3.1700 \(\int \frac{1}{(a+b x)^{19/4} \sqrt [4]{c+d x}} \, dx\)

Optimal. Leaf size=136 \[ \frac{512 d^3 (c+d x)^{3/4}}{1155 (a+b x)^{3/4} (b c-a d)^4}-\frac{128 d^2 (c+d x)^{3/4}}{385 (a+b x)^{7/4} (b c-a d)^3}+\frac{16 d (c+d x)^{3/4}}{55 (a+b x)^{11/4} (b c-a d)^2}-\frac{4 (c+d x)^{3/4}}{15 (a+b x)^{15/4} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.113177, 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 \[ \frac{512 d^3 (c+d x)^{3/4}}{1155 (a+b x)^{3/4} (b c-a d)^4}-\frac{128 d^2 (c+d x)^{3/4}}{385 (a+b x)^{7/4} (b c-a d)^3}+\frac{16 d (c+d x)^{3/4}}{55 (a+b x)^{11/4} (b c-a d)^2}-\frac{4 (c+d x)^{3/4}}{15 (a+b x)^{15/4} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.346, size = 121, normalized size = 0.89 \[ \frac{512 d^{3} \left (c + d x\right )^{\frac{3}{4}}}{1155 \left (a + b x\right )^{\frac{3}{4}} \left (a d - b c\right )^{4}} + \frac{128 d^{2} \left (c + d x\right )^{\frac{3}{4}}}{385 \left (a + b x\right )^{\frac{7}{4}} \left (a d - b c\right )^{3}} + \frac{16 d \left (c + d x\right )^{\frac{3}{4}}}{55 \left (a + b x\right )^{\frac{11}{4}} \left (a d - b c\right )^{2}} + \frac{4 \left (c + d x\right )^{\frac{3}{4}}}{15 \left (a + b x\right )^{\frac{15}{4}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.194258, size = 95, normalized size = 0.7 \[ \frac{4 (c+d x)^{3/4} \left (96 d^2 (a+b x)^2 (a d-b c)+84 d (a+b x) (b c-a d)^2-77 (b c-a d)^3+128 d^3 (a+b x)^3\right )}{1155 (a+b x)^{15/4} (b c-a d)^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 171, normalized size = 1.3 \[{\frac{512\,{x}^{3}{b}^{3}{d}^{3}+1920\,a{b}^{2}{d}^{3}{x}^{2}-384\,{b}^{3}c{d}^{2}{x}^{2}+2640\,{a}^{2}b{d}^{3}x-1440\,a{b}^{2}c{d}^{2}x+336\,{b}^{3}{c}^{2}dx+1540\,{a}^{3}{d}^{3}-1980\,{a}^{2}cb{d}^{2}+1260\,a{b}^{2}{c}^{2}d-308\,{b}^{3}{c}^{3}}{1155\,{a}^{4}{d}^{4}-4620\,{a}^{3}bc{d}^{3}+6930\,{a}^{2}{c}^{2}{b}^{2}{d}^{2}-4620\,a{b}^{3}{c}^{3}d+1155\,{b}^{4}{c}^{4}} \left ( dx+c \right ) ^{{\frac{3}{4}}} \left ( bx+a \right ) ^{-{\frac{15}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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+660*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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{19}{4}}{\left (d x + c\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.216609, size = 544, normalized size = 4. \[ \frac{4 \,{\left (128 \, b^{3} d^{4} x^{4} - 77 \, b^{3} c^{4} + 315 \, a b^{2} c^{3} d - 495 \, a^{2} b c^{2} d^{2} + 385 \, a^{3} c d^{3} + 32 \,{\left (b^{3} c d^{3} + 15 \, a b^{2} d^{4}\right )} x^{3} - 12 \,{\left (b^{3} c^{2} d^{2} - 10 \, a b^{2} c d^{3} - 55 \, a^{2} b d^{4}\right )} x^{2} +{\left (7 \, b^{3} c^{3} d - 45 \, a b^{2} c^{2} d^{2} + 165 \, a^{2} b c d^{3} + 385 \, a^{3} d^{4}\right )} x\right )}}{1155 \,{\left (a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4} +{\left (b^{7} c^{4} - 4 \, a b^{6} c^{3} d + 6 \, a^{2} b^{5} c^{2} d^{2} - 4 \, a^{3} b^{4} c d^{3} + a^{4} b^{3} d^{4}\right )} x^{3} + 3 \,{\left (a b^{6} c^{4} - 4 \, a^{2} b^{5} c^{3} d + 6 \, a^{3} b^{4} c^{2} d^{2} - 4 \, a^{4} b^{3} c d^{3} + a^{5} b^{2} d^{4}\right )} x^{2} + 3 \,{\left (a^{2} b^{5} c^{4} - 4 \, a^{3} b^{4} c^{3} d + 6 \, a^{4} b^{3} c^{2} d^{2} - 4 \, a^{5} b^{2} c d^{3} + a^{6} b d^{4}\right )} x\right )}{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{1}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/1155*(128*b^3*d^4*x^4 - 77*b^3*c^4 + 315*a*b^2*c^3*d - 495*a^2*b*c^2*d^2 + 385
*a^3*c*d^3 + 32*(b^3*c*d^3 + 15*a*b^2*d^4)*x^3 - 12*(b^3*c^2*d^2 - 10*a*b^2*c*d^
3 - 55*a^2*b*d^4)*x^2 + (7*b^3*c^3*d - 45*a*b^2*c^2*d^2 + 165*a^2*b*c*d^3 + 385*
a^3*d^4)*x)/((a^3*b^4*c^4 - 4*a^4*b^3*c^3*d + 6*a^5*b^2*c^2*d^2 - 4*a^6*b*c*d^3
+ a^7*d^4 + (b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4
*b^3*d^4)*x^3 + 3*(a*b^6*c^4 - 4*a^2*b^5*c^3*d + 6*a^3*b^4*c^2*d^2 - 4*a^4*b^3*c
*d^3 + a^5*b^2*d^4)*x^2 + 3*(a^2*b^5*c^4 - 4*a^3*b^4*c^3*d + 6*a^4*b^3*c^2*d^2 -
4*a^5*b^2*c*d^3 + a^6*b*d^4)*x)*(b*x + a)^(3/4)*(d*x + c)^(1/4))

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out