\(\int \frac {1}{(a+b x)^{19/4} \sqrt [4]{c+d x}} \, dx\) [594]

Optimal result

Integrand size = 19, antiderivative size = 136 \[ \int \frac {1}{(a+b x)^{19/4} \sqrt [4]{c+d x}} \, dx=-\frac {4 (c+d x)^{3/4}}{15 (b c-a d) (a+b x)^{15/4}}+\frac {16 d (c+d x)^{3/4}}{55 (b c-a d)^2 (a+b x)^{11/4}}-\frac {128 d^2 (c+d x)^{3/4}}{385 (b c-a d)^3 (a+b x)^{7/4}}+\frac {512 d^3 (c+d x)^{3/4}}{1155 (b c-a d)^4 (a+b x)^{3/4}} \] Output:

-4/15*(d*x+c)^(3/4)/(-a*d+b*c)/(b*x+a)^(15/4)+16/55*d*(d*x+c)^(3/4)/(-a*d+ 
b*c)^2/(b*x+a)^(11/4)-128/385*d^2*(d*x+c)^(3/4)/(-a*d+b*c)^3/(b*x+a)^(7/4) 
+512/1155*d^3*(d*x+c)^(3/4)/(-a*d+b*c)^4/(b*x+a)^(3/4)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.86 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(a+b x)^{19/4} \sqrt [4]{c+d x}} \, dx=\frac {4 (c+d x)^{3/4} \left (385 a^3 d^3+165 a^2 b d^2 (-3 c+4 d x)+15 a b^2 d \left (21 c^2-24 c d x+32 d^2 x^2\right )+b^3 \left (-77 c^3+84 c^2 d x-96 c d^2 x^2+128 d^3 x^3\right )\right )}{1155 (b c-a d)^4 (a+b x)^{15/4}} \] Input:

Integrate[1/((a + b*x)^(19/4)*(c + d*x)^(1/4)),x]
 

Output:

(4*(c + d*x)^(3/4)*(385*a^3*d^3 + 165*a^2*b*d^2*(-3*c + 4*d*x) + 15*a*b^2* 
d*(21*c^2 - 24*c*d*x + 32*d^2*x^2) + b^3*(-77*c^3 + 84*c^2*d*x - 96*c*d^2* 
x^2 + 128*d^3*x^3)))/(1155*(b*c - a*d)^4*(a + b*x)^(15/4))
 

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.19, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {55, 55, 55, 48}

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.

\(\displaystyle \int \frac {1}{(a+b x)^{19/4} \sqrt [4]{c+d x}} \, dx\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {4 d \int \frac {1}{(a+b x)^{15/4} \sqrt [4]{c+d x}}dx}{5 (b c-a d)}-\frac {4 (c+d x)^{3/4}}{15 (a+b x)^{15/4} (b c-a d)}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {4 d \left (-\frac {8 d \int \frac {1}{(a+b x)^{11/4} \sqrt [4]{c+d x}}dx}{11 (b c-a d)}-\frac {4 (c+d x)^{3/4}}{11 (a+b x)^{11/4} (b c-a d)}\right )}{5 (b c-a d)}-\frac {4 (c+d x)^{3/4}}{15 (a+b x)^{15/4} (b c-a d)}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {4 d \left (-\frac {8 d \left (-\frac {4 d \int \frac {1}{(a+b x)^{7/4} \sqrt [4]{c+d x}}dx}{7 (b c-a d)}-\frac {4 (c+d x)^{3/4}}{7 (a+b x)^{7/4} (b c-a d)}\right )}{11 (b c-a d)}-\frac {4 (c+d x)^{3/4}}{11 (a+b x)^{11/4} (b c-a d)}\right )}{5 (b c-a d)}-\frac {4 (c+d x)^{3/4}}{15 (a+b x)^{15/4} (b c-a d)}\)

\(\Big \downarrow \) 48

\(\displaystyle -\frac {4 d \left (-\frac {8 d \left (\frac {16 d (c+d x)^{3/4}}{21 (a+b x)^{3/4} (b c-a d)^2}-\frac {4 (c+d x)^{3/4}}{7 (a+b x)^{7/4} (b c-a d)}\right )}{11 (b c-a d)}-\frac {4 (c+d x)^{3/4}}{11 (a+b x)^{11/4} (b c-a d)}\right )}{5 (b c-a d)}-\frac {4 (c+d x)^{3/4}}{15 (a+b x)^{15/4} (b c-a d)}\)

Input:

Int[1/((a + b*x)^(19/4)*(c + d*x)^(1/4)),x]
 

Output:

(-4*(c + d*x)^(3/4))/(15*(b*c - a*d)*(a + b*x)^(15/4)) - (4*d*((-4*(c + d* 
x)^(3/4))/(11*(b*c - a*d)*(a + b*x)^(11/4)) - (8*d*((-4*(c + d*x)^(3/4))/( 
7*(b*c - a*d)*(a + b*x)^(7/4)) + (16*d*(c + d*x)^(3/4))/(21*(b*c - a*d)^2* 
(a + b*x)^(3/4))))/(11*(b*c - a*d))))/(5*(b*c - a*d))
 

Defintions of rubi rules used

rule 48

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

rule 55

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S 
implify[m + n + 2]/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^Simplify[m + 1]*( 
c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 
 2], 0] && NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ 
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] ||  !SumSimp 
lerQ[n, 1])
 

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.26

method	result	size
gosper	\(\frac {4 \left (x d +c \right )^{\frac {3}{4}} \left (128 d^{3} x^{3} b^{3}+480 x^{2} a \,b^{2} d^{3}-96 x^{2} b^{3} c \,d^{2}+660 x \,a^{2} b \,d^{3}-360 x a \,b^{2} c \,d^{2}+84 x \,b^{3} c^{2} d +385 a^{3} d^{3}-495 a^{2} b c \,d^{2}+315 a \,b^{2} c^{2} d -77 b^{3} c^{3}\right )}{1155 \left (b x +a \right )^{\frac {15}{4}} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}\)	\(171\)
orering	\(\frac {4 \left (x d +c \right )^{\frac {3}{4}} \left (128 d^{3} x^{3} b^{3}+480 x^{2} a \,b^{2} d^{3}-96 x^{2} b^{3} c \,d^{2}+660 x \,a^{2} b \,d^{3}-360 x a \,b^{2} c \,d^{2}+84 x \,b^{3} c^{2} d +385 a^{3} d^{3}-495 a^{2} b c \,d^{2}+315 a \,b^{2} c^{2} d -77 b^{3} c^{3}\right )}{1155 \left (b x +a \right )^{\frac {15}{4}} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}\)	\(171\)

Input:

int(1/(b*x+a)^(19/4)/(d*x+c)^(1/4),x,method=_RETURNVERBOSE)
 

Output:

4/1155*(d*x+c)^(3/4)*(128*b^3*d^3*x^3+480*a*b^2*d^3*x^2-96*b^3*c*d^2*x^2+6 
60*a^2*b*d^3*x-360*a*b^2*c*d^2*x+84*b^3*c^2*d*x+385*a^3*d^3-495*a^2*b*c*d^ 
2+315*a*b^2*c^2*d-77*b^3*c^3)/(b*x+a)^(15/4)/(a^4*d^4-4*a^3*b*c*d^3+6*a^2* 
b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 419 vs. \(2 (112) = 224\).

Time = 1.03 (sec) , antiderivative size = 419, normalized size of antiderivative = 3.08 \[ \int \frac {1}{(a+b x)^{19/4} \sqrt [4]{c+d x}} \, dx=\frac {4 \, {\left (128 \, b^{3} d^{3} x^{3} - 77 \, b^{3} c^{3} + 315 \, a b^{2} c^{2} d - 495 \, a^{2} b c d^{2} + 385 \, a^{3} d^{3} - 96 \, {\left (b^{3} c d^{2} - 5 \, a b^{2} d^{3}\right )} x^{2} + 12 \, {\left (7 \, b^{3} c^{2} d - 30 \, a b^{2} c d^{2} + 55 \, a^{2} b d^{3}\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{1155 \, {\left (a^{4} b^{4} c^{4} - 4 \, a^{5} b^{3} c^{3} d + 6 \, a^{6} b^{2} c^{2} d^{2} - 4 \, a^{7} b c d^{3} + a^{8} d^{4} + {\left (b^{8} c^{4} - 4 \, a b^{7} c^{3} d + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{3} b^{5} c d^{3} + a^{4} b^{4} d^{4}\right )} x^{4} + 4 \, {\left (a b^{7} c^{4} - 4 \, a^{2} b^{6} c^{3} d + 6 \, a^{3} b^{5} c^{2} d^{2} - 4 \, a^{4} b^{4} c d^{3} + a^{5} b^{3} d^{4}\right )} x^{3} + 6 \, {\left (a^{2} b^{6} c^{4} - 4 \, a^{3} b^{5} c^{3} d + 6 \, a^{4} b^{4} c^{2} d^{2} - 4 \, a^{5} b^{3} c d^{3} + a^{6} b^{2} d^{4}\right )} x^{2} + 4 \, {\left (a^{3} b^{5} c^{4} - 4 \, a^{4} b^{4} c^{3} d + 6 \, a^{5} b^{3} c^{2} d^{2} - 4 \, a^{6} b^{2} c d^{3} + a^{7} b d^{4}\right )} x\right )}} \] Input:

integrate(1/(b*x+a)^(19/4)/(d*x+c)^(1/4),x, algorithm="fricas")
 

Output:

4/1155*(128*b^3*d^3*x^3 - 77*b^3*c^3 + 315*a*b^2*c^2*d - 495*a^2*b*c*d^2 + 
 385*a^3*d^3 - 96*(b^3*c*d^2 - 5*a*b^2*d^3)*x^2 + 12*(7*b^3*c^2*d - 30*a*b 
^2*c*d^2 + 55*a^2*b*d^3)*x)*(b*x + a)^(1/4)*(d*x + c)^(3/4)/(a^4*b^4*c^4 - 
 4*a^5*b^3*c^3*d + 6*a^6*b^2*c^2*d^2 - 4*a^7*b*c*d^3 + a^8*d^4 + (b^8*c^4 
- 4*a*b^7*c^3*d + 6*a^2*b^6*c^2*d^2 - 4*a^3*b^5*c*d^3 + a^4*b^4*d^4)*x^4 + 
 4*(a*b^7*c^4 - 4*a^2*b^6*c^3*d + 6*a^3*b^5*c^2*d^2 - 4*a^4*b^4*c*d^3 + a^ 
5*b^3*d^4)*x^3 + 6*(a^2*b^6*c^4 - 4*a^3*b^5*c^3*d + 6*a^4*b^4*c^2*d^2 - 4* 
a^5*b^3*c*d^3 + a^6*b^2*d^4)*x^2 + 4*(a^3*b^5*c^4 - 4*a^4*b^4*c^3*d + 6*a^ 
5*b^3*c^2*d^2 - 4*a^6*b^2*c*d^3 + a^7*b*d^4)*x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x)^{19/4} \sqrt [4]{c+d x}} \, dx=\text {Timed out} \] Input:

integrate(1/(b*x+a)**(19/4)/(d*x+c)**(1/4),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{(a+b x)^{19/4} \sqrt [4]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {19}{4}} {\left (d x + c\right )}^{\frac {1}{4}}} \,d x } \] Input:

integrate(1/(b*x+a)^(19/4)/(d*x+c)^(1/4),x, algorithm="maxima")
 

Output:

integrate(1/((b*x + a)^(19/4)*(d*x + c)^(1/4)), x)
 

Giac [F]

\[ \int \frac {1}{(a+b x)^{19/4} \sqrt [4]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {19}{4}} {\left (d x + c\right )}^{\frac {1}{4}}} \,d x } \] Input:

integrate(1/(b*x+a)^(19/4)/(d*x+c)^(1/4),x, algorithm="giac")
 

Output:

integrate(1/((b*x + a)^(19/4)*(d*x + c)^(1/4)), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x)^{19/4} \sqrt [4]{c+d x}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{19/4}\,{\left (c+d\,x\right )}^{1/4}} \,d x \] Input:

int(1/((a + b*x)^(19/4)*(c + d*x)^(1/4)),x)
 

Output:

int(1/((a + b*x)^(19/4)*(c + d*x)^(1/4)), x)
 

Reduce [F]

\[ \int \frac {1}{(a+b x)^{19/4} \sqrt [4]{c+d x}} \, dx=\int \frac {1}{\left (d x +c \right )^{\frac {1}{4}} \left (b x +a \right )^{\frac {3}{4}} a^{4}+4 \left (d x +c \right )^{\frac {1}{4}} \left (b x +a \right )^{\frac {3}{4}} a^{3} b x +6 \left (d x +c \right )^{\frac {1}{4}} \left (b x +a \right )^{\frac {3}{4}} a^{2} b^{2} x^{2}+4 \left (d x +c \right )^{\frac {1}{4}} \left (b x +a \right )^{\frac {3}{4}} a \,b^{3} x^{3}+\left (d x +c \right )^{\frac {1}{4}} \left (b x +a \right )^{\frac {3}{4}} b^{4} x^{4}}d x \] Input:

int(1/(b*x+a)^(19/4)/(d*x+c)^(1/4),x)
 

Output:

int(1/((c + d*x)**(1/4)*(a + b*x)**(3/4)*a**4 + 4*(c + d*x)**(1/4)*(a + b* 
x)**(3/4)*a**3*b*x + 6*(c + d*x)**(1/4)*(a + b*x)**(3/4)*a**2*b**2*x**2 + 
4*(c + d*x)**(1/4)*(a + b*x)**(3/4)*a*b**3*x**3 + (c + d*x)**(1/4)*(a + b* 
x)**(3/4)*b**4*x**4),x)