\(\int \frac {(a+b x^3)^{14/3}}{(c+d x^3)^3} \, dx\) [161]

Optimal result

Integrand size = 21, antiderivative size = 541 \[ \int \frac {\left (a+b x^3\right )^{14/3}}{\left (c+d x^3\right )^3} \, dx=-\frac {b (2 b c-a d) \left (18 b^2 c^2-18 a b c d-5 a^2 d^2\right ) x \left (a+b x^3\right )^{2/3}}{18 c^2 d^4}+\frac {b \left (18 b^2 c^2-10 a b c d-5 a^2 d^2\right ) x \left (a+b x^3\right )^{5/3}}{18 c^2 d^3}-\frac {(b c-a d) x \left (a+b x^3\right )^{11/3}}{6 c d \left (c+d x^3\right )^2}-\frac {(b c-a d) (12 b c+5 a d) x \left (a+b x^3\right )^{8/3}}{18 c^2 d^2 \left (c+d x^3\right )}+\frac {b^{8/3} \left (54 b^2 c^2-126 a b c d+77 a^2 d^2\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} d^5}-\frac {(b c-a d)^{8/3} \left (54 b^2 c^2+18 a b c d+5 a^2 d^2\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} c^{8/3} d^5}-\frac {(b c-a d)^{8/3} \left (54 b^2 c^2+18 a b c d+5 a^2 d^2\right ) \log \left (c+d x^3\right )}{54 c^{8/3} d^5}+\frac {(b c-a d)^{8/3} \left (54 b^2 c^2+18 a b c d+5 a^2 d^2\right ) \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{18 c^{8/3} d^5}-\frac {b^{8/3} \left (54 b^2 c^2-126 a b c d+77 a^2 d^2\right ) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{18 d^5} \] Output:

-1/18*b*(-a*d+2*b*c)*(-5*a^2*d^2-18*a*b*c*d+18*b^2*c^2)*x*(b*x^3+a)^(2/3)/ 
c^2/d^4+1/18*b*(-5*a^2*d^2-10*a*b*c*d+18*b^2*c^2)*x*(b*x^3+a)^(5/3)/c^2/d^ 
3-1/6*(-a*d+b*c)*x*(b*x^3+a)^(11/3)/c/d/(d*x^3+c)^2-1/18*(-a*d+b*c)*(5*a*d 
+12*b*c)*x*(b*x^3+a)^(8/3)/c^2/d^2/(d*x^3+c)+1/27*b^(8/3)*(77*a^2*d^2-126* 
a*b*c*d+54*b^2*c^2)*arctan(1/3*(1+2*b^(1/3)*x/(b*x^3+a)^(1/3))*3^(1/2))*3^ 
(1/2)/d^5-1/27*(-a*d+b*c)^(8/3)*(5*a^2*d^2+18*a*b*c*d+54*b^2*c^2)*arctan(1 
/3*(1+2*(-a*d+b*c)^(1/3)*x/c^(1/3)/(b*x^3+a)^(1/3))*3^(1/2))*3^(1/2)/c^(8/ 
3)/d^5-1/54*(-a*d+b*c)^(8/3)*(5*a^2*d^2+18*a*b*c*d+54*b^2*c^2)*ln(d*x^3+c) 
/c^(8/3)/d^5+1/18*(-a*d+b*c)^(8/3)*(5*a^2*d^2+18*a*b*c*d+54*b^2*c^2)*ln((- 
a*d+b*c)^(1/3)*x/c^(1/3)-(b*x^3+a)^(1/3))/c^(8/3)/d^5-1/18*b^(8/3)*(77*a^2 
*d^2-126*a*b*c*d+54*b^2*c^2)*ln(-b^(1/3)*x+(b*x^3+a)^(1/3))/d^5
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 12.47 (sec) , antiderivative size = 1171, normalized size of antiderivative = 2.16 \[ \int \frac {\left (a+b x^3\right )^{14/3}}{\left (c+d x^3\right )^3} \, dx =\text {Too large to display} \] Input:

Integrate[(a + b*x^3)^(14/3)/(c + d*x^3)^3,x]
 

Output:

((6*x*(a + b*x^3)^(2/3)*(-2*b^3*(9*b*c - 13*a*d) + 3*b^4*d*x^3 + (3*(b*c - 
 a*d)^4)/(c*(c + d*x^3)^2) - ((b*c - a*d)^3*(21*b*c + 5*a*d))/(c^2*(c + d* 
x^3))))/d^4 + (162*b^5*c*x^4*(1 + (b*x^3)/a)^(1/3)*AppellF1[4/3, 1/3, 1, 7 
/3, -((b*x^3)/a), -((d*x^3)/c)])/(d^4*(a + b*x^3)^(1/3)) - (378*a*b^4*x^4* 
(1 + (b*x^3)/a)^(1/3)*AppellF1[4/3, 1/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c 
)])/(d^3*(a + b*x^3)^(1/3)) + (231*a^2*b^3*x^4*(1 + (b*x^3)/a)^(1/3)*Appel 
lF1[4/3, 1/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])/(c*d^2*(a + b*x^3)^(1/3 
)) + (10*a^5*(2*Sqrt[3]*ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(b + 
a*x^3)^(1/3)))/Sqrt[3]] - 2*Log[c^(1/3) - ((b*c - a*d)^(1/3)*x)/(b + a*x^3 
)^(1/3)] + Log[c^(2/3) + ((b*c - a*d)^(2/3)*x^2)/(b + a*x^3)^(2/3) + (c^(1 
/3)*(b*c - a*d)^(1/3)*x)/(b + a*x^3)^(1/3)]))/(c^(8/3)*(b*c - a*d)^(1/3)) 
+ (36*a*b^4*c^(4/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3 
)*(b + a*x^3)^(1/3)))/Sqrt[3]] - 2*Log[c^(1/3) - ((b*c - a*d)^(1/3)*x)/(b 
+ a*x^3)^(1/3)] + Log[c^(2/3) + ((b*c - a*d)^(2/3)*x^2)/(b + a*x^3)^(2/3) 
+ (c^(1/3)*(b*c - a*d)^(1/3)*x)/(b + a*x^3)^(1/3)]))/(d^4*(b*c - a*d)^(1/3 
)) - (72*a^2*b^3*c^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c 
^(1/3)*(b + a*x^3)^(1/3)))/Sqrt[3]] - 2*Log[c^(1/3) - ((b*c - a*d)^(1/3)*x 
)/(b + a*x^3)^(1/3)] + Log[c^(2/3) + ((b*c - a*d)^(2/3)*x^2)/(b + a*x^3)^( 
2/3) + (c^(1/3)*(b*c - a*d)^(1/3)*x)/(b + a*x^3)^(1/3)]))/(d^3*(b*c - a*d) 
^(1/3)) + (30*a^3*b^2*(2*Sqrt[3]*ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c...
 

Rubi [A] (verified)

Time = 1.39 (sec) , antiderivative size = 505, normalized size of antiderivative = 0.93, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {930, 1023, 27, 1025, 27, 1025, 27, 1026, 769, 901}

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.

\(\displaystyle \int \frac {\left (a+b x^3\right )^{14/3}}{\left (c+d x^3\right )^3} \, dx\)

\(\Big \downarrow \) 930

\(\displaystyle \frac {\int \frac {\left (b x^3+a\right )^{8/3} \left (6 b (2 b c-a d) x^3+a (b c+5 a d)\right )}{\left (d x^3+c\right )^2}dx}{6 c d}-\frac {x \left (a+b x^3\right )^{11/3} (b c-a d)}{6 c d \left (c+d x^3\right )^2}\)

\(\Big \downarrow \) 1023

\(\displaystyle \frac {\frac {x \left (a+b x^3\right )^{8/3} \left (\frac {5 a^2 d}{c}+7 a b-\frac {12 b^2 c}{d}\right )}{3 \left (c+d x^3\right )}-\frac {\int -\frac {2 \left (b x^3+a\right )^{5/3} \left (3 b \left (18 b^2 c^2-10 a b d c-5 a^2 d^2\right ) x^3+a \left (6 b^2 c^2-2 a b d c+5 a^2 d^2\right )\right )}{d x^3+c}dx}{3 c d}}{6 c d}-\frac {x \left (a+b x^3\right )^{11/3} (b c-a d)}{6 c d \left (c+d x^3\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 \int \frac {\left (b x^3+a\right )^{5/3} \left (3 b \left (18 b^2 c^2-10 a b d c-5 a^2 d^2\right ) x^3+a \left (6 b^2 c^2-2 a b d c+5 a^2 d^2\right )\right )}{d x^3+c}dx}{3 c d}+\frac {x \left (a+b x^3\right )^{8/3} \left (\frac {5 a^2 d}{c}+7 a b-\frac {12 b^2 c}{d}\right )}{3 \left (c+d x^3\right )}}{6 c d}-\frac {x \left (a+b x^3\right )^{11/3} (b c-a d)}{6 c d \left (c+d x^3\right )^2}\)

\(\Big \downarrow \) 1025

\(\displaystyle \frac {\frac {2 \left (\frac {\int -\frac {3 \left (b x^3+a\right )^{2/3} \left (3 b (2 b c-a d) \left (18 b^2 c^2-18 a b d c-5 a^2 d^2\right ) x^3+a \left (18 b^3 c^3-22 a b^2 d c^2-a^2 b d^2 c-10 a^3 d^3\right )\right )}{d x^3+c}dx}{6 d}+\frac {b x \left (a+b x^3\right )^{5/3} \left (-5 a^2 d^2-10 a b c d+18 b^2 c^2\right )}{2 d}\right )}{3 c d}+\frac {x \left (a+b x^3\right )^{8/3} \left (\frac {5 a^2 d}{c}+7 a b-\frac {12 b^2 c}{d}\right )}{3 \left (c+d x^3\right )}}{6 c d}-\frac {x \left (a+b x^3\right )^{11/3} (b c-a d)}{6 c d \left (c+d x^3\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 \left (\frac {b x \left (a+b x^3\right )^{5/3} \left (-5 a^2 d^2-10 a b c d+18 b^2 c^2\right )}{2 d}-\frac {\int \frac {\left (b x^3+a\right )^{2/3} \left (3 b (2 b c-a d) \left (18 b^2 c^2-18 a b d c-5 a^2 d^2\right ) x^3+a \left (18 b^3 c^3-22 a b^2 d c^2-a^2 b d^2 c-10 a^3 d^3\right )\right )}{d x^3+c}dx}{2 d}\right )}{3 c d}+\frac {x \left (a+b x^3\right )^{8/3} \left (\frac {5 a^2 d}{c}+7 a b-\frac {12 b^2 c}{d}\right )}{3 \left (c+d x^3\right )}}{6 c d}-\frac {x \left (a+b x^3\right )^{11/3} (b c-a d)}{6 c d \left (c+d x^3\right )^2}\)

\(\Big \downarrow \) 1025

\(\displaystyle \frac {\frac {2 \left (\frac {b x \left (a+b x^3\right )^{5/3} \left (-5 a^2 d^2-10 a b c d+18 b^2 c^2\right )}{2 d}-\frac {\frac {\int -\frac {6 \left (b^3 c^2 \left (54 b^2 c^2-126 a b d c+77 a^2 d^2\right ) x^3+a \left (18 b^4 c^4-36 a b^3 d c^3+15 a^2 b^2 d^2 c^2+3 a^3 b d^3 c+5 a^4 d^4\right )\right )}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{3 d}+\frac {b x \left (a+b x^3\right )^{2/3} (2 b c-a d) \left (-5 a^2 d^2-18 a b c d+18 b^2 c^2\right )}{d}}{2 d}\right )}{3 c d}+\frac {x \left (a+b x^3\right )^{8/3} \left (\frac {5 a^2 d}{c}+7 a b-\frac {12 b^2 c}{d}\right )}{3 \left (c+d x^3\right )}}{6 c d}-\frac {x \left (a+b x^3\right )^{11/3} (b c-a d)}{6 c d \left (c+d x^3\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 \left (\frac {b x \left (a+b x^3\right )^{5/3} \left (-5 a^2 d^2-10 a b c d+18 b^2 c^2\right )}{2 d}-\frac {\frac {b x \left (a+b x^3\right )^{2/3} (2 b c-a d) \left (-5 a^2 d^2-18 a b c d+18 b^2 c^2\right )}{d}-\frac {2 \int \frac {b^3 c^2 \left (54 b^2 c^2-126 a b d c+77 a^2 d^2\right ) x^3+a \left (18 b^4 c^4-36 a b^3 d c^3+15 a^2 b^2 d^2 c^2+3 a^3 b d^3 c+5 a^4 d^4\right )}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{d}}{2 d}\right )}{3 c d}+\frac {x \left (a+b x^3\right )^{8/3} \left (\frac {5 a^2 d}{c}+7 a b-\frac {12 b^2 c}{d}\right )}{3 \left (c+d x^3\right )}}{6 c d}-\frac {x \left (a+b x^3\right )^{11/3} (b c-a d)}{6 c d \left (c+d x^3\right )^2}\)

\(\Big \downarrow \) 1026

\(\displaystyle \frac {\frac {2 \left (\frac {b x \left (a+b x^3\right )^{5/3} \left (-5 a^2 d^2-10 a b c d+18 b^2 c^2\right )}{2 d}-\frac {\frac {b x \left (a+b x^3\right )^{2/3} (2 b c-a d) \left (-5 a^2 d^2-18 a b c d+18 b^2 c^2\right )}{d}-\frac {2 \left (\frac {b^3 c^2 \left (77 a^2 d^2-126 a b c d+54 b^2 c^2\right ) \int \frac {1}{\sqrt [3]{b x^3+a}}dx}{d}-\frac {(b c-a d)^3 \left (5 a^2 d^2+18 a b c d+54 b^2 c^2\right ) \int \frac {1}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{d}\right )}{d}}{2 d}\right )}{3 c d}+\frac {x \left (a+b x^3\right )^{8/3} \left (\frac {5 a^2 d}{c}+7 a b-\frac {12 b^2 c}{d}\right )}{3 \left (c+d x^3\right )}}{6 c d}-\frac {x \left (a+b x^3\right )^{11/3} (b c-a d)}{6 c d \left (c+d x^3\right )^2}\)

\(\Big \downarrow \) 769

\(\displaystyle \frac {\frac {2 \left (\frac {b x \left (a+b x^3\right )^{5/3} \left (-5 a^2 d^2-10 a b c d+18 b^2 c^2\right )}{2 d}-\frac {\frac {b x \left (a+b x^3\right )^{2/3} (2 b c-a d) \left (-5 a^2 d^2-18 a b c d+18 b^2 c^2\right )}{d}-\frac {2 \left (\frac {b^3 c^2 \left (77 a^2 d^2-126 a b c d+54 b^2 c^2\right ) \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}\right )}{d}-\frac {(b c-a d)^3 \left (5 a^2 d^2+18 a b c d+54 b^2 c^2\right ) \int \frac {1}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{d}\right )}{d}}{2 d}\right )}{3 c d}+\frac {x \left (a+b x^3\right )^{8/3} \left (\frac {5 a^2 d}{c}+7 a b-\frac {12 b^2 c}{d}\right )}{3 \left (c+d x^3\right )}}{6 c d}-\frac {x \left (a+b x^3\right )^{11/3} (b c-a d)}{6 c d \left (c+d x^3\right )^2}\)

\(\Big \downarrow \) 901

\(\displaystyle \frac {\frac {2 \left (\frac {b x \left (a+b x^3\right )^{5/3} \left (-5 a^2 d^2-10 a b c d+18 b^2 c^2\right )}{2 d}-\frac {\frac {b x \left (a+b x^3\right )^{2/3} (2 b c-a d) \left (-5 a^2 d^2-18 a b c d+18 b^2 c^2\right )}{d}-\frac {2 \left (\frac {b^3 c^2 \left (77 a^2 d^2-126 a b c d+54 b^2 c^2\right ) \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}\right )}{d}-\frac {(b c-a d)^3 \left (5 a^2 d^2+18 a b c d+54 b^2 c^2\right ) \left (\frac {\arctan \left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} c^{2/3} \sqrt [3]{b c-a d}}+\frac {\log \left (c+d x^3\right )}{6 c^{2/3} \sqrt [3]{b c-a d}}-\frac {\log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{2/3} \sqrt [3]{b c-a d}}\right )}{d}\right )}{d}}{2 d}\right )}{3 c d}+\frac {x \left (a+b x^3\right )^{8/3} \left (\frac {5 a^2 d}{c}+7 a b-\frac {12 b^2 c}{d}\right )}{3 \left (c+d x^3\right )}}{6 c d}-\frac {x \left (a+b x^3\right )^{11/3} (b c-a d)}{6 c d \left (c+d x^3\right )^2}\)

Input:

Int[(a + b*x^3)^(14/3)/(c + d*x^3)^3,x]
 

Output:

-1/6*((b*c - a*d)*x*(a + b*x^3)^(11/3))/(c*d*(c + d*x^3)^2) + (((7*a*b - ( 
12*b^2*c)/d + (5*a^2*d)/c)*x*(a + b*x^3)^(8/3))/(3*(c + d*x^3)) + (2*((b*( 
18*b^2*c^2 - 10*a*b*c*d - 5*a^2*d^2)*x*(a + b*x^3)^(5/3))/(2*d) - ((b*(2*b 
*c - a*d)*(18*b^2*c^2 - 18*a*b*c*d - 5*a^2*d^2)*x*(a + b*x^3)^(2/3))/d - ( 
2*(-(((b*c - a*d)^3*(54*b^2*c^2 + 18*a*b*c*d + 5*a^2*d^2)*(ArcTan[(1 + (2* 
(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*c^(2/3 
)*(b*c - a*d)^(1/3)) + Log[c + d*x^3]/(6*c^(2/3)*(b*c - a*d)^(1/3)) - Log[ 
((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)]/(2*c^(2/3)*(b*c - a*d)^ 
(1/3))))/d) + (b^3*c^2*(54*b^2*c^2 - 126*a*b*c*d + 77*a^2*d^2)*(ArcTan[(1 
+ (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*b^(1/3)) - Log[-(b^(1 
/3)*x) + (a + b*x^3)^(1/3)]/(2*b^(1/3))))/d))/d)/(2*d)))/(3*c*d))/(6*c*d)
 

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 769

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]* 
(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^ 
3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]
 

rule 901

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> Wit 
h[{q = Rt[(b*c - a*d)/c, 3]}, Simp[ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/S 
qrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q), x] 
 + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0]
 

rule 930

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[(a*d - c*b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(a*b*n*(p + 
1))), x] - Simp[1/(a*b*n*(p + 1))   Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q 
- 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*( 
p + q) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 
 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q, x]
 

rule 1023

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f 
_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c 
+ d*x^n)^q/(a*b*n*(p + 1))), x] + Simp[1/(a*b*n*(p + 1))   Int[(a + b*x^n)^ 
(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(b*e*n*(p + 1) + b*e - a*f) + d*(b*e*n*( 
p + 1) + (b*e - a*f)*(n*q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, 
 n}, x] && LtQ[p, -1] && GtQ[q, 0]
 

rule 1025

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + ( 
f_.)*(x_)^(n_)), x_Symbol] :> Simp[f*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/( 
b*(n*(p + q + 1) + 1))), x] + Simp[1/(b*(n*(p + q + 1) + 1))   Int[(a + b*x 
^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e 
- a*f) + f*n*q*(b*c - a*d) + b*d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[ 
{a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1, 0]
 

rule 1026

Int[(((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)* 
(x_)^(n_)), x_Symbol] :> Simp[f/d   Int[(a + b*x^n)^p, x], x] + Simp[(d*e - 
 c*f)/d   Int[(a + b*x^n)^p/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, 
 p, n}, x]
 

Maple [A] (verified)

Time = 2.86 (sec) , antiderivative size = 661, normalized size of antiderivative = 1.22

method	result	size
pseudoelliptic	\(-\frac {2 \left (\frac {5 \left (a d -b c \right )^{3} \left (a^{2} d^{2}+\frac {18}{5} a b c d +\frac {54}{5} b^{2} c^{2}\right ) \left (d \,x^{3}+c \right )^{2} \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{108}-\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} c^{3} \left (d \,x^{3}+c \right )^{2} \left (\frac {77 a^{2} b^{\frac {8}{3}} d^{2}}{54}+\left (b c -\frac {7 a d}{3}\right ) c \,b^{\frac {11}{3}}\right ) \ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}-\frac {5 \left (a d -b c \right )^{3} \left (a^{2} d^{2}+\frac {18}{5} a b c d +\frac {54}{5} b^{2} c^{2}\right ) \arctan \left (\frac {\sqrt {3}\, \left (-\frac {2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}}+x \right )}{3 x}\right ) \left (d \,x^{3}+c \right )^{2} \sqrt {3}}{54}-\frac {5 \left (a d -b c \right )^{3} \left (a^{2} d^{2}+\frac {18}{5} a b c d +\frac {54}{5} b^{2} c^{2}\right ) \left (d \,x^{3}+c \right )^{2} \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )}{54}+\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} c \left (c^{2} \left (d \,x^{3}+c \right )^{2} \sqrt {3}\, \left (\frac {77 a^{2} b^{\frac {8}{3}} d^{2}}{54}+\left (b c -\frac {7 a d}{3}\right ) c \,b^{\frac {11}{3}}\right ) \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right )+c^{2} \left (d \,x^{3}+c \right )^{2} \left (\frac {77 a^{2} b^{\frac {8}{3}} d^{2}}{54}+\left (b c -\frac {7 a d}{3}\right ) c \,b^{\frac {11}{3}}\right ) \ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )-\frac {2 \left (b \,x^{3}+a \right )^{\frac {2}{3}} d \left (-\frac {9 b^{4} c^{5}}{2}+9 \left (-\frac {3 b \,x^{3}}{4}+a \right ) d \,b^{3} c^{4}-\frac {15 d^{2} \left (\frac {2}{5} b^{2} x^{6}-\frac {11}{3} a b \,x^{3}+a^{2}\right ) b^{2} c^{3}}{4}-\frac {3 d^{3} \left (-\frac {1}{2} b^{3} x^{9}-\frac {13}{3} a \,b^{2} x^{6}+8 a^{2} b \,x^{3}+a^{3}\right ) b \,c^{2}}{4}+a^{3} d^{4} \left (\frac {3 b \,x^{3}}{4}+a \right ) c +\frac {5 a^{4} d^{5} x^{3}}{8}\right ) x}{9}\right )\right )}{\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} c^{3} d^{5} \left (d \,x^{3}+c \right )^{2}}\)	\(661\)

Input:

int((b*x^3+a)^(14/3)/(d*x^3+c)^3,x,method=_RETURNVERBOSE)
 

Output:

-2/((a*d-b*c)/c)^(1/3)*(5/108*(a*d-b*c)^3*(a^2*d^2+18/5*a*b*c*d+54/5*b^2*c 
^2)*(d*x^3+c)^2*ln((((a*d-b*c)/c)^(2/3)*x^2-((a*d-b*c)/c)^(1/3)*(b*x^3+a)^ 
(1/3)*x+(b*x^3+a)^(2/3))/x^2)-1/2*((a*d-b*c)/c)^(1/3)*c^3*(d*x^3+c)^2*(77/ 
54*a^2*b^(8/3)*d^2+(b*c-7/3*a*d)*c*b^(11/3))*ln((b^(2/3)*x^2+b^(1/3)*(b*x^ 
3+a)^(1/3)*x+(b*x^3+a)^(2/3))/x^2)-5/54*(a*d-b*c)^3*(a^2*d^2+18/5*a*b*c*d+ 
54/5*b^2*c^2)*arctan(1/3*3^(1/2)*(-2/((a*d-b*c)/c)^(1/3)*(b*x^3+a)^(1/3)+x 
)/x)*(d*x^3+c)^2*3^(1/2)-5/54*(a*d-b*c)^3*(a^2*d^2+18/5*a*b*c*d+54/5*b^2*c 
^2)*(d*x^3+c)^2*ln((((a*d-b*c)/c)^(1/3)*x+(b*x^3+a)^(1/3))/x)+((a*d-b*c)/c 
)^(1/3)*c*(c^2*(d*x^3+c)^2*3^(1/2)*(77/54*a^2*b^(8/3)*d^2+(b*c-7/3*a*d)*c* 
b^(11/3))*arctan(1/3*3^(1/2)*(b^(1/3)*x+2*(b*x^3+a)^(1/3))/b^(1/3)/x)+c^2* 
(d*x^3+c)^2*(77/54*a^2*b^(8/3)*d^2+(b*c-7/3*a*d)*c*b^(11/3))*ln((-b^(1/3)* 
x+(b*x^3+a)^(1/3))/x)-2/9*(b*x^3+a)^(2/3)*d*(-9/2*b^4*c^5+9*(-3/4*b*x^3+a) 
*d*b^3*c^4-15/4*d^2*(2/5*b^2*x^6-11/3*a*b*x^3+a^2)*b^2*c^3-3/4*d^3*(-1/2*b 
^3*x^9-13/3*a*b^2*x^6+8*a^2*b*x^3+a^3)*b*c^2+a^3*d^4*(3/4*b*x^3+a)*c+5/8*a 
^4*d^5*x^3)*x))/c^3/d^5/(d*x^3+c)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1555 vs. \(2 (473) = 946\).

Time = 88.77 (sec) , antiderivative size = 1555, normalized size of antiderivative = 2.87 \[ \int \frac {\left (a+b x^3\right )^{14/3}}{\left (c+d x^3\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((b*x^3+a)^(14/3)/(d*x^3+c)^3,x, algorithm="fricas")
 

Output:

-1/54*(2*sqrt(3)*(54*b^4*c^6 - 90*a*b^3*c^5*d + 23*a^2*b^2*c^4*d^2 + 8*a^3 
*b*c^3*d^3 + 5*a^4*c^2*d^4 + (54*b^4*c^4*d^2 - 90*a*b^3*c^3*d^3 + 23*a^2*b 
^2*c^2*d^4 + 8*a^3*b*c*d^5 + 5*a^4*d^6)*x^6 + 2*(54*b^4*c^5*d - 90*a*b^3*c 
^4*d^2 + 23*a^2*b^2*c^3*d^3 + 8*a^3*b*c^2*d^4 + 5*a^4*c*d^5)*x^3)*((b^2*c^ 
2 - 2*a*b*c*d + a^2*d^2)/c^2)^(1/3)*arctan(-1/3*(sqrt(3)*(b*c - a*d)*x + 2 
*sqrt(3)*(b*x^3 + a)^(1/3)*c*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)/c^2)^(1/3))/ 
((b*c - a*d)*x)) + 2*sqrt(3)*(54*b^4*c^6 - 126*a*b^3*c^5*d + 77*a^2*b^2*c^ 
4*d^2 + (54*b^4*c^4*d^2 - 126*a*b^3*c^3*d^3 + 77*a^2*b^2*c^2*d^4)*x^6 + 2* 
(54*b^4*c^5*d - 126*a*b^3*c^4*d^2 + 77*a^2*b^2*c^3*d^3)*x^3)*(-b^2)^(1/3)* 
arctan(-1/3*(sqrt(3)*b*x - 2*sqrt(3)*(b*x^3 + a)^(1/3)*(-b^2)^(1/3))/(b*x) 
) - 2*(54*b^4*c^6 - 90*a*b^3*c^5*d + 23*a^2*b^2*c^4*d^2 + 8*a^3*b*c^3*d^3 
+ 5*a^4*c^2*d^4 + (54*b^4*c^4*d^2 - 90*a*b^3*c^3*d^3 + 23*a^2*b^2*c^2*d^4 
+ 8*a^3*b*c*d^5 + 5*a^4*d^6)*x^6 + 2*(54*b^4*c^5*d - 90*a*b^3*c^4*d^2 + 23 
*a^2*b^2*c^3*d^3 + 8*a^3*b*c^2*d^4 + 5*a^4*c*d^5)*x^3)*((b^2*c^2 - 2*a*b*c 
*d + a^2*d^2)/c^2)^(1/3)*log((c*x*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)/c^2)^(2 
/3) - (b*x^3 + a)^(1/3)*(b*c - a*d))/x) - 2*(54*b^4*c^6 - 126*a*b^3*c^5*d 
+ 77*a^2*b^2*c^4*d^2 + (54*b^4*c^4*d^2 - 126*a*b^3*c^3*d^3 + 77*a^2*b^2*c^ 
2*d^4)*x^6 + 2*(54*b^4*c^5*d - 126*a*b^3*c^4*d^2 + 77*a^2*b^2*c^3*d^3)*x^3 
)*(-b^2)^(1/3)*log(-((-b^2)^(2/3)*x - (b*x^3 + a)^(1/3)*b)/x) + (54*b^4*c^ 
6 - 126*a*b^3*c^5*d + 77*a^2*b^2*c^4*d^2 + (54*b^4*c^4*d^2 - 126*a*b^3*...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^3\right )^{14/3}}{\left (c+d x^3\right )^3} \, dx=\text {Timed out} \] Input:

integrate((b*x**3+a)**(14/3)/(d*x**3+c)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b x^3\right )^{14/3}}{\left (c+d x^3\right )^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {14}{3}}}{{\left (d x^{3} + c\right )}^{3}} \,d x } \] Input:

integrate((b*x^3+a)^(14/3)/(d*x^3+c)^3,x, algorithm="maxima")
 

Output:

integrate((b*x^3 + a)^(14/3)/(d*x^3 + c)^3, x)
 

Giac [F]

\[ \int \frac {\left (a+b x^3\right )^{14/3}}{\left (c+d x^3\right )^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {14}{3}}}{{\left (d x^{3} + c\right )}^{3}} \,d x } \] Input:

integrate((b*x^3+a)^(14/3)/(d*x^3+c)^3,x, algorithm="giac")
 

Output:

integrate((b*x^3 + a)^(14/3)/(d*x^3 + c)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^3\right )^{14/3}}{\left (c+d x^3\right )^3} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{14/3}}{{\left (d\,x^3+c\right )}^3} \,d x \] Input:

int((a + b*x^3)^(14/3)/(c + d*x^3)^3,x)
 

Output:

int((a + b*x^3)^(14/3)/(c + d*x^3)^3, x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^3\right )^{14/3}}{\left (c+d x^3\right )^3} \, dx=\text {too large to display} \] Input:

int((b*x^3+a)^(14/3)/(d*x^3+c)^3,x)
 

Output:

( - 90*(a + b*x**3)**(2/3)*a**5*b*d**3*x - 90*(a + b*x**3)**(2/3)*a**4*b** 
2*c*d**2*x - 315*(a + b*x**3)**(2/3)*a**4*b**2*d**3*x**4 + 364*(a + b*x**3 
)**(2/3)*a**3*b**3*c**2*d*x + 725*(a + b*x**3)**(2/3)*a**3*b**3*c*d**2*x** 
4 + 130*(a + b*x**3)**(2/3)*a**3*b**3*d**3*x**7 - 168*(a + b*x**3)**(2/3)* 
a**2*b**4*c**3*x - 483*(a + b*x**3)**(2/3)*a**2*b**4*c**2*d*x**4 - 372*(a 
+ b*x**3)**(2/3)*a**2*b**4*c*d**2*x**7 + 15*(a + b*x**3)**(2/3)*a**2*b**4* 
d**3*x**10 + 126*(a + b*x**3)**(2/3)*a*b**5*c**3*x**4 + 378*(a + b*x**3)** 
(2/3)*a*b**5*c**2*d*x**7 - 36*(a + b*x**3)**(2/3)*a*b**5*c*d**2*x**10 - 10 
8*(a + b*x**3)**(2/3)*b**6*c**3*x**7 + 27*(a + b*x**3)**(2/3)*b**6*c**2*d* 
x**10 + 450*int((a + b*x**3)**(2/3)/(5*a**3*c**3*d**2 + 15*a**3*c**2*d**3* 
x**3 + 15*a**3*c*d**4*x**6 + 5*a**3*d**5*x**9 - 12*a**2*b*c**4*d - 31*a**2 
*b*c**3*d**2*x**3 - 21*a**2*b*c**2*d**3*x**6 + 3*a**2*b*c*d**4*x**9 + 5*a* 
*2*b*d**5*x**12 + 9*a*b**2*c**5 + 15*a*b**2*c**4*d*x**3 - 9*a*b**2*c**3*d* 
*2*x**6 - 27*a*b**2*c**2*d**3*x**9 - 12*a*b**2*c*d**4*x**12 + 9*b**3*c**5* 
x**3 + 27*b**3*c**4*d*x**6 + 27*b**3*c**3*d**2*x**9 + 9*b**3*c**2*d**3*x** 
12),x)*a**9*c**2*d**6 + 900*int((a + b*x**3)**(2/3)/(5*a**3*c**3*d**2 + 15 
*a**3*c**2*d**3*x**3 + 15*a**3*c*d**4*x**6 + 5*a**3*d**5*x**9 - 12*a**2*b* 
c**4*d - 31*a**2*b*c**3*d**2*x**3 - 21*a**2*b*c**2*d**3*x**6 + 3*a**2*b*c* 
d**4*x**9 + 5*a**2*b*d**5*x**12 + 9*a*b**2*c**5 + 15*a*b**2*c**4*d*x**3 - 
9*a*b**2*c**3*d**2*x**6 - 27*a*b**2*c**2*d**3*x**9 - 12*a*b**2*c*d**4*x...