3.1805 \(\int \frac{(a+b x)^{7/6}}{(c+d x)^{37/6}} \, dx\)

Optimal. Leaf size=136 \[ \frac{7776 b^3 (a+b x)^{13/6}}{191425 (c+d x)^{13/6} (b c-a d)^4}+\frac{1296 b^2 (a+b x)^{13/6}}{14725 (c+d x)^{19/6} (b c-a d)^3}+\frac{108 b (a+b x)^{13/6}}{775 (c+d x)^{25/6} (b c-a d)^2}+\frac{6 (a+b x)^{13/6}}{31 (c+d x)^{31/6} (b c-a d)} \]

[Out]

(6*(a + b*x)^(13/6))/(31*(b*c - a*d)*(c + d*x)^(31/6)) + (108*b*(a + b*x)^(13/6)
)/(775*(b*c - a*d)^2*(c + d*x)^(25/6)) + (1296*b^2*(a + b*x)^(13/6))/(14725*(b*c
- a*d)^3*(c + d*x)^(19/6)) + (7776*b^3*(a + b*x)^(13/6))/(191425*(b*c - a*d)^4*
(c + d*x)^(13/6))

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Rubi [A]  time = 0.120551, 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 \[ \frac{7776 b^3 (a+b x)^{13/6}}{191425 (c+d x)^{13/6} (b c-a d)^4}+\frac{1296 b^2 (a+b x)^{13/6}}{14725 (c+d x)^{19/6} (b c-a d)^3}+\frac{108 b (a+b x)^{13/6}}{775 (c+d x)^{25/6} (b c-a d)^2}+\frac{6 (a+b x)^{13/6}}{31 (c+d x)^{31/6} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(7/6)/(c + d*x)^(37/6),x]

[Out]

(6*(a + b*x)^(13/6))/(31*(b*c - a*d)*(c + d*x)^(31/6)) + (108*b*(a + b*x)^(13/6)
)/(775*(b*c - a*d)^2*(c + d*x)^(25/6)) + (1296*b^2*(a + b*x)^(13/6))/(14725*(b*c
- a*d)^3*(c + d*x)^(19/6)) + (7776*b^3*(a + b*x)^(13/6))/(191425*(b*c - a*d)^4*
(c + d*x)^(13/6))

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Rubi in Sympy [A]  time = 20.1018, size = 121, normalized size = 0.89 \[ \frac{7776 b^{3} \left (a + b x\right )^{\frac{13}{6}}}{191425 \left (c + d x\right )^{\frac{13}{6}} \left (a d - b c\right )^{4}} - \frac{1296 b^{2} \left (a + b x\right )^{\frac{13}{6}}}{14725 \left (c + d x\right )^{\frac{19}{6}} \left (a d - b c\right )^{3}} + \frac{108 b \left (a + b x\right )^{\frac{13}{6}}}{775 \left (c + d x\right )^{\frac{25}{6}} \left (a d - b c\right )^{2}} - \frac{6 \left (a + b x\right )^{\frac{13}{6}}}{31 \left (c + d x\right )^{\frac{31}{6}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(7/6)/(d*x+c)**(37/6),x)

[Out]

7776*b**3*(a + b*x)**(13/6)/(191425*(c + d*x)**(13/6)*(a*d - b*c)**4) - 1296*b**
2*(a + b*x)**(13/6)/(14725*(c + d*x)**(19/6)*(a*d - b*c)**3) + 108*b*(a + b*x)**
(13/6)/(775*(c + d*x)**(25/6)*(a*d - b*c)**2) - 6*(a + b*x)**(13/6)/(31*(c + d*x
)**(31/6)*(a*d - b*c))

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Mathematica [A]  time = 0.212472, size = 118, normalized size = 0.87 \[ \frac{6 (a+b x)^{13/6} \left (-6175 a^3 d^3+741 a^2 b d^2 (31 c+6 d x)-39 a b^2 d \left (775 c^2+372 c d x+72 d^2 x^2\right )+b^3 \left (14725 c^3+13950 c^2 d x+6696 c d^2 x^2+1296 d^3 x^3\right )\right )}{191425 (c+d x)^{31/6} (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(7/6)/(c + d*x)^(37/6),x]

[Out]

(6*(a + b*x)^(13/6)*(-6175*a^3*d^3 + 741*a^2*b*d^2*(31*c + 6*d*x) - 39*a*b^2*d*(
775*c^2 + 372*c*d*x + 72*d^2*x^2) + b^3*(14725*c^3 + 13950*c^2*d*x + 6696*c*d^2*
x^2 + 1296*d^3*x^3)))/(191425*(b*c - a*d)^4*(c + d*x)^(31/6))

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Maple [A]  time = 0.012, size = 171, normalized size = 1.3 \[ -{\frac{-7776\,{x}^{3}{b}^{3}{d}^{3}+16848\,a{b}^{2}{d}^{3}{x}^{2}-40176\,{b}^{3}c{d}^{2}{x}^{2}-26676\,{a}^{2}b{d}^{3}x+87048\,a{b}^{2}c{d}^{2}x-83700\,{b}^{3}{c}^{2}dx+37050\,{a}^{3}{d}^{3}-137826\,{a}^{2}cb{d}^{2}+181350\,a{b}^{2}{c}^{2}d-88350\,{b}^{3}{c}^{3}}{191425\,{a}^{4}{d}^{4}-765700\,{a}^{3}bc{d}^{3}+1148550\,{a}^{2}{c}^{2}{b}^{2}{d}^{2}-765700\,a{b}^{3}{c}^{3}d+191425\,{b}^{4}{c}^{4}} \left ( bx+a \right ) ^{{\frac{13}{6}}} \left ( dx+c \right ) ^{-{\frac{31}{6}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(7/6)/(d*x+c)^(37/6),x)

[Out]

-6/191425*(b*x+a)^(13/6)*(-1296*b^3*d^3*x^3+2808*a*b^2*d^3*x^2-6696*b^3*c*d^2*x^
2-4446*a^2*b*d^3*x+14508*a*b^2*c*d^2*x-13950*b^3*c^2*d*x+6175*a^3*d^3-22971*a^2*
b*c*d^2+30225*a*b^2*c^2*d-14725*b^3*c^3)/(d*x+c)^(31/6)/(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. \[ \int \frac{{\left (b x + a\right )}^{\frac{7}{6}}}{{\left (d x + c\right )}^{\frac{37}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/6)/(d*x + c)^(37/6),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(7/6)/(d*x + c)^(37/6), x)

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Fricas [A]  time = 0.23667, size = 876, normalized size = 6.44 \[ \frac{6 \,{\left (1296 \, b^{5} d^{3} x^{5} + 14725 \, a^{2} b^{3} c^{3} - 30225 \, a^{3} b^{2} c^{2} d + 22971 \, a^{4} b c d^{2} - 6175 \, a^{5} d^{3} + 216 \,{\left (31 \, b^{5} c d^{2} - a b^{4} d^{3}\right )} x^{4} + 18 \,{\left (775 \, b^{5} c^{2} d - 62 \, a b^{4} c d^{2} + 7 \, a^{2} b^{3} d^{3}\right )} x^{3} +{\left (14725 \, b^{5} c^{3} - 2325 \, a b^{4} c^{2} d + 651 \, a^{2} b^{3} c d^{2} - 91 \, a^{3} b^{2} d^{3}\right )} x^{2} + 2 \,{\left (14725 \, a b^{4} c^{3} - 23250 \, a^{2} b^{3} c^{2} d + 15717 \, a^{3} b^{2} c d^{2} - 3952 \, a^{4} b d^{3}\right )} x\right )}{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{5}{6}}}{191425 \,{\left (b^{4} c^{10} - 4 \, a b^{3} c^{9} d + 6 \, a^{2} b^{2} c^{8} d^{2} - 4 \, a^{3} b c^{7} d^{3} + a^{4} c^{6} d^{4} +{\left (b^{4} c^{4} d^{6} - 4 \, a b^{3} c^{3} d^{7} + 6 \, a^{2} b^{2} c^{2} d^{8} - 4 \, a^{3} b c d^{9} + a^{4} d^{10}\right )} x^{6} + 6 \,{\left (b^{4} c^{5} d^{5} - 4 \, a b^{3} c^{4} d^{6} + 6 \, a^{2} b^{2} c^{3} d^{7} - 4 \, a^{3} b c^{2} d^{8} + a^{4} c d^{9}\right )} x^{5} + 15 \,{\left (b^{4} c^{6} d^{4} - 4 \, a b^{3} c^{5} d^{5} + 6 \, a^{2} b^{2} c^{4} d^{6} - 4 \, a^{3} b c^{3} d^{7} + a^{4} c^{2} d^{8}\right )} x^{4} + 20 \,{\left (b^{4} c^{7} d^{3} - 4 \, a b^{3} c^{6} d^{4} + 6 \, a^{2} b^{2} c^{5} d^{5} - 4 \, a^{3} b c^{4} d^{6} + a^{4} c^{3} d^{7}\right )} x^{3} + 15 \,{\left (b^{4} c^{8} d^{2} - 4 \, a b^{3} c^{7} d^{3} + 6 \, a^{2} b^{2} c^{6} d^{4} - 4 \, a^{3} b c^{5} d^{5} + a^{4} c^{4} d^{6}\right )} x^{2} + 6 \,{\left (b^{4} c^{9} d - 4 \, a b^{3} c^{8} d^{2} + 6 \, a^{2} b^{2} c^{7} d^{3} - 4 \, a^{3} b c^{6} d^{4} + a^{4} c^{5} d^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/6)/(d*x + c)^(37/6),x, algorithm="fricas")

[Out]

6/191425*(1296*b^5*d^3*x^5 + 14725*a^2*b^3*c^3 - 30225*a^3*b^2*c^2*d + 22971*a^4
*b*c*d^2 - 6175*a^5*d^3 + 216*(31*b^5*c*d^2 - a*b^4*d^3)*x^4 + 18*(775*b^5*c^2*d
- 62*a*b^4*c*d^2 + 7*a^2*b^3*d^3)*x^3 + (14725*b^5*c^3 - 2325*a*b^4*c^2*d + 651
*a^2*b^3*c*d^2 - 91*a^3*b^2*d^3)*x^2 + 2*(14725*a*b^4*c^3 - 23250*a^2*b^3*c^2*d
+ 15717*a^3*b^2*c*d^2 - 3952*a^4*b*d^3)*x)*(b*x + a)^(1/6)*(d*x + c)^(5/6)/(b^4*
c^10 - 4*a*b^3*c^9*d + 6*a^2*b^2*c^8*d^2 - 4*a^3*b*c^7*d^3 + a^4*c^6*d^4 + (b^4*
c^4*d^6 - 4*a*b^3*c^3*d^7 + 6*a^2*b^2*c^2*d^8 - 4*a^3*b*c*d^9 + a^4*d^10)*x^6 +
6*(b^4*c^5*d^5 - 4*a*b^3*c^4*d^6 + 6*a^2*b^2*c^3*d^7 - 4*a^3*b*c^2*d^8 + a^4*c*d
^9)*x^5 + 15*(b^4*c^6*d^4 - 4*a*b^3*c^5*d^5 + 6*a^2*b^2*c^4*d^6 - 4*a^3*b*c^3*d^
7 + a^4*c^2*d^8)*x^4 + 20*(b^4*c^7*d^3 - 4*a*b^3*c^6*d^4 + 6*a^2*b^2*c^5*d^5 - 4
*a^3*b*c^4*d^6 + a^4*c^3*d^7)*x^3 + 15*(b^4*c^8*d^2 - 4*a*b^3*c^7*d^3 + 6*a^2*b^
2*c^6*d^4 - 4*a^3*b*c^5*d^5 + a^4*c^4*d^6)*x^2 + 6*(b^4*c^9*d - 4*a*b^3*c^8*d^2
+ 6*a^2*b^2*c^7*d^3 - 4*a^3*b*c^6*d^4 + a^4*c^5*d^5)*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((b*x+a)**(7/6)/(d*x+c)**(37/6),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((b*x + a)^(7/6)/(d*x + c)^(37/6),x, algorithm="giac")

[Out]

Timed out