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

Optimal. Leaf size=210 \[ -\frac{2 \sqrt{c+d x} \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (A d^3-2 B c d^2-4 c^3 D+3 c^2 C d\right )\right )}{d^5}+\frac{2 (b c-a d) \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^5 \sqrt{c+d x}}+\frac{2 (c+d x)^{3/2} \left (a d (C d-3 c D)-b \left (-B d^2-6 c^2 D+3 c C d\right )\right )}{3 d^5}+\frac{2 (c+d x)^{5/2} (a d D-4 b c D+b C d)}{5 d^5}+\frac{2 b D (c+d x)^{7/2}}{7 d^5} \]

[Out]

(2*(b*c - a*d)*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(d^5*Sqrt[c + d*x]) - (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))*Sqr
t[c + d*x])/d^5 + (2*(a*d*(C*d - 3*c*D) - b*(3*c*C*d - B*d^2 - 6*c^2*D))*(c + d*
x)^(3/2))/(3*d^5) + (2*(b*C*d - 4*b*c*D + a*d*D)*(c + d*x)^(5/2))/(5*d^5) + (2*b
*D*(c + d*x)^(7/2))/(7*d^5)

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Rubi [A]  time = 0.342634, 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 \[ -\frac{2 \sqrt{c+d x} \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (A d^3-2 B c d^2-4 c^3 D+3 c^2 C d\right )\right )}{d^5}+\frac{2 (b c-a d) \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^5 \sqrt{c+d x}}+\frac{2 (c+d x)^{3/2} \left (a d (C d-3 c D)-b \left (-B d^2-6 c^2 D+3 c C d\right )\right )}{3 d^5}+\frac{2 (c+d x)^{5/2} (a d D-4 b c D+b C d)}{5 d^5}+\frac{2 b D (c+d x)^{7/2}}{7 d^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*c - a*d)*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(d^5*Sqrt[c + d*x]) - (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))*Sqr
t[c + d*x])/d^5 + (2*(a*d*(C*d - 3*c*D) - b*(3*c*C*d - B*d^2 - 6*c^2*D))*(c + d*
x)^(3/2))/(3*d^5) + (2*(b*C*d - 4*b*c*D + a*d*D)*(c + d*x)^(5/2))/(5*d^5) + (2*b
*D*(c + d*x)^(7/2))/(7*d^5)

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Rubi in Sympy [A]  time = 81.1771, size = 224, normalized size = 1.07 \[ \frac{2 D b \left (c + d x\right )^{\frac{7}{2}}}{7 d^{5}} + \frac{2 \left (c + d x\right )^{\frac{5}{2}} \left (C b d + D a d - 4 D b c\right )}{5 d^{5}} + \frac{2 \left (c + d x\right )^{\frac{3}{2}} \left (B b d^{2} + C a d^{2} - 3 C b c d - 3 D a c d + 6 D b c^{2}\right )}{3 d^{5}} + \frac{2 \sqrt{c + d x} \left (A b d^{3} + B a d^{3} - 2 B b c d^{2} - 2 C a c d^{2} + 3 C b c^{2} d + 3 D a c^{2} d - 4 D b c^{3}\right )}{d^{5}} - \frac{2 \left (a d - b c\right ) \left (A d^{3} - B c d^{2} + C c^{2} d - D c^{3}\right )}{d^{5} \sqrt{c + d x}} \]

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)**(3/2),x)

[Out]

2*D*b*(c + d*x)**(7/2)/(7*d**5) + 2*(c + d*x)**(5/2)*(C*b*d + D*a*d - 4*D*b*c)/(
5*d**5) + 2*(c + d*x)**(3/2)*(B*b*d**2 + C*a*d**2 - 3*C*b*c*d - 3*D*a*c*d + 6*D*
b*c**2)/(3*d**5) + 2*sqrt(c + d*x)*(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 - 2*(a*d - b*c)*(A*d**3 -
B*c*d**2 + C*c**2*d - D*c**3)/(d**5*sqrt(c + d*x))

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Mathematica [A]  time = 0.373705, size = 188, normalized size = 0.9 \[ \frac{14 a d \left (d^3 \left (x \left (15 B+5 C x+3 D x^2\right )-15 A\right )+2 c d^2 (15 B-x (10 C+3 D x))+48 c^3 D-8 c^2 d (5 C-3 D x)\right )+b \left (4 c d^3 (105 A-x (70 B+3 x (7 C+4 D x)))+2 d^4 x (105 A+x (35 B+3 x (7 C+5 D x)))+16 c^2 d^2 (3 x (7 C+2 D x)-35 B)-768 c^4 D+96 c^3 d (7 C-4 D x)\right )}{105 d^5 \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

(14*a*d*(48*c^3*D - 8*c^2*d*(5*C - 3*D*x) + 2*c*d^2*(15*B - x*(10*C + 3*D*x)) +
d^3*(-15*A + x*(15*B + 5*C*x + 3*D*x^2))) + b*(-768*c^4*D + 96*c^3*d*(7*C - 4*D*
x) + 16*c^2*d^2*(-35*B + 3*x*(7*C + 2*D*x)) + 4*c*d^3*(105*A - x*(70*B + 3*x*(7*
C + 4*D*x))) + 2*d^4*x*(105*A + x*(35*B + 3*x*(7*C + 5*D*x)))))/(105*d^5*Sqrt[c
+ d*x])

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Maple [A]  time = 0.008, size = 241, normalized size = 1.2 \[ -{\frac{-30\,Db{x}^{4}{d}^{4}-42\,Cb{d}^{4}{x}^{3}-42\,Da{d}^{4}{x}^{3}+48\,Dbc{d}^{3}{x}^{3}-70\,Bb{d}^{4}{x}^{2}-70\,Ca{d}^{4}{x}^{2}+84\,Cbc{d}^{3}{x}^{2}+84\,Dac{d}^{3}{x}^{2}-96\,Db{c}^{2}{d}^{2}{x}^{2}-210\,Ab{d}^{4}x-210\,Ba{d}^{4}x+280\,Bbc{d}^{3}x+280\,Cac{d}^{3}x-336\,Cb{c}^{2}{d}^{2}x-336\,Da{c}^{2}{d}^{2}x+384\,Db{c}^{3}dx+210\,Aa{d}^{4}-420\,Abc{d}^{3}-420\,Bac{d}^{3}+560\,Bb{c}^{2}{d}^{2}+560\,Ca{c}^{2}{d}^{2}-672\,Cb{c}^{3}d-672\,Da{c}^{3}d+768\,Db{c}^{4}}{105\,{d}^{5}}{\frac{1}{\sqrt{dx+c}}}} \]

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)^(3/2),x)

[Out]

-2/105/(d*x+c)^(1/2)*(-15*D*b*d^4*x^4-21*C*b*d^4*x^3-21*D*a*d^4*x^3+24*D*b*c*d^3
*x^3-35*B*b*d^4*x^2-35*C*a*d^4*x^2+42*C*b*c*d^3*x^2+42*D*a*c*d^3*x^2-48*D*b*c^2*
d^2*x^2-105*A*b*d^4*x-105*B*a*d^4*x+140*B*b*c*d^3*x+140*C*a*c*d^3*x-168*C*b*c^2*
d^2*x-168*D*a*c^2*d^2*x+192*D*b*c^3*d*x+105*A*a*d^4-210*A*b*c*d^3-210*B*a*c*d^3+
280*B*b*c^2*d^2+280*C*a*c^2*d^2-336*C*b*c^3*d-336*D*a*c^3*d+384*D*b*c^4)/d^5

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Maxima [A]  time = 1.35549, size = 278, normalized size = 1.32 \[ \frac{2 \,{\left (\frac{15 \,{\left (d x + c\right )}^{\frac{7}{2}} D b - 21 \,{\left (4 \, D b c -{\left (D a + C b\right )} d\right )}{\left (d x + c\right )}^{\frac{5}{2}} + 35 \,{\left (6 \, D b c^{2} - 3 \,{\left (D a + C b\right )} c d +{\left (C a + B b\right )} d^{2}\right )}{\left (d x + c\right )}^{\frac{3}{2}} - 105 \,{\left (4 \, D b c^{3} - 3 \,{\left (D a + C b\right )} c^{2} d + 2 \,{\left (C a + B b\right )} c d^{2} -{\left (B a + A b\right )} d^{3}\right )} \sqrt{d x + c}}{d^{4}} - \frac{105 \,{\left (D b c^{4} + A a d^{4} -{\left (D a + C b\right )} c^{3} d +{\left (C a + B b\right )} c^{2} d^{2} -{\left (B a + A b\right )} c d^{3}\right )}}{\sqrt{d x + c} d^{4}}\right )}}{105 \, d} \]

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)^(3/2),x, algorithm="maxima")

[Out]

2/105*((15*(d*x + c)^(7/2)*D*b - 21*(4*D*b*c - (D*a + C*b)*d)*(d*x + c)^(5/2) +
35*(6*D*b*c^2 - 3*(D*a + C*b)*c*d + (C*a + B*b)*d^2)*(d*x + c)^(3/2) - 105*(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)*sqrt(d*x +
c))/d^4 - 105*(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)/(sqrt(d*x + c)*d^4))/d

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Fricas [A]  time = 0.212791, size = 265, normalized size = 1.26 \[ \frac{2 \,{\left (15 \, D b d^{4} x^{4} - 384 \, D b c^{4} - 105 \, A a d^{4} + 336 \,{\left (D a + C b\right )} c^{3} d - 280 \,{\left (C a + B b\right )} c^{2} d^{2} + 210 \,{\left (B a + A b\right )} c d^{3} - 3 \,{\left (8 \, D b c d^{3} - 7 \,{\left (D a + C b\right )} d^{4}\right )} x^{3} +{\left (48 \, D b c^{2} d^{2} - 42 \,{\left (D a + C b\right )} c d^{3} + 35 \,{\left (C a + B b\right )} d^{4}\right )} x^{2} -{\left (192 \, D b c^{3} d - 168 \,{\left (D a + C b\right )} c^{2} d^{2} + 140 \,{\left (C a + B b\right )} c d^{3} - 105 \,{\left (B a + A b\right )} d^{4}\right )} x\right )}}{105 \, \sqrt{d x + c} d^{5}} \]

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)^(3/2),x, algorithm="fricas")

[Out]

2/105*(15*D*b*d^4*x^4 - 384*D*b*c^4 - 105*A*a*d^4 + 336*(D*a + C*b)*c^3*d - 280*
(C*a + B*b)*c^2*d^2 + 210*(B*a + A*b)*c*d^3 - 3*(8*D*b*c*d^3 - 7*(D*a + C*b)*d^4
)*x^3 + (48*D*b*c^2*d^2 - 42*(D*a + C*b)*c*d^3 + 35*(C*a + B*b)*d^4)*x^2 - (192*
D*b*c^3*d - 168*(D*a + C*b)*c^2*d^2 + 140*(C*a + B*b)*c*d^3 - 105*(B*a + A*b)*d^
4)*x)/(sqrt(d*x + c)*d^5)

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Sympy [A]  time = 48.6013, size = 7874, normalized size = 37.5 \[ \text{result too large to display} \]

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)**(3/2),x)

[Out]

-2*A*a/(d*sqrt(c + d*x)) + A*b*Piecewise((4*c/(d**2*sqrt(c + d*x)) + 2*x/(d*sqrt
(c + d*x)), Ne(d, 0)), (x**2/(2*c**(3/2)), True)) + B*a*Piecewise((4*c/(d**2*sqr
t(c + d*x)) + 2*x/(d*sqrt(c + d*x)), Ne(d, 0)), (x**2/(2*c**(3/2)), True)) + B*b
*(-16*c**(19/2)*sqrt(1 + d*x/c)/(3*c**8*d**3 + 9*c**7*d**4*x + 9*c**6*d**5*x**2
+ 3*c**5*d**6*x**3) + 16*c**(19/2)/(3*c**8*d**3 + 9*c**7*d**4*x + 9*c**6*d**5*x*
*2 + 3*c**5*d**6*x**3) - 40*c**(17/2)*d*x*sqrt(1 + d*x/c)/(3*c**8*d**3 + 9*c**7*
d**4*x + 9*c**6*d**5*x**2 + 3*c**5*d**6*x**3) + 48*c**(17/2)*d*x/(3*c**8*d**3 +
9*c**7*d**4*x + 9*c**6*d**5*x**2 + 3*c**5*d**6*x**3) - 30*c**(15/2)*d**2*x**2*sq
rt(1 + d*x/c)/(3*c**8*d**3 + 9*c**7*d**4*x + 9*c**6*d**5*x**2 + 3*c**5*d**6*x**3
) + 48*c**(15/2)*d**2*x**2/(3*c**8*d**3 + 9*c**7*d**4*x + 9*c**6*d**5*x**2 + 3*c
**5*d**6*x**3) - 4*c**(13/2)*d**3*x**3*sqrt(1 + d*x/c)/(3*c**8*d**3 + 9*c**7*d**
4*x + 9*c**6*d**5*x**2 + 3*c**5*d**6*x**3) + 16*c**(13/2)*d**3*x**3/(3*c**8*d**3
+ 9*c**7*d**4*x + 9*c**6*d**5*x**2 + 3*c**5*d**6*x**3) + 2*c**(11/2)*d**4*x**4*
sqrt(1 + d*x/c)/(3*c**8*d**3 + 9*c**7*d**4*x + 9*c**6*d**5*x**2 + 3*c**5*d**6*x*
*3)) + C*a*(-16*c**(19/2)*sqrt(1 + d*x/c)/(3*c**8*d**3 + 9*c**7*d**4*x + 9*c**6*
d**5*x**2 + 3*c**5*d**6*x**3) + 16*c**(19/2)/(3*c**8*d**3 + 9*c**7*d**4*x + 9*c*
*6*d**5*x**2 + 3*c**5*d**6*x**3) - 40*c**(17/2)*d*x*sqrt(1 + d*x/c)/(3*c**8*d**3
+ 9*c**7*d**4*x + 9*c**6*d**5*x**2 + 3*c**5*d**6*x**3) + 48*c**(17/2)*d*x/(3*c*
*8*d**3 + 9*c**7*d**4*x + 9*c**6*d**5*x**2 + 3*c**5*d**6*x**3) - 30*c**(15/2)*d*
*2*x**2*sqrt(1 + d*x/c)/(3*c**8*d**3 + 9*c**7*d**4*x + 9*c**6*d**5*x**2 + 3*c**5
*d**6*x**3) + 48*c**(15/2)*d**2*x**2/(3*c**8*d**3 + 9*c**7*d**4*x + 9*c**6*d**5*
x**2 + 3*c**5*d**6*x**3) - 4*c**(13/2)*d**3*x**3*sqrt(1 + d*x/c)/(3*c**8*d**3 +
9*c**7*d**4*x + 9*c**6*d**5*x**2 + 3*c**5*d**6*x**3) + 16*c**(13/2)*d**3*x**3/(3
*c**8*d**3 + 9*c**7*d**4*x + 9*c**6*d**5*x**2 + 3*c**5*d**6*x**3) + 2*c**(11/2)*
d**4*x**4*sqrt(1 + d*x/c)/(3*c**8*d**3 + 9*c**7*d**4*x + 9*c**6*d**5*x**2 + 3*c*
*5*d**6*x**3)) + C*b*(32*c**(45/2)*sqrt(1 + d*x/c)/(5*c**20*d**4 + 30*c**19*d**5
*x + 75*c**18*d**6*x**2 + 100*c**17*d**7*x**3 + 75*c**16*d**8*x**4 + 30*c**15*d*
*9*x**5 + 5*c**14*d**10*x**6) - 32*c**(45/2)/(5*c**20*d**4 + 30*c**19*d**5*x + 7
5*c**18*d**6*x**2 + 100*c**17*d**7*x**3 + 75*c**16*d**8*x**4 + 30*c**15*d**9*x**
5 + 5*c**14*d**10*x**6) + 176*c**(43/2)*d*x*sqrt(1 + d*x/c)/(5*c**20*d**4 + 30*c
**19*d**5*x + 75*c**18*d**6*x**2 + 100*c**17*d**7*x**3 + 75*c**16*d**8*x**4 + 30
*c**15*d**9*x**5 + 5*c**14*d**10*x**6) - 192*c**(43/2)*d*x/(5*c**20*d**4 + 30*c*
*19*d**5*x + 75*c**18*d**6*x**2 + 100*c**17*d**7*x**3 + 75*c**16*d**8*x**4 + 30*
c**15*d**9*x**5 + 5*c**14*d**10*x**6) + 396*c**(41/2)*d**2*x**2*sqrt(1 + d*x/c)/
(5*c**20*d**4 + 30*c**19*d**5*x + 75*c**18*d**6*x**2 + 100*c**17*d**7*x**3 + 75*
c**16*d**8*x**4 + 30*c**15*d**9*x**5 + 5*c**14*d**10*x**6) - 480*c**(41/2)*d**2*
x**2/(5*c**20*d**4 + 30*c**19*d**5*x + 75*c**18*d**6*x**2 + 100*c**17*d**7*x**3
+ 75*c**16*d**8*x**4 + 30*c**15*d**9*x**5 + 5*c**14*d**10*x**6) + 462*c**(39/2)*
d**3*x**3*sqrt(1 + d*x/c)/(5*c**20*d**4 + 30*c**19*d**5*x + 75*c**18*d**6*x**2 +
100*c**17*d**7*x**3 + 75*c**16*d**8*x**4 + 30*c**15*d**9*x**5 + 5*c**14*d**10*x
**6) - 640*c**(39/2)*d**3*x**3/(5*c**20*d**4 + 30*c**19*d**5*x + 75*c**18*d**6*x
**2 + 100*c**17*d**7*x**3 + 75*c**16*d**8*x**4 + 30*c**15*d**9*x**5 + 5*c**14*d*
*10*x**6) + 290*c**(37/2)*d**4*x**4*sqrt(1 + d*x/c)/(5*c**20*d**4 + 30*c**19*d**
5*x + 75*c**18*d**6*x**2 + 100*c**17*d**7*x**3 + 75*c**16*d**8*x**4 + 30*c**15*d
**9*x**5 + 5*c**14*d**10*x**6) - 480*c**(37/2)*d**4*x**4/(5*c**20*d**4 + 30*c**1
9*d**5*x + 75*c**18*d**6*x**2 + 100*c**17*d**7*x**3 + 75*c**16*d**8*x**4 + 30*c*
*15*d**9*x**5 + 5*c**14*d**10*x**6) + 92*c**(35/2)*d**5*x**5*sqrt(1 + d*x/c)/(5*
c**20*d**4 + 30*c**19*d**5*x + 75*c**18*d**6*x**2 + 100*c**17*d**7*x**3 + 75*c**
16*d**8*x**4 + 30*c**15*d**9*x**5 + 5*c**14*d**10*x**6) - 192*c**(35/2)*d**5*x**
5/(5*c**20*d**4 + 30*c**19*d**5*x + 75*c**18*d**6*x**2 + 100*c**17*d**7*x**3 + 7
5*c**16*d**8*x**4 + 30*c**15*d**9*x**5 + 5*c**14*d**10*x**6) + 16*c**(33/2)*d**6
*x**6*sqrt(1 + d*x/c)/(5*c**20*d**4 + 30*c**19*d**5*x + 75*c**18*d**6*x**2 + 100
*c**17*d**7*x**3 + 75*c**16*d**8*x**4 + 30*c**15*d**9*x**5 + 5*c**14*d**10*x**6)
- 32*c**(33/2)*d**6*x**6/(5*c**20*d**4 + 30*c**19*d**5*x + 75*c**18*d**6*x**2 +
100*c**17*d**7*x**3 + 75*c**16*d**8*x**4 + 30*c**15*d**9*x**5 + 5*c**14*d**10*x
**6) + 6*c**(31/2)*d**7*x**7*sqrt(1 + d*x/c)/(5*c**20*d**4 + 30*c**19*d**5*x + 7
5*c**18*d**6*x**2 + 100*c**17*d**7*x**3 + 75*c**16*d**8*x**4 + 30*c**15*d**9*x**
5 + 5*c**14*d**10*x**6) + 2*c**(29/2)*d**8*x**8*sqrt(1 + d*x/c)/(5*c**20*d**4 +
30*c**19*d**5*x + 75*c**18*d**6*x**2 + 100*c**17*d**7*x**3 + 75*c**16*d**8*x**4
+ 30*c**15*d**9*x**5 + 5*c**14*d**10*x**6)) + D*a*(32*c**(45/2)*sqrt(1 + d*x/c)/
(5*c**20*d**4 + 30*c**19*d**5*x + 75*c**18*d**6*x**2 + 100*c**17*d**7*x**3 + 75*
c**16*d**8*x**4 + 30*c**15*d**9*x**5 + 5*c**14*d**10*x**6) - 32*c**(45/2)/(5*c**
20*d**4 + 30*c**19*d**5*x + 75*c**18*d**6*x**2 + 100*c**17*d**7*x**3 + 75*c**16*
d**8*x**4 + 30*c**15*d**9*x**5 + 5*c**14*d**10*x**6) + 176*c**(43/2)*d*x*sqrt(1
+ d*x/c)/(5*c**20*d**4 + 30*c**19*d**5*x + 75*c**18*d**6*x**2 + 100*c**17*d**7*x
**3 + 75*c**16*d**8*x**4 + 30*c**15*d**9*x**5 + 5*c**14*d**10*x**6) - 192*c**(43
/2)*d*x/(5*c**20*d**4 + 30*c**19*d**5*x + 75*c**18*d**6*x**2 + 100*c**17*d**7*x*
*3 + 75*c**16*d**8*x**4 + 30*c**15*d**9*x**5 + 5*c**14*d**10*x**6) + 396*c**(41/
2)*d**2*x**2*sqrt(1 + d*x/c)/(5*c**20*d**4 + 30*c**19*d**5*x + 75*c**18*d**6*x**
2 + 100*c**17*d**7*x**3 + 75*c**16*d**8*x**4 + 30*c**15*d**9*x**5 + 5*c**14*d**1
0*x**6) - 480*c**(41/2)*d**2*x**2/(5*c**20*d**4 + 30*c**19*d**5*x + 75*c**18*d**
6*x**2 + 100*c**17*d**7*x**3 + 75*c**16*d**8*x**4 + 30*c**15*d**9*x**5 + 5*c**14
*d**10*x**6) + 462*c**(39/2)*d**3*x**3*sqrt(1 + d*x/c)/(5*c**20*d**4 + 30*c**19*
d**5*x + 75*c**18*d**6*x**2 + 100*c**17*d**7*x**3 + 75*c**16*d**8*x**4 + 30*c**1
5*d**9*x**5 + 5*c**14*d**10*x**6) - 640*c**(39/2)*d**3*x**3/(5*c**20*d**4 + 30*c
**19*d**5*x + 75*c**18*d**6*x**2 + 100*c**17*d**7*x**3 + 75*c**16*d**8*x**4 + 30
*c**15*d**9*x**5 + 5*c**14*d**10*x**6) + 290*c**(37/2)*d**4*x**4*sqrt(1 + d*x/c)
/(5*c**20*d**4 + 30*c**19*d**5*x + 75*c**18*d**6*x**2 + 100*c**17*d**7*x**3 + 75
*c**16*d**8*x**4 + 30*c**15*d**9*x**5 + 5*c**14*d**10*x**6) - 480*c**(37/2)*d**4
*x**4/(5*c**20*d**4 + 30*c**19*d**5*x + 75*c**18*d**6*x**2 + 100*c**17*d**7*x**3
+ 75*c**16*d**8*x**4 + 30*c**15*d**9*x**5 + 5*c**14*d**10*x**6) + 92*c**(35/2)*
d**5*x**5*sqrt(1 + d*x/c)/(5*c**20*d**4 + 30*c**19*d**5*x + 75*c**18*d**6*x**2 +
100*c**17*d**7*x**3 + 75*c**16*d**8*x**4 + 30*c**15*d**9*x**5 + 5*c**14*d**10*x
**6) - 192*c**(35/2)*d**5*x**5/(5*c**20*d**4 + 30*c**19*d**5*x + 75*c**18*d**6*x
**2 + 100*c**17*d**7*x**3 + 75*c**16*d**8*x**4 + 30*c**15*d**9*x**5 + 5*c**14*d*
*10*x**6) + 16*c**(33/2)*d**6*x**6*sqrt(1 + d*x/c)/(5*c**20*d**4 + 30*c**19*d**5
*x + 75*c**18*d**6*x**2 + 100*c**17*d**7*x**3 + 75*c**16*d**8*x**4 + 30*c**15*d*
*9*x**5 + 5*c**14*d**10*x**6) - 32*c**(33/2)*d**6*x**6/(5*c**20*d**4 + 30*c**19*
d**5*x + 75*c**18*d**6*x**2 + 100*c**17*d**7*x**3 + 75*c**16*d**8*x**4 + 30*c**1
5*d**9*x**5 + 5*c**14*d**10*x**6) + 6*c**(31/2)*d**7*x**7*sqrt(1 + d*x/c)/(5*c**
20*d**4 + 30*c**19*d**5*x + 75*c**18*d**6*x**2 + 100*c**17*d**7*x**3 + 75*c**16*
d**8*x**4 + 30*c**15*d**9*x**5 + 5*c**14*d**10*x**6) + 2*c**(29/2)*d**8*x**8*sqr
t(1 + d*x/c)/(5*c**20*d**4 + 30*c**19*d**5*x + 75*c**18*d**6*x**2 + 100*c**17*d*
*7*x**3 + 75*c**16*d**8*x**4 + 30*c**15*d**9*x**5 + 5*c**14*d**10*x**6)) + D*b*(
-256*c**(87/2)*sqrt(1 + d*x/c)/(35*c**40*d**5 + 350*c**39*d**6*x + 1575*c**38*d*
*7*x**2 + 4200*c**37*d**8*x**3 + 7350*c**36*d**9*x**4 + 8820*c**35*d**10*x**5 +
7350*c**34*d**11*x**6 + 4200*c**33*d**12*x**7 + 1575*c**32*d**13*x**8 + 350*c**3
1*d**14*x**9 + 35*c**30*d**15*x**10) + 256*c**(87/2)/(35*c**40*d**5 + 350*c**39*
d**6*x + 1575*c**38*d**7*x**2 + 4200*c**37*d**8*x**3 + 7350*c**36*d**9*x**4 + 88
20*c**35*d**10*x**5 + 7350*c**34*d**11*x**6 + 4200*c**33*d**12*x**7 + 1575*c**32
*d**13*x**8 + 350*c**31*d**14*x**9 + 35*c**30*d**15*x**10) - 2432*c**(85/2)*d*x*
sqrt(1 + d*x/c)/(35*c**40*d**5 + 350*c**39*d**6*x + 1575*c**38*d**7*x**2 + 4200*
c**37*d**8*x**3 + 7350*c**36*d**9*x**4 + 8820*c**35*d**10*x**5 + 7350*c**34*d**1
1*x**6 + 4200*c**33*d**12*x**7 + 1575*c**32*d**13*x**8 + 350*c**31*d**14*x**9 +
35*c**30*d**15*x**10) + 2560*c**(85/2)*d*x/(35*c**40*d**5 + 350*c**39*d**6*x + 1
575*c**38*d**7*x**2 + 4200*c**37*d**8*x**3 + 7350*c**36*d**9*x**4 + 8820*c**35*d
**10*x**5 + 7350*c**34*d**11*x**6 + 4200*c**33*d**12*x**7 + 1575*c**32*d**13*x**
8 + 350*c**31*d**14*x**9 + 35*c**30*d**15*x**10) - 10336*c**(83/2)*d**2*x**2*sqr
t(1 + d*x/c)/(35*c**40*d**5 + 350*c**39*d**6*x + 1575*c**38*d**7*x**2 + 4200*c**
37*d**8*x**3 + 7350*c**36*d**9*x**4 + 8820*c**35*d**10*x**5 + 7350*c**34*d**11*x
**6 + 4200*c**33*d**12*x**7 + 1575*c**32*d**13*x**8 + 350*c**31*d**14*x**9 + 35*
c**30*d**15*x**10) + 11520*c**(83/2)*d**2*x**2/(35*c**40*d**5 + 350*c**39*d**6*x
+ 1575*c**38*d**7*x**2 + 4200*c**37*d**8*x**3 + 7350*c**36*d**9*x**4 + 8820*c**
35*d**10*x**5 + 7350*c**34*d**11*x**6 + 4200*c**33*d**12*x**7 + 1575*c**32*d**13
*x**8 + 350*c**31*d**14*x**9 + 35*c**30*d**15*x**10) - 25840*c**(81/2)*d**3*x**3
*sqrt(1 + d*x/c)/(35*c**40*d**5 + 350*c**39*d**6*x + 1575*c**38*d**7*x**2 + 4200
*c**37*d**8*x**3 + 7350*c**36*d**9*x**4 + 8820*c**35*d**10*x**5 + 7350*c**34*d**
11*x**6 + 4200*c**33*d**12*x**7 + 1575*c**32*d**13*x**8 + 350*c**31*d**14*x**9 +
35*c**30*d**15*x**10) + 30720*c**(81/2)*d**3*x**3/(35*c**40*d**5 + 350*c**39*d*
*6*x + 1575*c**38*d**7*x**2 + 4200*c**37*d**8*x**3 + 7350*c**36*d**9*x**4 + 8820
*c**35*d**10*x**5 + 7350*c**34*d**11*x**6 + 4200*c**33*d**12*x**7 + 1575*c**32*d
**13*x**8 + 350*c**31*d**14*x**9 + 35*c**30*d**15*x**10) - 41990*c**(79/2)*d**4*
x**4*sqrt(1 + d*x/c)/(35*c**40*d**5 + 350*c**39*d**6*x + 1575*c**38*d**7*x**2 +
4200*c**37*d**8*x**3 + 7350*c**36*d**9*x**4 + 8820*c**35*d**10*x**5 + 7350*c**34
*d**11*x**6 + 4200*c**33*d**12*x**7 + 1575*c**32*d**13*x**8 + 350*c**31*d**14*x*
*9 + 35*c**30*d**15*x**10) + 53760*c**(79/2)*d**4*x**4/(35*c**40*d**5 + 350*c**3
9*d**6*x + 1575*c**38*d**7*x**2 + 4200*c**37*d**8*x**3 + 7350*c**36*d**9*x**4 +
8820*c**35*d**10*x**5 + 7350*c**34*d**11*x**6 + 4200*c**33*d**12*x**7 + 1575*c**
32*d**13*x**8 + 350*c**31*d**14*x**9 + 35*c**30*d**15*x**10) - 46182*c**(77/2)*d
**5*x**5*sqrt(1 + d*x/c)/(35*c**40*d**5 + 350*c**39*d**6*x + 1575*c**38*d**7*x**
2 + 4200*c**37*d**8*x**3 + 7350*c**36*d**9*x**4 + 8820*c**35*d**10*x**5 + 7350*c
**34*d**11*x**6 + 4200*c**33*d**12*x**7 + 1575*c**32*d**13*x**8 + 350*c**31*d**1
4*x**9 + 35*c**30*d**15*x**10) + 64512*c**(77/2)*d**5*x**5/(35*c**40*d**5 + 350*
c**39*d**6*x + 1575*c**38*d**7*x**2 + 4200*c**37*d**8*x**3 + 7350*c**36*d**9*x**
4 + 8820*c**35*d**10*x**5 + 7350*c**34*d**11*x**6 + 4200*c**33*d**12*x**7 + 1575
*c**32*d**13*x**8 + 350*c**31*d**14*x**9 + 35*c**30*d**15*x**10) - 34584*c**(75/
2)*d**6*x**6*sqrt(1 + d*x/c)/(35*c**40*d**5 + 350*c**39*d**6*x + 1575*c**38*d**7
*x**2 + 4200*c**37*d**8*x**3 + 7350*c**36*d**9*x**4 + 8820*c**35*d**10*x**5 + 73
50*c**34*d**11*x**6 + 4200*c**33*d**12*x**7 + 1575*c**32*d**13*x**8 + 350*c**31*
d**14*x**9 + 35*c**30*d**15*x**10) + 53760*c**(75/2)*d**6*x**6/(35*c**40*d**5 +
350*c**39*d**6*x + 1575*c**38*d**7*x**2 + 4200*c**37*d**8*x**3 + 7350*c**36*d**9
*x**4 + 8820*c**35*d**10*x**5 + 7350*c**34*d**11*x**6 + 4200*c**33*d**12*x**7 +
1575*c**32*d**13*x**8 + 350*c**31*d**14*x**9 + 35*c**30*d**15*x**10) - 17112*c**
(73/2)*d**7*x**7*sqrt(1 + d*x/c)/(35*c**40*d**5 + 350*c**39*d**6*x + 1575*c**38*
d**7*x**2 + 4200*c**37*d**8*x**3 + 7350*c**36*d**9*x**4 + 8820*c**35*d**10*x**5
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5 + 350*c**39*d**6*x + 1575*c**38*d**7*x**2 + 4200*c**37*d**8*x**3 + 7350*c**36*
d**9*x**4 + 8820*c**35*d**10*x**5 + 7350*c**34*d**11*x**6 + 4200*c**33*d**12*x**
7 + 1575*c**32*d**13*x**8 + 350*c**31*d**14*x**9 + 35*c**30*d**15*x**10) - 4980*
c**(71/2)*d**8*x**8*sqrt(1 + d*x/c)/(35*c**40*d**5 + 350*c**39*d**6*x + 1575*c**
38*d**7*x**2 + 4200*c**37*d**8*x**3 + 7350*c**36*d**9*x**4 + 8820*c**35*d**10*x*
*5 + 7350*c**34*d**11*x**6 + 4200*c**33*d**12*x**7 + 1575*c**32*d**13*x**8 + 350
*c**31*d**14*x**9 + 35*c**30*d**15*x**10) + 11520*c**(71/2)*d**8*x**8/(35*c**40*
d**5 + 350*c**39*d**6*x + 1575*c**38*d**7*x**2 + 4200*c**37*d**8*x**3 + 7350*c**
36*d**9*x**4 + 8820*c**35*d**10*x**5 + 7350*c**34*d**11*x**6 + 4200*c**33*d**12*
x**7 + 1575*c**32*d**13*x**8 + 350*c**31*d**14*x**9 + 35*c**30*d**15*x**10) - 34
0*c**(69/2)*d**9*x**9*sqrt(1 + d*x/c)/(35*c**40*d**5 + 350*c**39*d**6*x + 1575*c
**38*d**7*x**2 + 4200*c**37*d**8*x**3 + 7350*c**36*d**9*x**4 + 8820*c**35*d**10*
x**5 + 7350*c**34*d**11*x**6 + 4200*c**33*d**12*x**7 + 1575*c**32*d**13*x**8 + 3
50*c**31*d**14*x**9 + 35*c**30*d**15*x**10) + 2560*c**(69/2)*d**9*x**9/(35*c**40
*d**5 + 350*c**39*d**6*x + 1575*c**38*d**7*x**2 + 4200*c**37*d**8*x**3 + 7350*c*
*36*d**9*x**4 + 8820*c**35*d**10*x**5 + 7350*c**34*d**11*x**6 + 4200*c**33*d**12
*x**7 + 1575*c**32*d**13*x**8 + 350*c**31*d**14*x**9 + 35*c**30*d**15*x**10) + 4
24*c**(67/2)*d**10*x**10*sqrt(1 + d*x/c)/(35*c**40*d**5 + 350*c**39*d**6*x + 157
5*c**38*d**7*x**2 + 4200*c**37*d**8*x**3 + 7350*c**36*d**9*x**4 + 8820*c**35*d**
10*x**5 + 7350*c**34*d**11*x**6 + 4200*c**33*d**12*x**7 + 1575*c**32*d**13*x**8
+ 350*c**31*d**14*x**9 + 35*c**30*d**15*x**10) + 256*c**(67/2)*d**10*x**10/(35*c
**40*d**5 + 350*c**39*d**6*x + 1575*c**38*d**7*x**2 + 4200*c**37*d**8*x**3 + 735
0*c**36*d**9*x**4 + 8820*c**35*d**10*x**5 + 7350*c**34*d**11*x**6 + 4200*c**33*d
**12*x**7 + 1575*c**32*d**13*x**8 + 350*c**31*d**14*x**9 + 35*c**30*d**15*x**10)
+ 248*c**(65/2)*d**11*x**11*sqrt(1 + d*x/c)/(35*c**40*d**5 + 350*c**39*d**6*x +
1575*c**38*d**7*x**2 + 4200*c**37*d**8*x**3 + 7350*c**36*d**9*x**4 + 8820*c**35
*d**10*x**5 + 7350*c**34*d**11*x**6 + 4200*c**33*d**12*x**7 + 1575*c**32*d**13*x
**8 + 350*c**31*d**14*x**9 + 35*c**30*d**15*x**10) + 74*c**(63/2)*d**12*x**12*sq
rt(1 + d*x/c)/(35*c**40*d**5 + 350*c**39*d**6*x + 1575*c**38*d**7*x**2 + 4200*c*
*37*d**8*x**3 + 7350*c**36*d**9*x**4 + 8820*c**35*d**10*x**5 + 7350*c**34*d**11*
x**6 + 4200*c**33*d**12*x**7 + 1575*c**32*d**13*x**8 + 350*c**31*d**14*x**9 + 35
*c**30*d**15*x**10) + 10*c**(61/2)*d**13*x**13*sqrt(1 + d*x/c)/(35*c**40*d**5 +
350*c**39*d**6*x + 1575*c**38*d**7*x**2 + 4200*c**37*d**8*x**3 + 7350*c**36*d**9
*x**4 + 8820*c**35*d**10*x**5 + 7350*c**34*d**11*x**6 + 4200*c**33*d**12*x**7 +
1575*c**32*d**13*x**8 + 350*c**31*d**14*x**9 + 35*c**30*d**15*x**10))

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GIAC/XCAS [A]  time = 0.216426, size = 436, normalized size = 2.08 \[ -\frac{2 \,{\left (D b c^{4} - D a c^{3} d - C b c^{3} d + C a c^{2} d^{2} + B b c^{2} d^{2} - B a c d^{3} - A b c d^{3} + A a d^{4}\right )}}{\sqrt{d x + c} d^{5}} + \frac{2 \,{\left (15 \,{\left (d x + c\right )}^{\frac{7}{2}} D b d^{30} - 84 \,{\left (d x + c\right )}^{\frac{5}{2}} D b c d^{30} + 210 \,{\left (d x + c\right )}^{\frac{3}{2}} D b c^{2} d^{30} - 420 \, \sqrt{d x + c} D b c^{3} d^{30} + 21 \,{\left (d x + c\right )}^{\frac{5}{2}} D a d^{31} + 21 \,{\left (d x + c\right )}^{\frac{5}{2}} C b d^{31} - 105 \,{\left (d x + c\right )}^{\frac{3}{2}} D a c d^{31} - 105 \,{\left (d x + c\right )}^{\frac{3}{2}} C b c d^{31} + 315 \, \sqrt{d x + c} D a c^{2} d^{31} + 315 \, \sqrt{d x + c} C b c^{2} d^{31} + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} C a d^{32} + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} B b d^{32} - 210 \, \sqrt{d x + c} C a c d^{32} - 210 \, \sqrt{d x + c} B b c d^{32} + 105 \, \sqrt{d x + c} B a d^{33} + 105 \, \sqrt{d x + c} A b d^{33}\right )}}{105 \, d^{35}} \]

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)^(3/2),x, algorithm="giac")

[Out]

-2*(D*b*c^4 - D*a*c^3*d - C*b*c^3*d + C*a*c^2*d^2 + B*b*c^2*d^2 - B*a*c*d^3 - A*
b*c*d^3 + A*a*d^4)/(sqrt(d*x + c)*d^5) + 2/105*(15*(d*x + c)^(7/2)*D*b*d^30 - 84
*(d*x + c)^(5/2)*D*b*c*d^30 + 210*(d*x + c)^(3/2)*D*b*c^2*d^30 - 420*sqrt(d*x +
c)*D*b*c^3*d^30 + 21*(d*x + c)^(5/2)*D*a*d^31 + 21*(d*x + c)^(5/2)*C*b*d^31 - 10
5*(d*x + c)^(3/2)*D*a*c*d^31 - 105*(d*x + c)^(3/2)*C*b*c*d^31 + 315*sqrt(d*x + c
)*D*a*c^2*d^31 + 315*sqrt(d*x + c)*C*b*c^2*d^31 + 35*(d*x + c)^(3/2)*C*a*d^32 +
35*(d*x + c)^(3/2)*B*b*d^32 - 210*sqrt(d*x + c)*C*a*c*d^32 - 210*sqrt(d*x + c)*B
*b*c*d^32 + 105*sqrt(d*x + c)*B*a*d^33 + 105*sqrt(d*x + c)*A*b*d^33)/d^35