3.1686 \(\int \frac{(c+d x)^{5/4}}{(a+b x)^{25/4}} \, dx\)

Optimal. Leaf size=136 \[ \frac{512 d^3 (c+d x)^{9/4}}{13923 (a+b x)^{9/4} (b c-a d)^4}-\frac{128 d^2 (c+d x)^{9/4}}{1547 (a+b x)^{13/4} (b c-a d)^3}+\frac{16 d (c+d x)^{9/4}}{119 (a+b x)^{17/4} (b c-a d)^2}-\frac{4 (c+d x)^{9/4}}{21 (a+b x)^{21/4} (b c-a d)} \]

[Out]

(-4*(c + d*x)^(9/4))/(21*(b*c - a*d)*(a + b*x)^(21/4)) + (16*d*(c + d*x)^(9/4))/
(119*(b*c - a*d)^2*(a + b*x)^(17/4)) - (128*d^2*(c + d*x)^(9/4))/(1547*(b*c - a*
d)^3*(a + b*x)^(13/4)) + (512*d^3*(c + d*x)^(9/4))/(13923*(b*c - a*d)^4*(a + b*x
)^(9/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.111014, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{512 d^3 (c+d x)^{9/4}}{13923 (a+b x)^{9/4} (b c-a d)^4}-\frac{128 d^2 (c+d x)^{9/4}}{1547 (a+b x)^{13/4} (b c-a d)^3}+\frac{16 d (c+d x)^{9/4}}{119 (a+b x)^{17/4} (b c-a d)^2}-\frac{4 (c+d x)^{9/4}}{21 (a+b x)^{21/4} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(5/4)/(a + b*x)^(25/4),x]

[Out]

(-4*(c + d*x)^(9/4))/(21*(b*c - a*d)*(a + b*x)^(21/4)) + (16*d*(c + d*x)^(9/4))/
(119*(b*c - a*d)^2*(a + b*x)^(17/4)) - (128*d^2*(c + d*x)^(9/4))/(1547*(b*c - a*
d)^3*(a + b*x)^(13/4)) + (512*d^3*(c + d*x)^(9/4))/(13923*(b*c - a*d)^4*(a + b*x
)^(9/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 19.3519, size = 121, normalized size = 0.89 \[ \frac{512 d^{3} \left (c + d x\right )^{\frac{9}{4}}}{13923 \left (a + b x\right )^{\frac{9}{4}} \left (a d - b c\right )^{4}} + \frac{128 d^{2} \left (c + d x\right )^{\frac{9}{4}}}{1547 \left (a + b x\right )^{\frac{13}{4}} \left (a d - b c\right )^{3}} + \frac{16 d \left (c + d x\right )^{\frac{9}{4}}}{119 \left (a + b x\right )^{\frac{17}{4}} \left (a d - b c\right )^{2}} + \frac{4 \left (c + d x\right )^{\frac{9}{4}}}{21 \left (a + b x\right )^{\frac{21}{4}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(5/4)/(b*x+a)**(25/4),x)

[Out]

512*d**3*(c + d*x)**(9/4)/(13923*(a + b*x)**(9/4)*(a*d - b*c)**4) + 128*d**2*(c
+ d*x)**(9/4)/(1547*(a + b*x)**(13/4)*(a*d - b*c)**3) + 16*d*(c + d*x)**(9/4)/(1
19*(a + b*x)**(17/4)*(a*d - b*c)**2) + 4*(c + d*x)**(9/4)/(21*(a + b*x)**(21/4)*
(a*d - b*c))

_______________________________________________________________________________________

Mathematica [A]  time = 0.222961, size = 118, normalized size = 0.87 \[ \frac{4 (c+d x)^{9/4} \left (1547 a^3 d^3+357 a^2 b d^2 (4 d x-9 c)+21 a b^2 d \left (117 c^2-72 c d x+32 d^2 x^2\right )+b^3 \left (-663 c^3+468 c^2 d x-288 c d^2 x^2+128 d^3 x^3\right )\right )}{13923 (a+b x)^{21/4} (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(5/4)/(a + b*x)^(25/4),x]

[Out]

(4*(c + d*x)^(9/4)*(1547*a^3*d^3 + 357*a^2*b*d^2*(-9*c + 4*d*x) + 21*a*b^2*d*(11
7*c^2 - 72*c*d*x + 32*d^2*x^2) + b^3*(-663*c^3 + 468*c^2*d*x - 288*c*d^2*x^2 + 1
28*d^3*x^3)))/(13923*(b*c - a*d)^4*(a + b*x)^(21/4))

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 171, normalized size = 1.3 \[{\frac{512\,{x}^{3}{b}^{3}{d}^{3}+2688\,a{b}^{2}{d}^{3}{x}^{2}-1152\,{b}^{3}c{d}^{2}{x}^{2}+5712\,{a}^{2}b{d}^{3}x-6048\,a{b}^{2}c{d}^{2}x+1872\,{b}^{3}{c}^{2}dx+6188\,{a}^{3}{d}^{3}-12852\,{a}^{2}cb{d}^{2}+9828\,a{b}^{2}{c}^{2}d-2652\,{b}^{3}{c}^{3}}{13923\,{a}^{4}{d}^{4}-55692\,{a}^{3}bc{d}^{3}+83538\,{a}^{2}{c}^{2}{b}^{2}{d}^{2}-55692\,a{b}^{3}{c}^{3}d+13923\,{b}^{4}{c}^{4}} \left ( dx+c \right ) ^{{\frac{9}{4}}} \left ( bx+a \right ) ^{-{\frac{21}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/4)/(b*x+a)^(25/4),x)

[Out]

4/13923*(d*x+c)^(9/4)*(128*b^3*d^3*x^3+672*a*b^2*d^3*x^2-288*b^3*c*d^2*x^2+1428*
a^2*b*d^3*x-1512*a*b^2*c*d^2*x+468*b^3*c^2*d*x+1547*a^3*d^3-3213*a^2*b*c*d^2+245
7*a*b^2*c^2*d-663*b^3*c^3)/(b*x+a)^(21/4)/(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)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{5}{4}}}{{\left (b x + a\right )}^{\frac{25}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/4)/(b*x + a)^(25/4),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(5/4)/(b*x + a)^(25/4), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.407286, size = 876, normalized size = 6.44 \[ \frac{4 \,{\left (128 \, b^{3} d^{5} x^{5} - 663 \, b^{3} c^{5} + 2457 \, a b^{2} c^{4} d - 3213 \, a^{2} b c^{3} d^{2} + 1547 \, a^{3} c^{2} d^{3} - 32 \,{\left (b^{3} c d^{4} - 21 \, a b^{2} d^{5}\right )} x^{4} + 4 \,{\left (5 \, b^{3} c^{2} d^{3} - 42 \, a b^{2} c d^{4} + 357 \, a^{2} b d^{5}\right )} x^{3} -{\left (15 \, b^{3} c^{3} d^{2} - 105 \, a b^{2} c^{2} d^{3} + 357 \, a^{2} b c d^{4} - 1547 \, a^{3} d^{5}\right )} x^{2} - 2 \,{\left (429 \, b^{3} c^{4} d - 1701 \, a b^{2} c^{3} d^{2} + 2499 \, a^{2} b c^{2} d^{3} - 1547 \, a^{3} c d^{4}\right )} x\right )}{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{1}{4}}}{13923 \,{\left (a^{6} b^{4} c^{4} - 4 \, a^{7} b^{3} c^{3} d + 6 \, a^{8} b^{2} c^{2} d^{2} - 4 \, a^{9} b c d^{3} + a^{10} d^{4} +{\left (b^{10} c^{4} - 4 \, a b^{9} c^{3} d + 6 \, a^{2} b^{8} c^{2} d^{2} - 4 \, a^{3} b^{7} c d^{3} + a^{4} b^{6} d^{4}\right )} x^{6} + 6 \,{\left (a b^{9} c^{4} - 4 \, a^{2} b^{8} c^{3} d + 6 \, a^{3} b^{7} c^{2} d^{2} - 4 \, a^{4} b^{6} c d^{3} + a^{5} b^{5} d^{4}\right )} x^{5} + 15 \,{\left (a^{2} b^{8} c^{4} - 4 \, a^{3} b^{7} c^{3} d + 6 \, a^{4} b^{6} c^{2} d^{2} - 4 \, a^{5} b^{5} c d^{3} + a^{6} b^{4} d^{4}\right )} x^{4} + 20 \,{\left (a^{3} b^{7} c^{4} - 4 \, a^{4} b^{6} c^{3} d + 6 \, a^{5} b^{5} c^{2} d^{2} - 4 \, a^{6} b^{4} c d^{3} + a^{7} b^{3} d^{4}\right )} x^{3} + 15 \,{\left (a^{4} b^{6} c^{4} - 4 \, a^{5} b^{5} c^{3} d + 6 \, a^{6} b^{4} c^{2} d^{2} - 4 \, a^{7} b^{3} c d^{3} + a^{8} b^{2} d^{4}\right )} x^{2} + 6 \,{\left (a^{5} b^{5} c^{4} - 4 \, a^{6} b^{4} c^{3} d + 6 \, a^{7} b^{3} c^{2} d^{2} - 4 \, a^{8} b^{2} c d^{3} + a^{9} b d^{4}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/4)/(b*x + a)^(25/4),x, algorithm="fricas")

[Out]

4/13923*(128*b^3*d^5*x^5 - 663*b^3*c^5 + 2457*a*b^2*c^4*d - 3213*a^2*b*c^3*d^2 +
1547*a^3*c^2*d^3 - 32*(b^3*c*d^4 - 21*a*b^2*d^5)*x^4 + 4*(5*b^3*c^2*d^3 - 42*a*
b^2*c*d^4 + 357*a^2*b*d^5)*x^3 - (15*b^3*c^3*d^2 - 105*a*b^2*c^2*d^3 + 357*a^2*b
*c*d^4 - 1547*a^3*d^5)*x^2 - 2*(429*b^3*c^4*d - 1701*a*b^2*c^3*d^2 + 2499*a^2*b*
c^2*d^3 - 1547*a^3*c*d^4)*x)*(b*x + a)^(3/4)*(d*x + c)^(1/4)/(a^6*b^4*c^4 - 4*a^
7*b^3*c^3*d + 6*a^8*b^2*c^2*d^2 - 4*a^9*b*c*d^3 + a^10*d^4 + (b^10*c^4 - 4*a*b^9
*c^3*d + 6*a^2*b^8*c^2*d^2 - 4*a^3*b^7*c*d^3 + a^4*b^6*d^4)*x^6 + 6*(a*b^9*c^4 -
4*a^2*b^8*c^3*d + 6*a^3*b^7*c^2*d^2 - 4*a^4*b^6*c*d^3 + a^5*b^5*d^4)*x^5 + 15*(
a^2*b^8*c^4 - 4*a^3*b^7*c^3*d + 6*a^4*b^6*c^2*d^2 - 4*a^5*b^5*c*d^3 + a^6*b^4*d^
4)*x^4 + 20*(a^3*b^7*c^4 - 4*a^4*b^6*c^3*d + 6*a^5*b^5*c^2*d^2 - 4*a^6*b^4*c*d^3
+ a^7*b^3*d^4)*x^3 + 15*(a^4*b^6*c^4 - 4*a^5*b^5*c^3*d + 6*a^6*b^4*c^2*d^2 - 4*
a^7*b^3*c*d^3 + a^8*b^2*d^4)*x^2 + 6*(a^5*b^5*c^4 - 4*a^6*b^4*c^3*d + 6*a^7*b^3*
c^2*d^2 - 4*a^8*b^2*c*d^3 + a^9*b*d^4)*x)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**(5/4)/(b*x+a)**(25/4),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{5}{4}}}{{\left (b x + a\right )}^{\frac{25}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/4)/(b*x + a)^(25/4),x, algorithm="giac")

[Out]

integrate((d*x + c)^(5/4)/(b*x + a)^(25/4), x)