3.20 \(\int \frac{(a+b x) \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=210 \[ \frac{2 \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (A d^3-2 B c d^2-4 c^3 D+3 c^2 C d\right )\right )}{d^5 \sqrt{c+d x}}+\frac{2 (b c-a d) \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^5 (c+d x)^{3/2}}+\frac{2 \sqrt{c+d x} \left (a d (C d-3 c D)-b \left (-B d^2-6 c^2 D+3 c C d\right )\right )}{d^5}+\frac{2 (c+d x)^{3/2} (a d D-4 b c D+b C d)}{3 d^5}+\frac{2 b D (c+d x)^{5/2}}{5 d^5} \]

[Out]

(2*(b*c - a*d)*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(3*d^5*(c + d*x)^(3/2)) + (2
*(a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(3*c^2*C*d - 2*B*c*d^2 + A*d^3 - 4*c^3*D))
)/(d^5*Sqrt[c + d*x]) + (2*(a*d*(C*d - 3*c*D) - b*(3*c*C*d - B*d^2 - 6*c^2*D))*S
qrt[c + d*x])/d^5 + (2*(b*C*d - 4*b*c*D + a*d*D)*(c + d*x)^(3/2))/(3*d^5) + (2*b
*D*(c + d*x)^(5/2))/(5*d^5)

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Rubi [A]  time = 0.345126, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033 \[ \frac{2 \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (A d^3-2 B c d^2-4 c^3 D+3 c^2 C d\right )\right )}{d^5 \sqrt{c+d x}}+\frac{2 (b c-a d) \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^5 (c+d x)^{3/2}}+\frac{2 \sqrt{c+d x} \left (a d (C d-3 c D)-b \left (-B d^2-6 c^2 D+3 c C d\right )\right )}{d^5}+\frac{2 (c+d x)^{3/2} (a d D-4 b c D+b C d)}{3 d^5}+\frac{2 b D (c+d x)^{5/2}}{5 d^5} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(5/2),x]

[Out]

(2*(b*c - a*d)*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(3*d^5*(c + d*x)^(3/2)) + (2
*(a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(3*c^2*C*d - 2*B*c*d^2 + A*d^3 - 4*c^3*D))
)/(d^5*Sqrt[c + d*x]) + (2*(a*d*(C*d - 3*c*D) - b*(3*c*C*d - B*d^2 - 6*c^2*D))*S
qrt[c + d*x])/d^5 + (2*(b*C*d - 4*b*c*D + a*d*D)*(c + d*x)^(3/2))/(3*d^5) + (2*b
*D*(c + d*x)^(5/2))/(5*d^5)

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Rubi in Sympy [A]  time = 80.6565, size = 226, normalized size = 1.08 \[ \frac{2 D b \left (c + d x\right )^{\frac{5}{2}}}{5 d^{5}} + \frac{2 \left (c + d x\right )^{\frac{3}{2}} \left (C b d + D a d - 4 D b c\right )}{3 d^{5}} + \frac{2 \sqrt{c + d x} \left (B b d^{2} + C a d^{2} - 3 C b c d - 3 D a c d + 6 D b c^{2}\right )}{d^{5}} - \frac{2 \left (A b d^{3} + B a d^{3} - 2 B b c d^{2} - 2 C a c d^{2} + 3 C b c^{2} d + 3 D a c^{2} d - 4 D b c^{3}\right )}{d^{5} \sqrt{c + d x}} - \frac{2 \left (a d - b c\right ) \left (A d^{3} - B c d^{2} + C c^{2} d - D c^{3}\right )}{3 d^{5} \left (c + d x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(5/2),x)

[Out]

2*D*b*(c + d*x)**(5/2)/(5*d**5) + 2*(c + d*x)**(3/2)*(C*b*d + D*a*d - 4*D*b*c)/(
3*d**5) + 2*sqrt(c + d*x)*(B*b*d**2 + C*a*d**2 - 3*C*b*c*d - 3*D*a*c*d + 6*D*b*c
**2)/d**5 - 2*(A*b*d**3 + B*a*d**3 - 2*B*b*c*d**2 - 2*C*a*c*d**2 + 3*C*b*c**2*d
+ 3*D*a*c**2*d - 4*D*b*c**3)/(d**5*sqrt(c + d*x)) - 2*(a*d - b*c)*(A*d**3 - B*c*
d**2 + C*c**2*d - D*c**3)/(3*d**5*(c + d*x)**(3/2))

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Mathematica [A]  time = 0.29109, size = 177, normalized size = 0.84 \[ \frac{2 \left (b \left (-2 c d^3 \left (5 A+x \left (-30 B+15 C x+4 D x^2\right )\right )+d^4 x \left (x \left (15 B+5 C x+3 D x^2\right )-15 A\right )+8 c^2 d^2 (5 B+3 x (2 D x-5 C))+128 c^4 D+c^3 (192 d D x-80 C d)\right )-5 a d \left (d^3 \left (A+3 B x+x^2 (-(3 C+D x))\right )+2 c d^2 (B+3 x (D x-2 C))+16 c^3 D-8 c^2 d (C-3 D x)\right )\right )}{15 d^5 (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(5/2),x]

[Out]

(2*(-5*a*d*(16*c^3*D - 8*c^2*d*(C - 3*D*x) + 2*c*d^2*(B + 3*x*(-2*C + D*x)) + d^
3*(A + 3*B*x - x^2*(3*C + D*x))) + b*(128*c^4*D + c^3*(-80*C*d + 192*d*D*x) + 8*
c^2*d^2*(5*B + 3*x*(-5*C + 2*D*x)) + d^4*x*(-15*A + x*(15*B + 5*C*x + 3*D*x^2))
- 2*c*d^3*(5*A + x*(-30*B + 15*C*x + 4*D*x^2)))))/(15*d^5*(c + d*x)^(3/2))

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Maple [A]  time = 0.007, size = 241, normalized size = 1.2 \[ -{\frac{-6\,Db{x}^{4}{d}^{4}-10\,Cb{d}^{4}{x}^{3}-10\,Da{d}^{4}{x}^{3}+16\,Dbc{d}^{3}{x}^{3}-30\,Bb{d}^{4}{x}^{2}-30\,Ca{d}^{4}{x}^{2}+60\,Cbc{d}^{3}{x}^{2}+60\,Dac{d}^{3}{x}^{2}-96\,Db{c}^{2}{d}^{2}{x}^{2}+30\,Ab{d}^{4}x+30\,Ba{d}^{4}x-120\,Bbc{d}^{3}x-120\,Cac{d}^{3}x+240\,Cb{c}^{2}{d}^{2}x+240\,Da{c}^{2}{d}^{2}x-384\,Db{c}^{3}dx+10\,Aa{d}^{4}+20\,Abc{d}^{3}+20\,Bac{d}^{3}-80\,Bb{c}^{2}{d}^{2}-80\,Ca{c}^{2}{d}^{2}+160\,Cb{c}^{3}d+160\,Da{c}^{3}d-256\,Db{c}^{4}}{15\,{d}^{5}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(5/2),x)

[Out]

-2/15/(d*x+c)^(3/2)*(-3*D*b*d^4*x^4-5*C*b*d^4*x^3-5*D*a*d^4*x^3+8*D*b*c*d^3*x^3-
15*B*b*d^4*x^2-15*C*a*d^4*x^2+30*C*b*c*d^3*x^2+30*D*a*c*d^3*x^2-48*D*b*c^2*d^2*x
^2+15*A*b*d^4*x+15*B*a*d^4*x-60*B*b*c*d^3*x-60*C*a*c*d^3*x+120*C*b*c^2*d^2*x+120
*D*a*c^2*d^2*x-192*D*b*c^3*d*x+5*A*a*d^4+10*A*b*c*d^3+10*B*a*c*d^3-40*B*b*c^2*d^
2-40*C*a*c^2*d^2+80*C*b*c^3*d+80*D*a*c^3*d-128*D*b*c^4)/d^5

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Maxima [A]  time = 1.36125, size = 275, normalized size = 1.31 \[ \frac{2 \,{\left (\frac{3 \,{\left (d x + c\right )}^{\frac{5}{2}} D b - 5 \,{\left (4 \, D b c -{\left (D a + C b\right )} d\right )}{\left (d x + c\right )}^{\frac{3}{2}} + 15 \,{\left (6 \, D b c^{2} - 3 \,{\left (D a + C b\right )} c d +{\left (C a + B b\right )} d^{2}\right )} \sqrt{d x + c}}{d^{4}} - \frac{5 \,{\left (D b c^{4} + A a d^{4} -{\left (D a + C b\right )} c^{3} d +{\left (C a + B b\right )} c^{2} d^{2} -{\left (B a + A b\right )} c d^{3} - 3 \,{\left (4 \, D b c^{3} - 3 \,{\left (D a + C b\right )} c^{2} d + 2 \,{\left (C a + B b\right )} c d^{2} -{\left (B a + A b\right )} d^{3}\right )}{\left (d x + c\right )}\right )}}{{\left (d x + c\right )}^{\frac{3}{2}} d^{4}}\right )}}{15 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^3 + C*x^2 + B*x + A)*(b*x + a)/(d*x + c)^(5/2),x, algorithm="maxima")

[Out]

2/15*((3*(d*x + c)^(5/2)*D*b - 5*(4*D*b*c - (D*a + C*b)*d)*(d*x + c)^(3/2) + 15*
(6*D*b*c^2 - 3*(D*a + C*b)*c*d + (C*a + B*b)*d^2)*sqrt(d*x + c))/d^4 - 5*(D*b*c^
4 + A*a*d^4 - (D*a + C*b)*c^3*d + (C*a + B*b)*c^2*d^2 - (B*a + A*b)*c*d^3 - 3*(4
*D*b*c^3 - 3*(D*a + C*b)*c^2*d + 2*(C*a + B*b)*c*d^2 - (B*a + A*b)*d^3)*(d*x + c
))/((d*x + c)^(3/2)*d^4))/d

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Fricas [A]  time = 0.21685, size = 279, normalized size = 1.33 \[ \frac{2 \,{\left (3 \, D b d^{4} x^{4} + 128 \, D b c^{4} - 5 \, A a d^{4} - 80 \,{\left (D a + C b\right )} c^{3} d + 40 \,{\left (C a + B b\right )} c^{2} d^{2} - 10 \,{\left (B a + A b\right )} c d^{3} -{\left (8 \, D b c d^{3} - 5 \,{\left (D a + C b\right )} d^{4}\right )} x^{3} + 3 \,{\left (16 \, D b c^{2} d^{2} - 10 \,{\left (D a + C b\right )} c d^{3} + 5 \,{\left (C a + B b\right )} d^{4}\right )} x^{2} + 3 \,{\left (64 \, D b c^{3} d - 40 \,{\left (D a + C b\right )} c^{2} d^{2} + 20 \,{\left (C a + B b\right )} c d^{3} - 5 \,{\left (B a + A b\right )} d^{4}\right )} x\right )}}{15 \,{\left (d^{6} x + c d^{5}\right )} \sqrt{d x + c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^3 + C*x^2 + B*x + A)*(b*x + a)/(d*x + c)^(5/2),x, algorithm="fricas")

[Out]

2/15*(3*D*b*d^4*x^4 + 128*D*b*c^4 - 5*A*a*d^4 - 80*(D*a + C*b)*c^3*d + 40*(C*a +
B*b)*c^2*d^2 - 10*(B*a + A*b)*c*d^3 - (8*D*b*c*d^3 - 5*(D*a + C*b)*d^4)*x^3 + 3
*(16*D*b*c^2*d^2 - 10*(D*a + C*b)*c*d^3 + 5*(C*a + B*b)*d^4)*x^2 + 3*(64*D*b*c^3
*d - 40*(D*a + C*b)*c^2*d^2 + 20*(C*a + B*b)*c*d^3 - 5*(B*a + A*b)*d^4)*x)/((d^6
*x + c*d^5)*sqrt(d*x + c))

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Sympy [A]  time = 44.4195, size = 4235, normalized size = 20.17 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(5/2),x)

[Out]

-2*A*a/(3*d*(c + d*x)**(3/2)) + A*b*Piecewise((-4*c/(3*c*d**2*sqrt(c + d*x) + 3*
d**3*x*sqrt(c + d*x)) - 6*d*x/(3*c*d**2*sqrt(c + d*x) + 3*d**3*x*sqrt(c + d*x)),
Ne(d, 0)), (x**2/(2*c**(5/2)), True)) + B*a*Piecewise((-4*c/(3*c*d**2*sqrt(c +
d*x) + 3*d**3*x*sqrt(c + d*x)) - 6*d*x/(3*c*d**2*sqrt(c + d*x) + 3*d**3*x*sqrt(c
+ d*x)), Ne(d, 0)), (x**2/(2*c**(5/2)), True)) + B*b*Piecewise((16*c**2/(3*c*d*
*3*sqrt(c + d*x) + 3*d**4*x*sqrt(c + d*x)) + 24*c*d*x/(3*c*d**3*sqrt(c + d*x) +
3*d**4*x*sqrt(c + d*x)) + 6*d**2*x**2/(3*c*d**3*sqrt(c + d*x) + 3*d**4*x*sqrt(c
+ d*x)), Ne(d, 0)), (x**3/(3*c**(5/2)), True)) + C*a*Piecewise((16*c**2/(3*c*d**
3*sqrt(c + d*x) + 3*d**4*x*sqrt(c + d*x)) + 24*c*d*x/(3*c*d**3*sqrt(c + d*x) + 3
*d**4*x*sqrt(c + d*x)) + 6*d**2*x**2/(3*c*d**3*sqrt(c + d*x) + 3*d**4*x*sqrt(c +
d*x)), Ne(d, 0)), (x**3/(3*c**(5/2)), True)) + C*b*Piecewise((-32*c**3/(3*c*d**
4*sqrt(c + d*x) + 3*d**5*x*sqrt(c + d*x)) - 48*c**2*d*x/(3*c*d**4*sqrt(c + d*x)
+ 3*d**5*x*sqrt(c + d*x)) - 12*c*d**2*x**2/(3*c*d**4*sqrt(c + d*x) + 3*d**5*x*sq
rt(c + d*x)) + 2*d**3*x**3/(3*c*d**4*sqrt(c + d*x) + 3*d**5*x*sqrt(c + d*x)), Ne
(d, 0)), (x**4/(4*c**(5/2)), True)) + D*a*Piecewise((-32*c**3/(3*c*d**4*sqrt(c +
d*x) + 3*d**5*x*sqrt(c + d*x)) - 48*c**2*d*x/(3*c*d**4*sqrt(c + d*x) + 3*d**5*x
*sqrt(c + d*x)) - 12*c*d**2*x**2/(3*c*d**4*sqrt(c + d*x) + 3*d**5*x*sqrt(c + d*x
)) + 2*d**3*x**3/(3*c*d**4*sqrt(c + d*x) + 3*d**5*x*sqrt(c + d*x)), Ne(d, 0)), (
x**4/(4*c**(5/2)), True)) + D*b*(256*c**(85/2)*sqrt(1 + d*x/c)/(15*c**40*d**5 +
150*c**39*d**6*x + 675*c**38*d**7*x**2 + 1800*c**37*d**8*x**3 + 3150*c**36*d**9*
x**4 + 3780*c**35*d**10*x**5 + 3150*c**34*d**11*x**6 + 1800*c**33*d**12*x**7 + 6
75*c**32*d**13*x**8 + 150*c**31*d**14*x**9 + 15*c**30*d**15*x**10) - 256*c**(85/
2)/(15*c**40*d**5 + 150*c**39*d**6*x + 675*c**38*d**7*x**2 + 1800*c**37*d**8*x**
3 + 3150*c**36*d**9*x**4 + 3780*c**35*d**10*x**5 + 3150*c**34*d**11*x**6 + 1800*
c**33*d**12*x**7 + 675*c**32*d**13*x**8 + 150*c**31*d**14*x**9 + 15*c**30*d**15*
x**10) + 2432*c**(83/2)*d*x*sqrt(1 + d*x/c)/(15*c**40*d**5 + 150*c**39*d**6*x +
675*c**38*d**7*x**2 + 1800*c**37*d**8*x**3 + 3150*c**36*d**9*x**4 + 3780*c**35*d
**10*x**5 + 3150*c**34*d**11*x**6 + 1800*c**33*d**12*x**7 + 675*c**32*d**13*x**8
+ 150*c**31*d**14*x**9 + 15*c**30*d**15*x**10) - 2560*c**(83/2)*d*x/(15*c**40*d
**5 + 150*c**39*d**6*x + 675*c**38*d**7*x**2 + 1800*c**37*d**8*x**3 + 3150*c**36
*d**9*x**4 + 3780*c**35*d**10*x**5 + 3150*c**34*d**11*x**6 + 1800*c**33*d**12*x*
*7 + 675*c**32*d**13*x**8 + 150*c**31*d**14*x**9 + 15*c**30*d**15*x**10) + 10336
*c**(81/2)*d**2*x**2*sqrt(1 + d*x/c)/(15*c**40*d**5 + 150*c**39*d**6*x + 675*c**
38*d**7*x**2 + 1800*c**37*d**8*x**3 + 3150*c**36*d**9*x**4 + 3780*c**35*d**10*x*
*5 + 3150*c**34*d**11*x**6 + 1800*c**33*d**12*x**7 + 675*c**32*d**13*x**8 + 150*
c**31*d**14*x**9 + 15*c**30*d**15*x**10) - 11520*c**(81/2)*d**2*x**2/(15*c**40*d
**5 + 150*c**39*d**6*x + 675*c**38*d**7*x**2 + 1800*c**37*d**8*x**3 + 3150*c**36
*d**9*x**4 + 3780*c**35*d**10*x**5 + 3150*c**34*d**11*x**6 + 1800*c**33*d**12*x*
*7 + 675*c**32*d**13*x**8 + 150*c**31*d**14*x**9 + 15*c**30*d**15*x**10) + 25840
*c**(79/2)*d**3*x**3*sqrt(1 + d*x/c)/(15*c**40*d**5 + 150*c**39*d**6*x + 675*c**
38*d**7*x**2 + 1800*c**37*d**8*x**3 + 3150*c**36*d**9*x**4 + 3780*c**35*d**10*x*
*5 + 3150*c**34*d**11*x**6 + 1800*c**33*d**12*x**7 + 675*c**32*d**13*x**8 + 150*
c**31*d**14*x**9 + 15*c**30*d**15*x**10) - 30720*c**(79/2)*d**3*x**3/(15*c**40*d
**5 + 150*c**39*d**6*x + 675*c**38*d**7*x**2 + 1800*c**37*d**8*x**3 + 3150*c**36
*d**9*x**4 + 3780*c**35*d**10*x**5 + 3150*c**34*d**11*x**6 + 1800*c**33*d**12*x*
*7 + 675*c**32*d**13*x**8 + 150*c**31*d**14*x**9 + 15*c**30*d**15*x**10) + 41990
*c**(77/2)*d**4*x**4*sqrt(1 + d*x/c)/(15*c**40*d**5 + 150*c**39*d**6*x + 675*c**
38*d**7*x**2 + 1800*c**37*d**8*x**3 + 3150*c**36*d**9*x**4 + 3780*c**35*d**10*x*
*5 + 3150*c**34*d**11*x**6 + 1800*c**33*d**12*x**7 + 675*c**32*d**13*x**8 + 150*
c**31*d**14*x**9 + 15*c**30*d**15*x**10) - 53760*c**(77/2)*d**4*x**4/(15*c**40*d
**5 + 150*c**39*d**6*x + 675*c**38*d**7*x**2 + 1800*c**37*d**8*x**3 + 3150*c**36
*d**9*x**4 + 3780*c**35*d**10*x**5 + 3150*c**34*d**11*x**6 + 1800*c**33*d**12*x*
*7 + 675*c**32*d**13*x**8 + 150*c**31*d**14*x**9 + 15*c**30*d**15*x**10) + 46192
*c**(75/2)*d**5*x**5*sqrt(1 + d*x/c)/(15*c**40*d**5 + 150*c**39*d**6*x + 675*c**
38*d**7*x**2 + 1800*c**37*d**8*x**3 + 3150*c**36*d**9*x**4 + 3780*c**35*d**10*x*
*5 + 3150*c**34*d**11*x**6 + 1800*c**33*d**12*x**7 + 675*c**32*d**13*x**8 + 150*
c**31*d**14*x**9 + 15*c**30*d**15*x**10) - 64512*c**(75/2)*d**5*x**5/(15*c**40*d
**5 + 150*c**39*d**6*x + 675*c**38*d**7*x**2 + 1800*c**37*d**8*x**3 + 3150*c**36
*d**9*x**4 + 3780*c**35*d**10*x**5 + 3150*c**34*d**11*x**6 + 1800*c**33*d**12*x*
*7 + 675*c**32*d**13*x**8 + 150*c**31*d**14*x**9 + 15*c**30*d**15*x**10) + 34664
*c**(73/2)*d**6*x**6*sqrt(1 + d*x/c)/(15*c**40*d**5 + 150*c**39*d**6*x + 675*c**
38*d**7*x**2 + 1800*c**37*d**8*x**3 + 3150*c**36*d**9*x**4 + 3780*c**35*d**10*x*
*5 + 3150*c**34*d**11*x**6 + 1800*c**33*d**12*x**7 + 675*c**32*d**13*x**8 + 150*
c**31*d**14*x**9 + 15*c**30*d**15*x**10) - 53760*c**(73/2)*d**6*x**6/(15*c**40*d
**5 + 150*c**39*d**6*x + 675*c**38*d**7*x**2 + 1800*c**37*d**8*x**3 + 3150*c**36
*d**9*x**4 + 3780*c**35*d**10*x**5 + 3150*c**34*d**11*x**6 + 1800*c**33*d**12*x*
*7 + 675*c**32*d**13*x**8 + 150*c**31*d**14*x**9 + 15*c**30*d**15*x**10) + 17392
*c**(71/2)*d**7*x**7*sqrt(1 + d*x/c)/(15*c**40*d**5 + 150*c**39*d**6*x + 675*c**
38*d**7*x**2 + 1800*c**37*d**8*x**3 + 3150*c**36*d**9*x**4 + 3780*c**35*d**10*x*
*5 + 3150*c**34*d**11*x**6 + 1800*c**33*d**12*x**7 + 675*c**32*d**13*x**8 + 150*
c**31*d**14*x**9 + 15*c**30*d**15*x**10) - 30720*c**(71/2)*d**7*x**7/(15*c**40*d
**5 + 150*c**39*d**6*x + 675*c**38*d**7*x**2 + 1800*c**37*d**8*x**3 + 3150*c**36
*d**9*x**4 + 3780*c**35*d**10*x**5 + 3150*c**34*d**11*x**6 + 1800*c**33*d**12*x*
*7 + 675*c**32*d**13*x**8 + 150*c**31*d**14*x**9 + 15*c**30*d**15*x**10) + 5540*
c**(69/2)*d**8*x**8*sqrt(1 + d*x/c)/(15*c**40*d**5 + 150*c**39*d**6*x + 675*c**3
8*d**7*x**2 + 1800*c**37*d**8*x**3 + 3150*c**36*d**9*x**4 + 3780*c**35*d**10*x**
5 + 3150*c**34*d**11*x**6 + 1800*c**33*d**12*x**7 + 675*c**32*d**13*x**8 + 150*c
**31*d**14*x**9 + 15*c**30*d**15*x**10) - 11520*c**(69/2)*d**8*x**8/(15*c**40*d*
*5 + 150*c**39*d**6*x + 675*c**38*d**7*x**2 + 1800*c**37*d**8*x**3 + 3150*c**36*
d**9*x**4 + 3780*c**35*d**10*x**5 + 3150*c**34*d**11*x**6 + 1800*c**33*d**12*x**
7 + 675*c**32*d**13*x**8 + 150*c**31*d**14*x**9 + 15*c**30*d**15*x**10) + 1040*c
**(67/2)*d**9*x**9*sqrt(1 + d*x/c)/(15*c**40*d**5 + 150*c**39*d**6*x + 675*c**38
*d**7*x**2 + 1800*c**37*d**8*x**3 + 3150*c**36*d**9*x**4 + 3780*c**35*d**10*x**5
+ 3150*c**34*d**11*x**6 + 1800*c**33*d**12*x**7 + 675*c**32*d**13*x**8 + 150*c*
*31*d**14*x**9 + 15*c**30*d**15*x**10) - 2560*c**(67/2)*d**9*x**9/(15*c**40*d**5
+ 150*c**39*d**6*x + 675*c**38*d**7*x**2 + 1800*c**37*d**8*x**3 + 3150*c**36*d*
*9*x**4 + 3780*c**35*d**10*x**5 + 3150*c**34*d**11*x**6 + 1800*c**33*d**12*x**7
+ 675*c**32*d**13*x**8 + 150*c**31*d**14*x**9 + 15*c**30*d**15*x**10) + 136*c**(
65/2)*d**10*x**10*sqrt(1 + d*x/c)/(15*c**40*d**5 + 150*c**39*d**6*x + 675*c**38*
d**7*x**2 + 1800*c**37*d**8*x**3 + 3150*c**36*d**9*x**4 + 3780*c**35*d**10*x**5
+ 3150*c**34*d**11*x**6 + 1800*c**33*d**12*x**7 + 675*c**32*d**13*x**8 + 150*c**
31*d**14*x**9 + 15*c**30*d**15*x**10) - 256*c**(65/2)*d**10*x**10/(15*c**40*d**5
+ 150*c**39*d**6*x + 675*c**38*d**7*x**2 + 1800*c**37*d**8*x**3 + 3150*c**36*d*
*9*x**4 + 3780*c**35*d**10*x**5 + 3150*c**34*d**11*x**6 + 1800*c**33*d**12*x**7
+ 675*c**32*d**13*x**8 + 150*c**31*d**14*x**9 + 15*c**30*d**15*x**10) + 32*c**(6
3/2)*d**11*x**11*sqrt(1 + d*x/c)/(15*c**40*d**5 + 150*c**39*d**6*x + 675*c**38*d
**7*x**2 + 1800*c**37*d**8*x**3 + 3150*c**36*d**9*x**4 + 3780*c**35*d**10*x**5 +
3150*c**34*d**11*x**6 + 1800*c**33*d**12*x**7 + 675*c**32*d**13*x**8 + 150*c**3
1*d**14*x**9 + 15*c**30*d**15*x**10) + 6*c**(61/2)*d**12*x**12*sqrt(1 + d*x/c)/(
15*c**40*d**5 + 150*c**39*d**6*x + 675*c**38*d**7*x**2 + 1800*c**37*d**8*x**3 +
3150*c**36*d**9*x**4 + 3780*c**35*d**10*x**5 + 3150*c**34*d**11*x**6 + 1800*c**3
3*d**12*x**7 + 675*c**32*d**13*x**8 + 150*c**31*d**14*x**9 + 15*c**30*d**15*x**1
0))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.209497, size = 408, normalized size = 1.94 \[ \frac{2 \,{\left (12 \,{\left (d x + c\right )} D b c^{3} - D b c^{4} - 9 \,{\left (d x + c\right )} D a c^{2} d - 9 \,{\left (d x + c\right )} C b c^{2} d + D a c^{3} d + C b c^{3} d + 6 \,{\left (d x + c\right )} C a c d^{2} + 6 \,{\left (d x + c\right )} B b c d^{2} - C a c^{2} d^{2} - B b c^{2} d^{2} - 3 \,{\left (d x + c\right )} B a d^{3} - 3 \,{\left (d x + c\right )} A b d^{3} + B a c d^{3} + A b c d^{3} - A a d^{4}\right )}}{3 \,{\left (d x + c\right )}^{\frac{3}{2}} d^{5}} + \frac{2 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} D b d^{20} - 20 \,{\left (d x + c\right )}^{\frac{3}{2}} D b c d^{20} + 90 \, \sqrt{d x + c} D b c^{2} d^{20} + 5 \,{\left (d x + c\right )}^{\frac{3}{2}} D a d^{21} + 5 \,{\left (d x + c\right )}^{\frac{3}{2}} C b d^{21} - 45 \, \sqrt{d x + c} D a c d^{21} - 45 \, \sqrt{d x + c} C b c d^{21} + 15 \, \sqrt{d x + c} C a d^{22} + 15 \, \sqrt{d x + c} B b d^{22}\right )}}{15 \, d^{25}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^3 + C*x^2 + B*x + A)*(b*x + a)/(d*x + c)^(5/2),x, algorithm="giac")

[Out]

2/3*(12*(d*x + c)*D*b*c^3 - D*b*c^4 - 9*(d*x + c)*D*a*c^2*d - 9*(d*x + c)*C*b*c^
2*d + D*a*c^3*d + C*b*c^3*d + 6*(d*x + c)*C*a*c*d^2 + 6*(d*x + c)*B*b*c*d^2 - C*
a*c^2*d^2 - B*b*c^2*d^2 - 3*(d*x + c)*B*a*d^3 - 3*(d*x + c)*A*b*d^3 + B*a*c*d^3
+ A*b*c*d^3 - A*a*d^4)/((d*x + c)^(3/2)*d^5) + 2/15*(3*(d*x + c)^(5/2)*D*b*d^20
- 20*(d*x + c)^(3/2)*D*b*c*d^20 + 90*sqrt(d*x + c)*D*b*c^2*d^20 + 5*(d*x + c)^(3
/2)*D*a*d^21 + 5*(d*x + c)^(3/2)*C*b*d^21 - 45*sqrt(d*x + c)*D*a*c*d^21 - 45*sqr
t(d*x + c)*C*b*c*d^21 + 15*sqrt(d*x + c)*C*a*d^22 + 15*sqrt(d*x + c)*B*b*d^22)/d
^25