3.1684 \(\int \frac{(c+d x)^{5/4}}{(a+b x)^{17/4}} \, dx\)

Optimal. Leaf size=66 \[ \frac{16 d (c+d x)^{9/4}}{117 (a+b x)^{9/4} (b c-a d)^2}-\frac{4 (c+d x)^{9/4}}{13 (a+b x)^{13/4} (b c-a d)} \]

[Out]

(-4*(c + d*x)^(9/4))/(13*(b*c - a*d)*(a + b*x)^(13/4)) + (16*d*(c + d*x)^(9/4))/
(117*(b*c - a*d)^2*(a + b*x)^(9/4))

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Rubi [A]  time = 0.0471783, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{16 d (c+d x)^{9/4}}{117 (a+b x)^{9/4} (b c-a d)^2}-\frac{4 (c+d x)^{9/4}}{13 (a+b x)^{13/4} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(5/4)/(a + b*x)^(17/4),x]

[Out]

(-4*(c + d*x)^(9/4))/(13*(b*c - a*d)*(a + b*x)^(13/4)) + (16*d*(c + d*x)^(9/4))/
(117*(b*c - a*d)^2*(a + b*x)^(9/4))

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Rubi in Sympy [A]  time = 6.76898, size = 56, normalized size = 0.85 \[ \frac{16 d \left (c + d x\right )^{\frac{9}{4}}}{117 \left (a + b x\right )^{\frac{9}{4}} \left (a d - b c\right )^{2}} + \frac{4 \left (c + d x\right )^{\frac{9}{4}}}{13 \left (a + b x\right )^{\frac{13}{4}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(5/4)/(b*x+a)**(17/4),x)

[Out]

16*d*(c + d*x)**(9/4)/(117*(a + b*x)**(9/4)*(a*d - b*c)**2) + 4*(c + d*x)**(9/4)
/(13*(a + b*x)**(13/4)*(a*d - b*c))

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Mathematica [A]  time = 0.104648, size = 46, normalized size = 0.7 \[ \frac{4 (c+d x)^{9/4} (13 a d-9 b c+4 b d x)}{117 (a+b x)^{13/4} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(5/4)/(a + b*x)^(17/4),x]

[Out]

(4*(c + d*x)^(9/4)*(-9*b*c + 13*a*d + 4*b*d*x))/(117*(b*c - a*d)^2*(a + b*x)^(13
/4))

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Maple [A]  time = 0.008, size = 54, normalized size = 0.8 \[{\frac{16\,bdx+52\,ad-36\,bc}{117\,{a}^{2}{d}^{2}-234\,abcd+117\,{b}^{2}{c}^{2}} \left ( dx+c \right ) ^{{\frac{9}{4}}} \left ( bx+a \right ) ^{-{\frac{13}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/4)/(b*x+a)^(17/4),x)

[Out]

4/117*(d*x+c)^(9/4)*(4*b*d*x+13*a*d-9*b*c)/(b*x+a)^(13/4)/(a^2*d^2-2*a*b*c*d+b^2
*c^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{5}{4}}}{{\left (b x + a\right )}^{\frac{17}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/4)/(b*x + a)^(17/4),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(5/4)/(b*x + a)^(17/4), x)

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Fricas [A]  time = 0.262955, size = 317, normalized size = 4.8 \[ \frac{4 \,{\left (4 \, b d^{3} x^{3} - 9 \, b c^{3} + 13 \, a c^{2} d -{\left (b c d^{2} - 13 \, a d^{3}\right )} x^{2} - 2 \,{\left (7 \, b c^{2} d - 13 \, a c d^{2}\right )} x\right )}{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{1}{4}}}{117 \,{\left (a^{4} b^{2} c^{2} - 2 \, a^{5} b c d + a^{6} d^{2} +{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{4} + 4 \,{\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x^{3} + 6 \,{\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} x^{2} + 4 \,{\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/4)/(b*x + a)^(17/4),x, algorithm="fricas")

[Out]

4/117*(4*b*d^3*x^3 - 9*b*c^3 + 13*a*c^2*d - (b*c*d^2 - 13*a*d^3)*x^2 - 2*(7*b*c^
2*d - 13*a*c*d^2)*x)*(b*x + a)^(3/4)*(d*x + c)^(1/4)/(a^4*b^2*c^2 - 2*a^5*b*c*d
+ a^6*d^2 + (b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*x^4 + 4*(a*b^5*c^2 - 2*a^2*b^4
*c*d + a^3*b^3*d^2)*x^3 + 6*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*x^2 + 4*
(a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)*x)

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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((d*x+c)**(5/4)/(b*x+a)**(17/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{5}{4}}}{{\left (b x + a\right )}^{\frac{17}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/4)/(b*x + a)^(17/4),x, algorithm="giac")

[Out]

integrate((d*x + c)^(5/4)/(b*x + a)^(17/4), x)