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

Optimal. Leaf size=110 \[ -\frac{5 d (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{7/2}}+\frac{5 d \sqrt{c+d x} (b c-a d)}{b^3}-\frac{(c+d x)^{5/2}}{b (a+b x)}+\frac{5 d (c+d x)^{3/2}}{3 b^2} \]

[Out]

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

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Rubi [A]  time = 0.152977, antiderivative size = 110, 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 \[ -\frac{5 d (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{7/2}}+\frac{5 d \sqrt{c+d x} (b c-a d)}{b^3}-\frac{(c+d x)^{5/2}}{b (a+b x)}+\frac{5 d (c+d x)^{3/2}}{3 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.6999, size = 97, normalized size = 0.88 \[ - \frac{\left (c + d x\right )^{\frac{5}{2}}}{b \left (a + b x\right )} + \frac{5 d \left (c + d x\right )^{\frac{3}{2}}}{3 b^{2}} - \frac{5 d \sqrt{c + d x} \left (a d - b c\right )}{b^{3}} + \frac{5 d \left (a d - b c\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.182145, size = 104, normalized size = 0.95 \[ \frac{\sqrt{c+d x} \left (-\frac{3 (b c-a d)^2}{a+b x}+2 d (7 b c-6 a d)+2 b d^2 x\right )}{3 b^3}-\frac{5 d (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.007, size = 258, normalized size = 2.4 \[{\frac{2\,d}{3\,{b}^{2}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-4\,{\frac{{d}^{2}a\sqrt{dx+c}}{{b}^{3}}}+4\,{\frac{d\sqrt{dx+c}c}{{b}^{2}}}-{\frac{{a}^{2}{d}^{3}}{{b}^{3} \left ( bdx+ad \right ) }\sqrt{dx+c}}+2\,{\frac{\sqrt{dx+c}ac{d}^{2}}{{b}^{2} \left ( bdx+ad \right ) }}-{\frac{d{c}^{2}}{b \left ( bdx+ad \right ) }\sqrt{dx+c}}+5\,{\frac{{a}^{2}{d}^{3}}{{b}^{3}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-10\,{\frac{ac{d}^{2}}{{b}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+5\,{\frac{d{c}^{2}}{b\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/2)/(b*x+a)^2,x)

[Out]

2/3*d*(d*x+c)^(3/2)/b^2-4/b^3*a*d^2*(d*x+c)^(1/2)+4*d/b^2*(d*x+c)^(1/2)*c-1/b^3*
(d*x+c)^(1/2)/(b*d*x+a*d)*a^2*d^3+2/b^2*(d*x+c)^(1/2)/(b*d*x+a*d)*a*c*d^2-d/b*(d
*x+c)^(1/2)/(b*d*x+a*d)*c^2+5/b^3/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a
*d-b*c)*b)^(1/2))*a^2*d^3-10/b^2/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*
d-b*c)*b)^(1/2))*a*c*d^2+5*d/b/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-
b*c)*b)^(1/2))*c^2

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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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.261849, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (a b c d - a^{2} d^{2} +{\left (b^{2} c d - a b d^{2}\right )} x\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x + 2 \, b c - a d + 2 \, \sqrt{d x + c} b \sqrt{\frac{b c - a d}{b}}}{b x + a}\right ) - 2 \,{\left (2 \, b^{2} d^{2} x^{2} - 3 \, b^{2} c^{2} + 20 \, a b c d - 15 \, a^{2} d^{2} + 2 \,{\left (7 \, b^{2} c d - 5 \, a b d^{2}\right )} x\right )} \sqrt{d x + c}}{6 \,{\left (b^{4} x + a b^{3}\right )}}, -\frac{15 \,{\left (a b c d - a^{2} d^{2} +{\left (b^{2} c d - a b d^{2}\right )} x\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) -{\left (2 \, b^{2} d^{2} x^{2} - 3 \, b^{2} c^{2} + 20 \, a b c d - 15 \, a^{2} d^{2} + 2 \,{\left (7 \, b^{2} c d - 5 \, a b d^{2}\right )} x\right )} \sqrt{d x + c}}{3 \,{\left (b^{4} x + a b^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/6*(15*(a*b*c*d - a^2*d^2 + (b^2*c*d - a*b*d^2)*x)*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*(2*b^
2*d^2*x^2 - 3*b^2*c^2 + 20*a*b*c*d - 15*a^2*d^2 + 2*(7*b^2*c*d - 5*a*b*d^2)*x)*s
qrt(d*x + c))/(b^4*x + a*b^3), -1/3*(15*(a*b*c*d - a^2*d^2 + (b^2*c*d - a*b*d^2)
*x)*sqrt(-(b*c - a*d)/b)*arctan(sqrt(d*x + c)/sqrt(-(b*c - a*d)/b)) - (2*b^2*d^2
*x^2 - 3*b^2*c^2 + 20*a*b*c*d - 15*a^2*d^2 + 2*(7*b^2*c*d - 5*a*b*d^2)*x)*sqrt(d
*x + c))/(b^4*x + a*b^3)]

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Sympy [A]  time = 109.835, size = 1622, normalized size = 14.75 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**3*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) + a**3*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) - a**3*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) + 6*a**2*
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) - 3*a**2*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))/(2*b**2) + 3*a**2*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))/(2*b**2) + 6*a**
2*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(sqrt(c + d*x)/sqrt(-a*d/b + c))/(b*s
qrt(-a*d/b + c)), (a*d/b - c < 0) & (c + d*x < -a*d/b + c)))/b**3 - 6*a*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*a*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*a*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) - 12*a*c*d**2*Piecewi
se((atan(sqrt(c + d*x)/sqrt(a*d/b - c))/(b*sqrt(a*d/b - c)), a*d/b - c > 0), (-a
coth(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 - 4*a*d**2*sqrt(c + d*x)/b**
3 - c**3*d*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 + c**3*d*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 + 2*c**3*d*sqrt(c + d*x)/(2*a**2*d**2 - 2
*a*b*c*d + 2*a*b*d**2*x - 2*b**2*c*d*x) + 6*c**2*d*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)/sqr
t(-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 + 4*c*d*sqrt(c + d*x)/b**2 + 2*d*(c + d*x)**(3/2)/(3*
b**2)

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GIAC/XCAS [A]  time = 0.219088, size = 244, normalized size = 2.22 \[ \frac{5 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b^{3}} - \frac{\sqrt{d x + c} b^{2} c^{2} d - 2 \, \sqrt{d x + c} a b c d^{2} + \sqrt{d x + c} a^{2} d^{3}}{{\left ({\left (d x + c\right )} b - b c + a d\right )} b^{3}} + \frac{2 \,{\left ({\left (d x + c\right )}^{\frac{3}{2}} b^{4} d + 6 \, \sqrt{d x + c} b^{4} c d - 6 \, \sqrt{d x + c} a b^{3} d^{2}\right )}}{3 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d
))/(sqrt(-b^2*c + a*b*d)*b^3) - (sqrt(d*x + c)*b^2*c^2*d - 2*sqrt(d*x + c)*a*b*c
*d^2 + sqrt(d*x + c)*a^2*d^3)/(((d*x + c)*b - b*c + a*d)*b^3) + 2/3*((d*x + c)^(
3/2)*b^4*d + 6*sqrt(d*x + c)*b^4*c*d - 6*sqrt(d*x + c)*a*b^3*d^2)/b^6