3.1407 \(\int \frac{(c+d x)^{5/2}}{(a+b x)^3} \, dx\)

Optimal. Leaf size=119 \[ -\frac{15 d^2 \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{4 b^{7/2}}-\frac{5 d (c+d x)^{3/2}}{4 b^2 (a+b x)}-\frac{(c+d x)^{5/2}}{2 b (a+b x)^2}+\frac{15 d^2 \sqrt{c+d x}}{4 b^3} \]

[Out]

(15*d^2*Sqrt[c + d*x])/(4*b^3) - (5*d*(c + d*x)^(3/2))/(4*b^2*(a + b*x)) - (c +
d*x)^(5/2)/(2*b*(a + b*x)^2) - (15*d^2*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c +
d*x])/Sqrt[b*c - a*d]])/(4*b^(7/2))

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Rubi [A]  time = 0.133166, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{15 d^2 \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{4 b^{7/2}}-\frac{5 d (c+d x)^{3/2}}{4 b^2 (a+b x)}-\frac{(c+d x)^{5/2}}{2 b (a+b x)^2}+\frac{15 d^2 \sqrt{c+d x}}{4 b^3} \]

Antiderivative was successfully verified.

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

[Out]

(15*d^2*Sqrt[c + d*x])/(4*b^3) - (5*d*(c + d*x)^(3/2))/(4*b^2*(a + b*x)) - (c +
d*x)^(5/2)/(2*b*(a + b*x)^2) - (15*d^2*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c +
d*x])/Sqrt[b*c - a*d]])/(4*b^(7/2))

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Rubi in Sympy [A]  time = 25.4154, size = 105, normalized size = 0.88 \[ - \frac{\left (c + d x\right )^{\frac{5}{2}}}{2 b \left (a + b x\right )^{2}} - \frac{5 d \left (c + d x\right )^{\frac{3}{2}}}{4 b^{2} \left (a + b x\right )} + \frac{15 d^{2} \sqrt{c + d x}}{4 b^{3}} - \frac{15 d^{2} \sqrt{a d - b c} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{4 b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(c + d*x)**(5/2)/(2*b*(a + b*x)**2) - 5*d*(c + d*x)**(3/2)/(4*b**2*(a + b*x)) +
15*d**2*sqrt(c + d*x)/(4*b**3) - 15*d**2*sqrt(a*d - b*c)*atan(sqrt(b)*sqrt(c +
d*x)/sqrt(a*d - b*c))/(4*b**(7/2))

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Mathematica [A]  time = 0.21668, size = 119, normalized size = 1. \[ \sqrt{c+d x} \left (-\frac{9 d (b c-a d)}{4 b^3 (a+b x)}-\frac{(b c-a d)^2}{2 b^3 (a+b x)^2}+\frac{2 d^2}{b^3}\right )-\frac{15 d^2 \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{4 b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[c + d*x]*((2*d^2)/b^3 - (b*c - a*d)^2/(2*b^3*(a + b*x)^2) - (9*d*(b*c - a*d
))/(4*b^3*(a + b*x))) - (15*d^2*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/
Sqrt[b*c - a*d]])/(4*b^(7/2))

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Maple [B]  time = 0.022, size = 238, normalized size = 2. \[ 2\,{\frac{{d}^{2}\sqrt{dx+c}}{{b}^{3}}}+{\frac{9\,{d}^{3}a}{4\,{b}^{2} \left ( bdx+ad \right ) ^{2}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{9\,{d}^{2}c}{4\,b \left ( bdx+ad \right ) ^{2}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{d}^{4}{a}^{2}}{4\,{b}^{3} \left ( bdx+ad \right ) ^{2}}\sqrt{dx+c}}-{\frac{7\,{d}^{3}ac}{2\,{b}^{2} \left ( bdx+ad \right ) ^{2}}\sqrt{dx+c}}+{\frac{7\,{d}^{2}{c}^{2}}{4\,b \left ( bdx+ad \right ) ^{2}}\sqrt{dx+c}}-{\frac{15\,{d}^{3}a}{4\,{b}^{3}}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}}+{\frac{15\,{d}^{2}c}{4\,{b}^{2}}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/2)/(b*x+a)^3,x)

[Out]

2*d^2*(d*x+c)^(1/2)/b^3+9/4*d^3/b^2/(b*d*x+a*d)^2*(d*x+c)^(3/2)*a-9/4*d^2/b/(b*d
*x+a*d)^2*(d*x+c)^(3/2)*c+7/4*d^4/b^3/(b*d*x+a*d)^2*(d*x+c)^(1/2)*a^2-7/2*d^3/b^
2/(b*d*x+a*d)^2*(d*x+c)^(1/2)*a*c+7/4*d^2/b/(b*d*x+a*d)^2*(d*x+c)^(1/2)*c^2-15/4
*d^3/b^3/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*a+15/4*
d^2/b^2/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228196, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x + 2 \, b c - a d - 2 \, \sqrt{d x + c} b \sqrt{\frac{b c - a d}{b}}}{b x + a}\right ) + 2 \,{\left (8 \, b^{2} d^{2} x^{2} - 2 \, b^{2} c^{2} - 5 \, a b c d + 15 \, a^{2} d^{2} -{\left (9 \, b^{2} c d - 25 \, a b d^{2}\right )} x\right )} \sqrt{d x + c}}{8 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}, -\frac{15 \,{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) -{\left (8 \, b^{2} d^{2} x^{2} - 2 \, b^{2} c^{2} - 5 \, a b c d + 15 \, a^{2} d^{2} -{\left (9 \, b^{2} c d - 25 \, a b d^{2}\right )} x\right )} \sqrt{d x + c}}{4 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(15*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*sqrt((b*c - a*d)/b)*log((b*d*x +
2*b*c - a*d - 2*sqrt(d*x + c)*b*sqrt((b*c - a*d)/b))/(b*x + a)) + 2*(8*b^2*d^2*x
^2 - 2*b^2*c^2 - 5*a*b*c*d + 15*a^2*d^2 - (9*b^2*c*d - 25*a*b*d^2)*x)*sqrt(d*x +
c))/(b^5*x^2 + 2*a*b^4*x + a^2*b^3), -1/4*(15*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*
d^2)*sqrt(-(b*c - a*d)/b)*arctan(sqrt(d*x + c)/sqrt(-(b*c - a*d)/b)) - (8*b^2*d^
2*x^2 - 2*b^2*c^2 - 5*a*b*c*d + 15*a^2*d^2 - (9*b^2*c*d - 25*a*b*d^2)*x)*sqrt(d*
x + c))/(b^5*x^2 + 2*a*b^4*x + a^2*b^3)]

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Sympy [A]  time = 121.672, size = 3842, normalized size = 32.29 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-10*a**4*d**6*sqrt(c + d*x)/(8*a**4*b**3*d**4 - 16*a**3*b**4*c*d**3 + 16*a**3*b*
*4*d**4*x - 48*a**2*b**5*c*d**3*x + 8*a**2*b**5*d**2*(c + d*x)**2 + 16*a*b**6*c*
*3*d + 48*a*b**6*c**2*d**2*x - 16*a*b**6*c*d*(c + d*x)**2 - 8*b**7*c**4 - 16*b**
7*c**3*d*x + 8*b**7*c**2*(c + d*x)**2) + 40*a**3*c*d**5*sqrt(c + d*x)/(8*a**4*b*
*2*d**4 - 16*a**3*b**3*c*d**3 + 16*a**3*b**3*d**4*x - 48*a**2*b**4*c*d**3*x + 8*
a**2*b**4*d**2*(c + d*x)**2 + 16*a*b**5*c**3*d + 48*a*b**5*c**2*d**2*x - 16*a*b*
*5*c*d*(c + d*x)**2 - 8*b**6*c**4 - 16*b**6*c**3*d*x + 8*b**6*c**2*(c + d*x)**2)
- 6*a**3*d**5*(c + d*x)**(3/2)/(8*a**4*b**2*d**4 - 16*a**3*b**3*c*d**3 + 16*a**
3*b**3*d**4*x - 48*a**2*b**4*c*d**3*x + 8*a**2*b**4*d**2*(c + d*x)**2 + 16*a*b**
5*c**3*d + 48*a*b**5*c**2*d**2*x - 16*a*b**5*c*d*(c + d*x)**2 - 8*b**6*c**4 - 16
*b**6*c**3*d*x + 8*b**6*c**2*(c + d*x)**2) + 3*a**3*d**5*sqrt(-1/(b*(a*d - b*c)*
*5))*log(-a**3*d**3*sqrt(-1/(b*(a*d - b*c)**5)) + 3*a**2*b*c*d**2*sqrt(-1/(b*(a*
d - b*c)**5)) - 3*a*b**2*c**2*d*sqrt(-1/(b*(a*d - b*c)**5)) + b**3*c**3*sqrt(-1/
(b*(a*d - b*c)**5)) + sqrt(c + d*x))/(8*b**3) - 3*a**3*d**5*sqrt(-1/(b*(a*d - b*
c)**5))*log(a**3*d**3*sqrt(-1/(b*(a*d - b*c)**5)) - 3*a**2*b*c*d**2*sqrt(-1/(b*(
a*d - b*c)**5)) + 3*a*b**2*c**2*d*sqrt(-1/(b*(a*d - b*c)**5)) - b**3*c**3*sqrt(-
1/(b*(a*d - b*c)**5)) + sqrt(c + d*x))/(8*b**3) - 60*a**2*c**2*d**4*sqrt(c + d*x
)/(8*a**4*b*d**4 - 16*a**3*b**2*c*d**3 + 16*a**3*b**2*d**4*x - 48*a**2*b**3*c*d*
*3*x + 8*a**2*b**3*d**2*(c + d*x)**2 + 16*a*b**4*c**3*d + 48*a*b**4*c**2*d**2*x
- 16*a*b**4*c*d*(c + d*x)**2 - 8*b**5*c**4 - 16*b**5*c**3*d*x + 8*b**5*c**2*(c +
d*x)**2) + 18*a**2*c*d**4*(c + d*x)**(3/2)/(8*a**4*b*d**4 - 16*a**3*b**2*c*d**3
+ 16*a**3*b**2*d**4*x - 48*a**2*b**3*c*d**3*x + 8*a**2*b**3*d**2*(c + d*x)**2 +
16*a*b**4*c**3*d + 48*a*b**4*c**2*d**2*x - 16*a*b**4*c*d*(c + d*x)**2 - 8*b**5*
c**4 - 16*b**5*c**3*d*x + 8*b**5*c**2*(c + d*x)**2) + 6*a**2*d**4*sqrt(c + d*x)/
(2*a**2*b**3*d**2 - 2*a*b**4*c*d + 2*a*b**4*d**2*x - 2*b**5*c*d*x) - 9*a**2*c*d*
*4*sqrt(-1/(b*(a*d - b*c)**5))*log(-a**3*d**3*sqrt(-1/(b*(a*d - b*c)**5)) + 3*a*
*2*b*c*d**2*sqrt(-1/(b*(a*d - b*c)**5)) - 3*a*b**2*c**2*d*sqrt(-1/(b*(a*d - b*c)
**5)) + b**3*c**3*sqrt(-1/(b*(a*d - b*c)**5)) + sqrt(c + d*x))/(8*b**2) + 9*a**2
*c*d**4*sqrt(-1/(b*(a*d - b*c)**5))*log(a**3*d**3*sqrt(-1/(b*(a*d - b*c)**5)) -
3*a**2*b*c*d**2*sqrt(-1/(b*(a*d - b*c)**5)) + 3*a*b**2*c**2*d*sqrt(-1/(b*(a*d -
b*c)**5)) - b**3*c**3*sqrt(-1/(b*(a*d - b*c)**5)) + sqrt(c + d*x))/(8*b**2) - 3*
a**2*d**4*sqrt(-1/(b*(a*d - b*c)**3))*log(-a**2*d**2*sqrt(-1/(b*(a*d - b*c)**3))
+ 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) - b**2*c**2*sqrt(-1/(b*(a*d - b*c)**3))
+ sqrt(c + d*x))/(2*b**3) + 3*a**2*d**4*sqrt(-1/(b*(a*d - b*c)**3))*log(a**2*d*
*2*sqrt(-1/(b*(a*d - b*c)**3)) - 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) + b**2*c*
*2*sqrt(-1/(b*(a*d - b*c)**3)) + sqrt(c + d*x))/(2*b**3) + 40*a*c**3*d**3*sqrt(c
+ d*x)/(8*a**4*d**4 - 16*a**3*b*c*d**3 + 16*a**3*b*d**4*x - 48*a**2*b**2*c*d**3
*x + 8*a**2*b**2*d**2*(c + d*x)**2 + 16*a*b**3*c**3*d + 48*a*b**3*c**2*d**2*x -
16*a*b**3*c*d*(c + d*x)**2 - 8*b**4*c**4 - 16*b**4*c**3*d*x + 8*b**4*c**2*(c + d
*x)**2) - 18*a*c**2*d**3*(c + d*x)**(3/2)/(8*a**4*d**4 - 16*a**3*b*c*d**3 + 16*a
**3*b*d**4*x - 48*a**2*b**2*c*d**3*x + 8*a**2*b**2*d**2*(c + d*x)**2 + 16*a*b**3
*c**3*d + 48*a*b**3*c**2*d**2*x - 16*a*b**3*c*d*(c + d*x)**2 - 8*b**4*c**4 - 16*
b**4*c**3*d*x + 8*b**4*c**2*(c + d*x)**2) - 12*a*c*d**3*sqrt(c + d*x)/(2*a**2*b*
*2*d**2 - 2*a*b**3*c*d + 2*a*b**3*d**2*x - 2*b**4*c*d*x) + 9*a*c**2*d**3*sqrt(-1
/(b*(a*d - b*c)**5))*log(-a**3*d**3*sqrt(-1/(b*(a*d - b*c)**5)) + 3*a**2*b*c*d**
2*sqrt(-1/(b*(a*d - b*c)**5)) - 3*a*b**2*c**2*d*sqrt(-1/(b*(a*d - b*c)**5)) + b*
*3*c**3*sqrt(-1/(b*(a*d - b*c)**5)) + sqrt(c + d*x))/(8*b) - 9*a*c**2*d**3*sqrt(
-1/(b*(a*d - b*c)**5))*log(a**3*d**3*sqrt(-1/(b*(a*d - b*c)**5)) - 3*a**2*b*c*d*
*2*sqrt(-1/(b*(a*d - b*c)**5)) + 3*a*b**2*c**2*d*sqrt(-1/(b*(a*d - b*c)**5)) - b
**3*c**3*sqrt(-1/(b*(a*d - b*c)**5)) + sqrt(c + d*x))/(8*b) + 3*a*c*d**3*sqrt(-1
/(b*(a*d - b*c)**3))*log(-a**2*d**2*sqrt(-1/(b*(a*d - b*c)**3)) + 2*a*b*c*d*sqrt
(-1/(b*(a*d - b*c)**3)) - b**2*c**2*sqrt(-1/(b*(a*d - b*c)**3)) + sqrt(c + d*x))
/b**2 - 3*a*c*d**3*sqrt(-1/(b*(a*d - b*c)**3))*log(a**2*d**2*sqrt(-1/(b*(a*d - b
*c)**3)) - 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) + b**2*c**2*sqrt(-1/(b*(a*d - b
*c)**3)) + sqrt(c + d*x))/b**2 - 6*a*d**3*Piecewise((atan(sqrt(c + d*x)/sqrt(a*d
/b - c))/(b*sqrt(a*d/b - c)), a*d/b - c > 0), (-acoth(sqrt(c + d*x)/sqrt(-a*d/b
+ c))/(b*sqrt(-a*d/b + c)), (a*d/b - c < 0) & (c + d*x > -a*d/b + c)), (-atanh(s
qrt(c + d*x)/sqrt(-a*d/b + c))/(b*sqrt(-a*d/b + c)), (a*d/b - c < 0) & (c + d*x
< -a*d/b + c)))/b**3 - 10*b*c**4*d**2*sqrt(c + d*x)/(8*a**4*d**4 - 16*a**3*b*c*d
**3 + 16*a**3*b*d**4*x - 48*a**2*b**2*c*d**3*x + 8*a**2*b**2*d**2*(c + d*x)**2 +
16*a*b**3*c**3*d + 48*a*b**3*c**2*d**2*x - 16*a*b**3*c*d*(c + d*x)**2 - 8*b**4*
c**4 - 16*b**4*c**3*d*x + 8*b**4*c**2*(c + d*x)**2) + 6*b*c**3*d**2*(c + d*x)**(
3/2)/(8*a**4*d**4 - 16*a**3*b*c*d**3 + 16*a**3*b*d**4*x - 48*a**2*b**2*c*d**3*x
+ 8*a**2*b**2*d**2*(c + d*x)**2 + 16*a*b**3*c**3*d + 48*a*b**3*c**2*d**2*x - 16*
a*b**3*c*d*(c + d*x)**2 - 8*b**4*c**4 - 16*b**4*c**3*d*x + 8*b**4*c**2*(c + d*x)
**2) - 3*c**3*d**2*sqrt(-1/(b*(a*d - b*c)**5))*log(-a**3*d**3*sqrt(-1/(b*(a*d -
b*c)**5)) + 3*a**2*b*c*d**2*sqrt(-1/(b*(a*d - b*c)**5)) - 3*a*b**2*c**2*d*sqrt(-
1/(b*(a*d - b*c)**5)) + b**3*c**3*sqrt(-1/(b*(a*d - b*c)**5)) + sqrt(c + d*x))/8
+ 3*c**3*d**2*sqrt(-1/(b*(a*d - b*c)**5))*log(a**3*d**3*sqrt(-1/(b*(a*d - b*c)*
*5)) - 3*a**2*b*c*d**2*sqrt(-1/(b*(a*d - b*c)**5)) + 3*a*b**2*c**2*d*sqrt(-1/(b*
(a*d - b*c)**5)) - b**3*c**3*sqrt(-1/(b*(a*d - b*c)**5)) + sqrt(c + d*x))/8 + 6*
c**2*d**2*sqrt(c + d*x)/(2*a**2*b*d**2 - 2*a*b**2*c*d + 2*a*b**2*d**2*x - 2*b**3
*c*d*x) - 3*c**2*d**2*sqrt(-1/(b*(a*d - b*c)**3))*log(-a**2*d**2*sqrt(-1/(b*(a*d
- b*c)**3)) + 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) - b**2*c**2*sqrt(-1/(b*(a*d
- b*c)**3)) + sqrt(c + d*x))/(2*b) + 3*c**2*d**2*sqrt(-1/(b*(a*d - b*c)**3))*lo
g(a**2*d**2*sqrt(-1/(b*(a*d - b*c)**3)) - 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3))
+ b**2*c**2*sqrt(-1/(b*(a*d - b*c)**3)) + sqrt(c + d*x))/(2*b) + 6*c*d**2*Piecew
ise((atan(sqrt(c + d*x)/sqrt(a*d/b - c))/(b*sqrt(a*d/b - c)), a*d/b - c > 0), (-
acoth(sqrt(c + d*x)/sqrt(-a*d/b + c))/(b*sqrt(-a*d/b + c)), (a*d/b - c < 0) & (c
+ d*x > -a*d/b + c)), (-atanh(sqrt(c + d*x)/sqrt(-a*d/b + c))/(b*sqrt(-a*d/b +
c)), (a*d/b - c < 0) & (c + d*x < -a*d/b + c)))/b**2 + 2*d**2*sqrt(c + d*x)/b**3

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GIAC/XCAS [A]  time = 0.233061, size = 231, normalized size = 1.94 \[ \frac{2 \, \sqrt{d x + c} d^{2}}{b^{3}} + \frac{15 \,{\left (b c d^{2} - a d^{3}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{4 \, \sqrt{-b^{2} c + a b d} b^{3}} - \frac{9 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{2} c d^{2} - 7 \, \sqrt{d x + c} b^{2} c^{2} d^{2} - 9 \,{\left (d x + c\right )}^{\frac{3}{2}} a b d^{3} + 14 \, \sqrt{d x + c} a b c d^{3} - 7 \, \sqrt{d x + c} a^{2} d^{4}}{4 \,{\left ({\left (d x + c\right )} b - b c + a d\right )}^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(d*x + c)*d^2/b^3 + 15/4*(b*c*d^2 - a*d^3)*arctan(sqrt(d*x + c)*b/sqrt(-b^
2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*b^3) - 1/4*(9*(d*x + c)^(3/2)*b^2*c*d^2 - 7*
sqrt(d*x + c)*b^2*c^2*d^2 - 9*(d*x + c)^(3/2)*a*b*d^3 + 14*sqrt(d*x + c)*a*b*c*d
^3 - 7*sqrt(d*x + c)*a^2*d^4)/(((d*x + c)*b - b*c + a*d)^2*b^3)