Integrand size = 15, antiderivative size = 103 \[ \int \frac {x^{7/4}}{(a+b x)^{7/4}} \, dx=\frac {4 a x^{3/4}}{3 b^2 (a+b x)^{3/4}}+\frac {x^{3/4} \sqrt [4]{a+b x}}{b^2}+\frac {7 a \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{x}}{\sqrt [4]{a+b x}}\right )}{2 b^{11/4}}-\frac {7 a \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt [4]{x}}{\sqrt [4]{a+b x}}\right )}{2 b^{11/4}} \] Output:

4/3*a*x^(3/4)/b^2/(b*x+a)^(3/4)+x^(3/4)*(b*x+a)^(1/4)/b^2+7/2*a*arctan(b^( 
1/4)*x^(1/4)/(b*x+a)^(1/4))/b^(11/4)-7/2*a*arctanh(b^(1/4)*x^(1/4)/(b*x+a) 
^(1/4))/b^(11/4)
 

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.84 \[ \int \frac {x^{7/4}}{(a+b x)^{7/4}} \, dx=\frac {\frac {2 b^{3/4} x^{3/4} (7 a+3 b x)}{(a+b x)^{3/4}}+21 a \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{x}}{\sqrt [4]{a+b x}}\right )-21 a \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt [4]{x}}{\sqrt [4]{a+b x}}\right )}{6 b^{11/4}} \] Input:

Integrate[x^(7/4)/(a + b*x)^(7/4),x]
 

Output:

((2*b^(3/4)*x^(3/4)*(7*a + 3*b*x))/(a + b*x)^(3/4) + 21*a*ArcTan[(b^(1/4)* 
x^(1/4))/(a + b*x)^(1/4)] - 21*a*ArcTanh[(b^(1/4)*x^(1/4))/(a + b*x)^(1/4) 
])/(6*b^(11/4))
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {57, 60, 73, 854, 827, 216, 219}

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.

\(\displaystyle \int \frac {x^{7/4}}{(a+b x)^{7/4}} \, dx\)

\(\Big \downarrow \) 57

\(\displaystyle \frac {7 \int \frac {x^{3/4}}{(a+b x)^{3/4}}dx}{3 b}-\frac {4 x^{7/4}}{3 b (a+b x)^{3/4}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {7 \left (\frac {x^{3/4} \sqrt [4]{a+b x}}{b}-\frac {3 a \int \frac {1}{\sqrt [4]{x} (a+b x)^{3/4}}dx}{4 b}\right )}{3 b}-\frac {4 x^{7/4}}{3 b (a+b x)^{3/4}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {7 \left (\frac {x^{3/4} \sqrt [4]{a+b x}}{b}-\frac {3 a \int \frac {\sqrt {x}}{(a+b x)^{3/4}}d\sqrt [4]{x}}{b}\right )}{3 b}-\frac {4 x^{7/4}}{3 b (a+b x)^{3/4}}\)

\(\Big \downarrow \) 854

\(\displaystyle \frac {7 \left (\frac {x^{3/4} \sqrt [4]{a+b x}}{b}-\frac {3 a \int \frac {\sqrt {x}}{1-b x}d\frac {\sqrt [4]{x}}{\sqrt [4]{a+b x}}}{b}\right )}{3 b}-\frac {4 x^{7/4}}{3 b (a+b x)^{3/4}}\)

\(\Big \downarrow \) 827

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {7 \left (\frac {x^{3/4} \sqrt [4]{a+b x}}{b}-\frac {3 a \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt [4]{x}}{\sqrt [4]{a+b x}}\right )}{2 b^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{x}}{\sqrt [4]{a+b x}}\right )}{2 b^{3/4}}\right )}{b}\right )}{3 b}-\frac {4 x^{7/4}}{3 b (a+b x)^{3/4}}\)

Input:

Int[x^(7/4)/(a + b*x)^(7/4),x]
 

Output:

(-4*x^(7/4))/(3*b*(a + b*x)^(3/4)) + (7*((x^(3/4)*(a + b*x)^(1/4))/b - (3* 
a*(-1/2*ArcTan[(b^(1/4)*x^(1/4))/(a + b*x)^(1/4)]/b^(3/4) + ArcTanh[(b^(1/ 
4)*x^(1/4))/(a + b*x)^(1/4)]/(2*b^(3/4))))/b))/(3*b)
 

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 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 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 827

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 
 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b)   Int[1/(r + s*x^2), x], 
x] - Simp[s/(2*b)   Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ 
[a/b, 0]
 

rule 854

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 
 1)/n)   Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n 
)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 
2^(-1)] && IntegersQ[m, p + (m + 1)/n]
 

Maple [F]

\[\int \frac {x^{\frac {7}{4}}}{\left (b x +a \right )^{\frac {7}{4}}}d x\]

Input:

int(x^(7/4)/(b*x+a)^(7/4),x)
 

Output:

int(x^(7/4)/(b*x+a)^(7/4),x)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.57 \[ \int \frac {x^{7/4}}{(a+b x)^{7/4}} \, dx=-\frac {21 \, {\left (b^{3} x + a b^{2}\right )} \left (\frac {a^{4}}{b^{11}}\right )^{\frac {1}{4}} \log \left (\frac {7 \, {\left (b^{3} x \left (\frac {a^{4}}{b^{11}}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}} a x^{\frac {3}{4}}\right )}}{x}\right ) - 21 \, {\left (b^{3} x + a b^{2}\right )} \left (\frac {a^{4}}{b^{11}}\right )^{\frac {1}{4}} \log \left (-\frac {7 \, {\left (b^{3} x \left (\frac {a^{4}}{b^{11}}\right )^{\frac {1}{4}} - {\left (b x + a\right )}^{\frac {1}{4}} a x^{\frac {3}{4}}\right )}}{x}\right ) + 21 \, {\left (-i \, b^{3} x - i \, a b^{2}\right )} \left (\frac {a^{4}}{b^{11}}\right )^{\frac {1}{4}} \log \left (-\frac {7 \, {\left (i \, b^{3} x \left (\frac {a^{4}}{b^{11}}\right )^{\frac {1}{4}} - {\left (b x + a\right )}^{\frac {1}{4}} a x^{\frac {3}{4}}\right )}}{x}\right ) + 21 \, {\left (i \, b^{3} x + i \, a b^{2}\right )} \left (\frac {a^{4}}{b^{11}}\right )^{\frac {1}{4}} \log \left (-\frac {7 \, {\left (-i \, b^{3} x \left (\frac {a^{4}}{b^{11}}\right )^{\frac {1}{4}} - {\left (b x + a\right )}^{\frac {1}{4}} a x^{\frac {3}{4}}\right )}}{x}\right ) - 4 \, {\left (3 \, b x + 7 \, a\right )} {\left (b x + a\right )}^{\frac {1}{4}} x^{\frac {3}{4}}}{12 \, {\left (b^{3} x + a b^{2}\right )}} \] Input:

integrate(x^(7/4)/(b*x+a)^(7/4),x, algorithm="fricas")
 

Output:

-1/12*(21*(b^3*x + a*b^2)*(a^4/b^11)^(1/4)*log(7*(b^3*x*(a^4/b^11)^(1/4) + 
 (b*x + a)^(1/4)*a*x^(3/4))/x) - 21*(b^3*x + a*b^2)*(a^4/b^11)^(1/4)*log(- 
7*(b^3*x*(a^4/b^11)^(1/4) - (b*x + a)^(1/4)*a*x^(3/4))/x) + 21*(-I*b^3*x - 
 I*a*b^2)*(a^4/b^11)^(1/4)*log(-7*(I*b^3*x*(a^4/b^11)^(1/4) - (b*x + a)^(1 
/4)*a*x^(3/4))/x) + 21*(I*b^3*x + I*a*b^2)*(a^4/b^11)^(1/4)*log(-7*(-I*b^3 
*x*(a^4/b^11)^(1/4) - (b*x + a)^(1/4)*a*x^(3/4))/x) - 4*(3*b*x + 7*a)*(b*x 
 + a)^(1/4)*x^(3/4))/(b^3*x + a*b^2)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 7.94 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.35 \[ \int \frac {x^{7/4}}{(a+b x)^{7/4}} \, dx=\frac {x^{\frac {11}{4}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{4}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{a^{\frac {7}{4}} \Gamma \left (\frac {15}{4}\right )} \] Input:

integrate(x**(7/4)/(b*x+a)**(7/4),x)
 

Output:

x**(11/4)*gamma(11/4)*hyper((7/4, 11/4), (15/4,), b*x*exp_polar(I*pi)/a)/( 
a**(7/4)*gamma(15/4))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.17 \[ \int \frac {x^{7/4}}{(a+b x)^{7/4}} \, dx=\frac {4 \, a b - \frac {7 \, {\left (b x + a\right )} a}{x}}{3 \, {\left (\frac {{\left (b x + a\right )}^{\frac {3}{4}} b^{3}}{x^{\frac {3}{4}}} - \frac {{\left (b x + a\right )}^{\frac {7}{4}} b^{2}}{x^{\frac {7}{4}}}\right )}} - \frac {7 \, {\left (\frac {2 \, a \arctan \left (\frac {{\left (b x + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} x^{\frac {1}{4}}}\right )}{b^{\frac {3}{4}}} - \frac {a \log \left (-\frac {b^{\frac {1}{4}} - \frac {{\left (b x + a\right )}^{\frac {1}{4}}}{x^{\frac {1}{4}}}}{b^{\frac {1}{4}} + \frac {{\left (b x + a\right )}^{\frac {1}{4}}}{x^{\frac {1}{4}}}}\right )}{b^{\frac {3}{4}}}\right )}}{4 \, b^{2}} \] Input:

integrate(x^(7/4)/(b*x+a)^(7/4),x, algorithm="maxima")
 

Output:

1/3*(4*a*b - 7*(b*x + a)*a/x)/((b*x + a)^(3/4)*b^3/x^(3/4) - (b*x + a)^(7/ 
4)*b^2/x^(7/4)) - 7/4*(2*a*arctan((b*x + a)^(1/4)/(b^(1/4)*x^(1/4)))/b^(3/ 
4) - a*log(-(b^(1/4) - (b*x + a)^(1/4)/x^(1/4))/(b^(1/4) + (b*x + a)^(1/4) 
/x^(1/4)))/b^(3/4))/b^2
 

Giac [F]

\[ \int \frac {x^{7/4}}{(a+b x)^{7/4}} \, dx=\int { \frac {x^{\frac {7}{4}}}{{\left (b x + a\right )}^{\frac {7}{4}}} \,d x } \] Input:

integrate(x^(7/4)/(b*x+a)^(7/4),x, algorithm="giac")
 

Output:

integrate(x^(7/4)/(b*x + a)^(7/4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{7/4}}{(a+b x)^{7/4}} \, dx=\int \frac {x^{7/4}}{{\left (a+b\,x\right )}^{7/4}} \,d x \] Input:

int(x^(7/4)/(a + b*x)^(7/4),x)
 

Output:

int(x^(7/4)/(a + b*x)^(7/4), x)
 

Reduce [F]

\[ \int \frac {x^{7/4}}{(a+b x)^{7/4}} \, dx=\int \frac {x^{\frac {7}{4}}}{\left (b x +a \right )^{\frac {3}{4}} a +\left (b x +a \right )^{\frac {3}{4}} b x}d x \] Input:

int(x^(7/4)/(b*x+a)^(7/4),x)
 

Output:

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