3.1434 \(\int \frac{(a+b x)^5}{(c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=152 \[ -\frac{2 b^4 (c+d x)^{5/2} (b c-a d)}{d^6}+\frac{20 b^3 (c+d x)^{3/2} (b c-a d)^2}{3 d^6}-\frac{20 b^2 \sqrt{c+d x} (b c-a d)^3}{d^6}-\frac{10 b (b c-a d)^4}{d^6 \sqrt{c+d x}}+\frac{2 (b c-a d)^5}{3 d^6 (c+d x)^{3/2}}+\frac{2 b^5 (c+d x)^{7/2}}{7 d^6} \]

[Out]

(2*(b*c - a*d)^5)/(3*d^6*(c + d*x)^(3/2)) - (10*b*(b*c - a*d)^4)/(d^6*Sqrt[c + d
*x]) - (20*b^2*(b*c - a*d)^3*Sqrt[c + d*x])/d^6 + (20*b^3*(b*c - a*d)^2*(c + d*x
)^(3/2))/(3*d^6) - (2*b^4*(b*c - a*d)*(c + d*x)^(5/2))/d^6 + (2*b^5*(c + d*x)^(7
/2))/(7*d^6)

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Rubi [A]  time = 0.148486, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{2 b^4 (c+d x)^{5/2} (b c-a d)}{d^6}+\frac{20 b^3 (c+d x)^{3/2} (b c-a d)^2}{3 d^6}-\frac{20 b^2 \sqrt{c+d x} (b c-a d)^3}{d^6}-\frac{10 b (b c-a d)^4}{d^6 \sqrt{c+d x}}+\frac{2 (b c-a d)^5}{3 d^6 (c+d x)^{3/2}}+\frac{2 b^5 (c+d x)^{7/2}}{7 d^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^5/(c + d*x)^(5/2),x]

[Out]

(2*(b*c - a*d)^5)/(3*d^6*(c + d*x)^(3/2)) - (10*b*(b*c - a*d)^4)/(d^6*Sqrt[c + d
*x]) - (20*b^2*(b*c - a*d)^3*Sqrt[c + d*x])/d^6 + (20*b^3*(b*c - a*d)^2*(c + d*x
)^(3/2))/(3*d^6) - (2*b^4*(b*c - a*d)*(c + d*x)^(5/2))/d^6 + (2*b^5*(c + d*x)^(7
/2))/(7*d^6)

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Rubi in Sympy [A]  time = 38.031, size = 141, normalized size = 0.93 \[ \frac{2 b^{5} \left (c + d x\right )^{\frac{7}{2}}}{7 d^{6}} + \frac{2 b^{4} \left (c + d x\right )^{\frac{5}{2}} \left (a d - b c\right )}{d^{6}} + \frac{20 b^{3} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}}{3 d^{6}} + \frac{20 b^{2} \sqrt{c + d x} \left (a d - b c\right )^{3}}{d^{6}} - \frac{10 b \left (a d - b c\right )^{4}}{d^{6} \sqrt{c + d x}} - \frac{2 \left (a d - b c\right )^{5}}{3 d^{6} \left (c + d x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5/(d*x+c)**(5/2),x)

[Out]

2*b**5*(c + d*x)**(7/2)/(7*d**6) + 2*b**4*(c + d*x)**(5/2)*(a*d - b*c)/d**6 + 20
*b**3*(c + d*x)**(3/2)*(a*d - b*c)**2/(3*d**6) + 20*b**2*sqrt(c + d*x)*(a*d - b*
c)**3/d**6 - 10*b*(a*d - b*c)**4/(d**6*sqrt(c + d*x)) - 2*(a*d - b*c)**5/(3*d**6
*(c + d*x)**(3/2))

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Mathematica [A]  time = 0.284035, size = 157, normalized size = 1.03 \[ \frac{2 \sqrt{c+d x} \left (b^3 d x \left (70 a^2 d^2-98 a b c d+37 b^2 c^2\right )+b^2 \left (210 a^3 d^3-560 a^2 b c d^2+511 a b^2 c^2 d-158 b^3 c^3\right )-3 b^4 d^2 x^2 (4 b c-7 a d)-\frac{105 b (b c-a d)^4}{c+d x}+\frac{7 (b c-a d)^5}{(c+d x)^2}+3 b^5 d^3 x^3\right )}{21 d^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^5/(c + d*x)^(5/2),x]

[Out]

(2*Sqrt[c + d*x]*(b^2*(-158*b^3*c^3 + 511*a*b^2*c^2*d - 560*a^2*b*c*d^2 + 210*a^
3*d^3) + b^3*d*(37*b^2*c^2 - 98*a*b*c*d + 70*a^2*d^2)*x - 3*b^4*d^2*(4*b*c - 7*a
*d)*x^2 + 3*b^5*d^3*x^3 + (7*(b*c - a*d)^5)/(c + d*x)^2 - (105*b*(b*c - a*d)^4)/
(c + d*x)))/(21*d^6)

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Maple [B]  time = 0.009, size = 273, normalized size = 1.8 \[ -{\frac{-6\,{b}^{5}{x}^{5}{d}^{5}-42\,a{b}^{4}{d}^{5}{x}^{4}+12\,{b}^{5}c{d}^{4}{x}^{4}-140\,{a}^{2}{b}^{3}{d}^{5}{x}^{3}+112\,a{b}^{4}c{d}^{4}{x}^{3}-32\,{b}^{5}{c}^{2}{d}^{3}{x}^{3}-420\,{a}^{3}{b}^{2}{d}^{5}{x}^{2}+840\,{a}^{2}{b}^{3}c{d}^{4}{x}^{2}-672\,a{b}^{4}{c}^{2}{d}^{3}{x}^{2}+192\,{b}^{5}{c}^{3}{d}^{2}{x}^{2}+210\,{a}^{4}b{d}^{5}x-1680\,{a}^{3}{b}^{2}c{d}^{4}x+3360\,{a}^{2}{b}^{3}{c}^{2}{d}^{3}x-2688\,a{b}^{4}{c}^{3}{d}^{2}x+768\,{b}^{5}{c}^{4}dx+14\,{a}^{5}{d}^{5}+140\,{a}^{4}bc{d}^{4}-1120\,{a}^{3}{b}^{2}{c}^{2}{d}^{3}+2240\,{a}^{2}{b}^{3}{c}^{3}{d}^{2}-1792\,a{b}^{4}{c}^{4}d+512\,{b}^{5}{c}^{5}}{21\,{d}^{6}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5/(d*x+c)^(5/2),x)

[Out]

-2/21/(d*x+c)^(3/2)*(-3*b^5*d^5*x^5-21*a*b^4*d^5*x^4+6*b^5*c*d^4*x^4-70*a^2*b^3*
d^5*x^3+56*a*b^4*c*d^4*x^3-16*b^5*c^2*d^3*x^3-210*a^3*b^2*d^5*x^2+420*a^2*b^3*c*
d^4*x^2-336*a*b^4*c^2*d^3*x^2+96*b^5*c^3*d^2*x^2+105*a^4*b*d^5*x-840*a^3*b^2*c*d
^4*x+1680*a^2*b^3*c^2*d^3*x-1344*a*b^4*c^3*d^2*x+384*b^5*c^4*d*x+7*a^5*d^5+70*a^
4*b*c*d^4-560*a^3*b^2*c^2*d^3+1120*a^2*b^3*c^3*d^2-896*a*b^4*c^4*d+256*b^5*c^5)/
d^6

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Maxima [A]  time = 1.35839, size = 358, normalized size = 2.36 \[ \frac{2 \,{\left (\frac{3 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{5} - 21 \,{\left (b^{5} c - a b^{4} d\right )}{\left (d x + c\right )}^{\frac{5}{2}} + 70 \,{\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )}{\left (d x + c\right )}^{\frac{3}{2}} - 210 \,{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \sqrt{d x + c}}{d^{5}} + \frac{7 \,{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5} - 15 \,{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )}{\left (d x + c\right )}\right )}}{{\left (d x + c\right )}^{\frac{3}{2}} d^{5}}\right )}}{21 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(d*x + c)^(5/2),x, algorithm="maxima")

[Out]

2/21*((3*(d*x + c)^(7/2)*b^5 - 21*(b^5*c - a*b^4*d)*(d*x + c)^(5/2) + 70*(b^5*c^
2 - 2*a*b^4*c*d + a^2*b^3*d^2)*(d*x + c)^(3/2) - 210*(b^5*c^3 - 3*a*b^4*c^2*d +
3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*sqrt(d*x + c))/d^5 + 7*(b^5*c^5 - 5*a*b^4*c^4*d +
10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5 - 15*(b^5*c^4
- 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*(d*x + c))/(
(d*x + c)^(3/2)*d^5))/d

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Fricas [A]  time = 0.206617, size = 367, normalized size = 2.41 \[ \frac{2 \,{\left (3 \, b^{5} d^{5} x^{5} - 256 \, b^{5} c^{5} + 896 \, a b^{4} c^{4} d - 1120 \, a^{2} b^{3} c^{3} d^{2} + 560 \, a^{3} b^{2} c^{2} d^{3} - 70 \, a^{4} b c d^{4} - 7 \, a^{5} d^{5} - 3 \,{\left (2 \, b^{5} c d^{4} - 7 \, a b^{4} d^{5}\right )} x^{4} + 2 \,{\left (8 \, b^{5} c^{2} d^{3} - 28 \, a b^{4} c d^{4} + 35 \, a^{2} b^{3} d^{5}\right )} x^{3} - 6 \,{\left (16 \, b^{5} c^{3} d^{2} - 56 \, a b^{4} c^{2} d^{3} + 70 \, a^{2} b^{3} c d^{4} - 35 \, a^{3} b^{2} d^{5}\right )} x^{2} - 3 \,{\left (128 \, b^{5} c^{4} d - 448 \, a b^{4} c^{3} d^{2} + 560 \, a^{2} b^{3} c^{2} d^{3} - 280 \, a^{3} b^{2} c d^{4} + 35 \, a^{4} b d^{5}\right )} x\right )}}{21 \,{\left (d^{7} x + c d^{6}\right )} \sqrt{d x + c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(d*x + c)^(5/2),x, algorithm="fricas")

[Out]

2/21*(3*b^5*d^5*x^5 - 256*b^5*c^5 + 896*a*b^4*c^4*d - 1120*a^2*b^3*c^3*d^2 + 560
*a^3*b^2*c^2*d^3 - 70*a^4*b*c*d^4 - 7*a^5*d^5 - 3*(2*b^5*c*d^4 - 7*a*b^4*d^5)*x^
4 + 2*(8*b^5*c^2*d^3 - 28*a*b^4*c*d^4 + 35*a^2*b^3*d^5)*x^3 - 6*(16*b^5*c^3*d^2
- 56*a*b^4*c^2*d^3 + 70*a^2*b^3*c*d^4 - 35*a^3*b^2*d^5)*x^2 - 3*(128*b^5*c^4*d -
448*a*b^4*c^3*d^2 + 560*a^2*b^3*c^2*d^3 - 280*a^3*b^2*c*d^4 + 35*a^4*b*d^5)*x)/
((d^7*x + c*d^6)*sqrt(d*x + c))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{5}}{\left (c + d x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5/(d*x+c)**(5/2),x)

[Out]

Integral((a + b*x)**5/(c + d*x)**(5/2), x)

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GIAC/XCAS [A]  time = 0.230209, size = 452, normalized size = 2.97 \[ -\frac{2 \,{\left (15 \,{\left (d x + c\right )} b^{5} c^{4} - b^{5} c^{5} - 60 \,{\left (d x + c\right )} a b^{4} c^{3} d + 5 \, a b^{4} c^{4} d + 90 \,{\left (d x + c\right )} a^{2} b^{3} c^{2} d^{2} - 10 \, a^{2} b^{3} c^{3} d^{2} - 60 \,{\left (d x + c\right )} a^{3} b^{2} c d^{3} + 10 \, a^{3} b^{2} c^{2} d^{3} + 15 \,{\left (d x + c\right )} a^{4} b d^{4} - 5 \, a^{4} b c d^{4} + a^{5} d^{5}\right )}}{3 \,{\left (d x + c\right )}^{\frac{3}{2}} d^{6}} + \frac{2 \,{\left (3 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{5} d^{36} - 21 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{5} c d^{36} + 70 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{5} c^{2} d^{36} - 210 \, \sqrt{d x + c} b^{5} c^{3} d^{36} + 21 \,{\left (d x + c\right )}^{\frac{5}{2}} a b^{4} d^{37} - 140 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{4} c d^{37} + 630 \, \sqrt{d x + c} a b^{4} c^{2} d^{37} + 70 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} b^{3} d^{38} - 630 \, \sqrt{d x + c} a^{2} b^{3} c d^{38} + 210 \, \sqrt{d x + c} a^{3} b^{2} d^{39}\right )}}{21 \, d^{42}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(d*x + c)^(5/2),x, algorithm="giac")

[Out]

-2/3*(15*(d*x + c)*b^5*c^4 - b^5*c^5 - 60*(d*x + c)*a*b^4*c^3*d + 5*a*b^4*c^4*d
+ 90*(d*x + c)*a^2*b^3*c^2*d^2 - 10*a^2*b^3*c^3*d^2 - 60*(d*x + c)*a^3*b^2*c*d^3
+ 10*a^3*b^2*c^2*d^3 + 15*(d*x + c)*a^4*b*d^4 - 5*a^4*b*c*d^4 + a^5*d^5)/((d*x
+ c)^(3/2)*d^6) + 2/21*(3*(d*x + c)^(7/2)*b^5*d^36 - 21*(d*x + c)^(5/2)*b^5*c*d^
36 + 70*(d*x + c)^(3/2)*b^5*c^2*d^36 - 210*sqrt(d*x + c)*b^5*c^3*d^36 + 21*(d*x
+ c)^(5/2)*a*b^4*d^37 - 140*(d*x + c)^(3/2)*a*b^4*c*d^37 + 630*sqrt(d*x + c)*a*b
^4*c^2*d^37 + 70*(d*x + c)^(3/2)*a^2*b^3*d^38 - 630*sqrt(d*x + c)*a^2*b^3*c*d^38
+ 210*sqrt(d*x + c)*a^3*b^2*d^39)/d^42