3.17.27 \(\int \frac {1}{(a+b x)^{7/3} (c+d x)^{4/3}} \, dx\) [1627]

3.17.27.1 Optimal result

Integrand size = 19, antiderivative size = 1370 \[ \int \frac {1}{(a+b x)^{7/3} (c+d x)^{4/3}} \, dx=-\frac {3}{4 (b c-a d) (a+b x)^{4/3} \sqrt [3]{c+d x}}+\frac {15 d}{4 (b c-a d)^2 \sqrt [3]{a+b x} \sqrt [3]{c+d x}}+\frac {15 d^2 (a+b x)^{2/3}}{2 (b c-a d)^3 \sqrt [3]{c+d x}}-\frac {15 \sqrt [3]{b} d^{4/3} \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \sqrt {(a d+b (c+2 d x))^2}}{2 \sqrt [3]{2} (b c-a d)^3 \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}+\frac {15 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{b} d^{4/3} \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right )|-7-4 \sqrt {3}\right )}{4 \sqrt [3]{2} (b c-a d)^{7/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} \sqrt {(a d+b (c+2 d x))^2}}-\frac {5\ 3^{3/4} \sqrt [3]{b} d^{4/3} \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right ),-7-4 \sqrt {3}\right )}{2^{5/6} (b c-a d)^{7/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} \sqrt {(a d+b (c+2 d x))^2}} \]

output

-3/4/(-a*d+b*c)/(b*x+a)^(4/3)/(d*x+c)^(1/3)+15/4*d/(-a*d+b*c)^2/(b*x+a)^(1 
/3)/(d*x+c)^(1/3)+15/2*d^2*(b*x+a)^(2/3)/(-a*d+b*c)^3/(d*x+c)^(1/3)-15/4*b 
^(1/3)*d^(4/3)*((b*x+a)*(d*x+c))^(1/3)*((2*b*d*x+a*d+b*c)^2)^(1/2)*((a*d+b 
*(2*d*x+c))^2)^(1/2)*2^(2/3)/(-a*d+b*c)^3/(b*x+a)^(1/3)/(d*x+c)^(1/3)/(2*b 
*d*x+a*d+b*c)/(2^(2/3)*b^(1/3)*d^(1/3)*((b*x+a)*(d*x+c))^(1/3)+(-a*d+b*c)^ 
(2/3)*(1+3^(1/2)))-5/2*3^(3/4)*b^(1/3)*d^(4/3)*((b*x+a)*(d*x+c))^(1/3)*((- 
a*d+b*c)^(2/3)+2^(2/3)*b^(1/3)*d^(1/3)*((b*x+a)*(d*x+c))^(1/3))*EllipticF( 
(2^(2/3)*b^(1/3)*d^(1/3)*((b*x+a)*(d*x+c))^(1/3)+(-a*d+b*c)^(2/3)*(1-3^(1/ 
2)))/(2^(2/3)*b^(1/3)*d^(1/3)*((b*x+a)*(d*x+c))^(1/3)+(-a*d+b*c)^(2/3)*(1+ 
3^(1/2))),I*3^(1/2)+2*I)*((2*b*d*x+a*d+b*c)^2)^(1/2)*(((-a*d+b*c)^(4/3)-2^ 
(2/3)*b^(1/3)*d^(1/3)*(-a*d+b*c)^(2/3)*((b*x+a)*(d*x+c))^(1/3)+2*2^(1/3)*b 
^(2/3)*d^(2/3)*((b*x+a)*(d*x+c))^(2/3))/(2^(2/3)*b^(1/3)*d^(1/3)*((b*x+a)* 
(d*x+c))^(1/3)+(-a*d+b*c)^(2/3)*(1+3^(1/2)))^2)^(1/2)*2^(1/6)/(-a*d+b*c)^( 
7/3)/(b*x+a)^(1/3)/(d*x+c)^(1/3)/(2*b*d*x+a*d+b*c)/((a*d+b*(2*d*x+c))^2)^( 
1/2)/((-a*d+b*c)^(2/3)*((-a*d+b*c)^(2/3)+2^(2/3)*b^(1/3)*d^(1/3)*((b*x+a)* 
(d*x+c))^(1/3))/(2^(2/3)*b^(1/3)*d^(1/3)*((b*x+a)*(d*x+c))^(1/3)+(-a*d+b*c 
)^(2/3)*(1+3^(1/2)))^2)^(1/2)+15/8*3^(1/4)*b^(1/3)*d^(4/3)*((b*x+a)*(d*x+c 
))^(1/3)*((-a*d+b*c)^(2/3)+2^(2/3)*b^(1/3)*d^(1/3)*((b*x+a)*(d*x+c))^(1/3) 
)*EllipticE((2^(2/3)*b^(1/3)*d^(1/3)*((b*x+a)*(d*x+c))^(1/3)+(-a*d+b*c)^(2 
/3)*(1-3^(1/2)))/(2^(2/3)*b^(1/3)*d^(1/3)*((b*x+a)*(d*x+c))^(1/3)+(-a*d...
 

3.17.27.2 Mathematica [C] (verified)

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

Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.05 \[ \int \frac {1}{(a+b x)^{7/3} (c+d x)^{4/3}} \, dx=-\frac {3 \left (\frac {b (c+d x)}{b c-a d}\right )^{4/3} \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},\frac {4}{3},-\frac {1}{3},\frac {d (a+b x)}{-b c+a d}\right )}{4 b (a+b x)^{4/3} (c+d x)^{4/3}} \]

input

Integrate[1/((a + b*x)^(7/3)*(c + d*x)^(4/3)),x]
 

output

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

3.17.27.3 Rubi [C] (verified)

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

Time = 0.17 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.06, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {80, 79}

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.

\(\displaystyle \int \frac {1}{(a+b x)^{7/3} (c+d x)^{4/3}} \, dx\)

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

\(\displaystyle -\frac {3 \sqrt [3]{\frac {b (c+d x)}{b c-a d}} \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},\frac {4}{3},-\frac {1}{3},-\frac {d (a+b x)}{b c-a d}\right )}{4 (a+b x)^{4/3} \sqrt [3]{c+d x} (b c-a d)}\)

input

Int[1/((a + b*x)^(7/3)*(c + d*x)^(4/3)),x]
 

output

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

3.17.27.3.1 Defintions of rubi rules used

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 80

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c 
 + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) 
^FracPart[n])   Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) 
), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integ 
erQ[n] && (RationalQ[m] ||  !SimplerQ[n + 1, m + 1])
 

3.17.27.4 Maple [F]

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

input

int(1/(b*x+a)^(7/3)/(d*x+c)^(4/3),x)
 

output

int(1/(b*x+a)^(7/3)/(d*x+c)^(4/3),x)
 

3.17.27.5 Fricas [F]

\[ \int \frac {1}{(a+b x)^{7/3} (c+d x)^{4/3}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {7}{3}} {\left (d x + c\right )}^{\frac {4}{3}}} \,d x } \]

input

integrate(1/(b*x+a)^(7/3)/(d*x+c)^(4/3),x, algorithm="fricas")
 

output

integral((b*x + a)^(2/3)*(d*x + c)^(2/3)/(b^3*d^2*x^5 + a^3*c^2 + (2*b^3*c 
*d + 3*a*b^2*d^2)*x^4 + (b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*x^3 + (3*a*b 
^2*c^2 + 6*a^2*b*c*d + a^3*d^2)*x^2 + (3*a^2*b*c^2 + 2*a^3*c*d)*x), x)
 

3.17.27.6 Sympy [F]

\[ \int \frac {1}{(a+b x)^{7/3} (c+d x)^{4/3}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {7}{3}} \left (c + d x\right )^{\frac {4}{3}}}\, dx \]

input

integrate(1/(b*x+a)**(7/3)/(d*x+c)**(4/3),x)
 

output

Integral(1/((a + b*x)**(7/3)*(c + d*x)**(4/3)), x)
 

3.17.27.7 Maxima [F]

\[ \int \frac {1}{(a+b x)^{7/3} (c+d x)^{4/3}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {7}{3}} {\left (d x + c\right )}^{\frac {4}{3}}} \,d x } \]

input

integrate(1/(b*x+a)^(7/3)/(d*x+c)^(4/3),x, algorithm="maxima")
 

output

integrate(1/((b*x + a)^(7/3)*(d*x + c)^(4/3)), x)
 

3.17.27.8 Giac [F]

\[ \int \frac {1}{(a+b x)^{7/3} (c+d x)^{4/3}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {7}{3}} {\left (d x + c\right )}^{\frac {4}{3}}} \,d x } \]

input

integrate(1/(b*x+a)^(7/3)/(d*x+c)^(4/3),x, algorithm="giac")
 

output

integrate(1/((b*x + a)^(7/3)*(d*x + c)^(4/3)), x)
 

3.17.27.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x)^{7/3} (c+d x)^{4/3}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{7/3}\,{\left (c+d\,x\right )}^{4/3}} \,d x \]

input

int(1/((a + b*x)^(7/3)*(c + d*x)^(4/3)),x)
 

output

int(1/((a + b*x)^(7/3)*(c + d*x)^(4/3)), x)