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

Optimal. Leaf size=178 \[ -\frac{(2 b c-7 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{9/2}}+\frac{\sqrt{c+d x} (2 b c-7 a d) (b c-a d)}{b^4}+\frac{(c+d x)^{3/2} (2 b c-7 a d)}{3 b^3}+\frac{(c+d x)^{5/2} (2 b c-7 a d)}{5 b^2 (b c-a d)}+\frac{a (c+d x)^{7/2}}{b (a+b x) (b c-a d)} \]

[Out]

((2*b*c - 7*a*d)*(b*c - a*d)*Sqrt[c + d*x])/b^4 + ((2*b*c - 7*a*d)*(c + d*x)^(3/
2))/(3*b^3) + ((2*b*c - 7*a*d)*(c + d*x)^(5/2))/(5*b^2*(b*c - a*d)) + (a*(c + d*
x)^(7/2))/(b*(b*c - a*d)*(a + b*x)) - ((2*b*c - 7*a*d)*(b*c - a*d)^(3/2)*ArcTanh
[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/b^(9/2)

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Rubi [A]  time = 0.2832, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{(2 b c-7 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{9/2}}+\frac{\sqrt{c+d x} (2 b c-7 a d) (b c-a d)}{b^4}+\frac{(c+d x)^{3/2} (2 b c-7 a d)}{3 b^3}+\frac{(c+d x)^{5/2} (2 b c-7 a d)}{5 b^2 (b c-a d)}+\frac{a (c+d x)^{7/2}}{b (a+b x) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

((2*b*c - 7*a*d)*(b*c - a*d)*Sqrt[c + d*x])/b^4 + ((2*b*c - 7*a*d)*(c + d*x)^(3/
2))/(3*b^3) + ((2*b*c - 7*a*d)*(c + d*x)^(5/2))/(5*b^2*(b*c - a*d)) + (a*(c + d*
x)^(7/2))/(b*(b*c - a*d)*(a + b*x)) - ((2*b*c - 7*a*d)*(b*c - a*d)^(3/2)*ArcTanh
[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/b^(9/2)

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Rubi in Sympy [A]  time = 38.3309, size = 162, normalized size = 0.91 \[ - \frac{a \left (c + d x\right )^{\frac{7}{2}}}{b \left (a + b x\right ) \left (a d - b c\right )} + \frac{2 \left (c + d x\right )^{\frac{5}{2}} \left (\frac{7 a d}{2} - b c\right )}{5 b^{2} \left (a d - b c\right )} - \frac{2 \left (c + d x\right )^{\frac{3}{2}} \left (\frac{7 a d}{2} - b c\right )}{3 b^{3}} + \frac{\sqrt{c + d x} \left (a d - b c\right ) \left (7 a d - 2 b c\right )}{b^{4}} - \frac{2 \left (a d - b c\right )^{\frac{3}{2}} \left (\frac{7 a d}{2} - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*(c + d*x)**(7/2)/(b*(a + b*x)*(a*d - b*c)) + 2*(c + d*x)**(5/2)*(7*a*d/2 - b*
c)/(5*b**2*(a*d - b*c)) - 2*(c + d*x)**(3/2)*(7*a*d/2 - b*c)/(3*b**3) + sqrt(c +
d*x)*(a*d - b*c)*(7*a*d - 2*b*c)/b**4 - 2*(a*d - b*c)**(3/2)*(7*a*d/2 - b*c)*at
an(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c))/b**(9/2)

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Mathematica [A]  time = 0.319908, size = 141, normalized size = 0.79 \[ \frac{\sqrt{c+d x} \left (90 a^2 d^2+2 b d x (11 b c-10 a d)+\frac{15 a (b c-a d)^2}{a+b x}-140 a b c d+46 b^2 c^2+6 b^2 d^2 x^2\right )}{15 b^4}-\frac{(2 b c-7 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x]*(46*b^2*c^2 - 140*a*b*c*d + 90*a^2*d^2 + 2*b*d*(11*b*c - 10*a*d)*
x + 6*b^2*d^2*x^2 + (15*a*(b*c - a*d)^2)/(a + b*x)))/(15*b^4) - ((2*b*c - 7*a*d)
*(b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/b^(9/2)

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Maple [B]  time = 0.023, size = 348, normalized size = 2. \[{\frac{2}{5\,{b}^{2}} \left ( dx+c \right ) ^{{\frac{5}{2}}}}-{\frac{4\,ad}{3\,{b}^{3}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+{\frac{2\,c}{3\,{b}^{2}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+6\,{\frac{{a}^{2}{d}^{2}\sqrt{dx+c}}{{b}^{4}}}-8\,{\frac{acd\sqrt{dx+c}}{{b}^{3}}}+2\,{\frac{{c}^{2}\sqrt{dx+c}}{{b}^{2}}}+{\frac{{a}^{3}{d}^{3}}{{b}^{4} \left ( bdx+ad \right ) }\sqrt{dx+c}}-2\,{\frac{\sqrt{dx+c}{a}^{2}c{d}^{2}}{{b}^{3} \left ( bdx+ad \right ) }}+{\frac{a{c}^{2}d}{{b}^{2} \left ( bdx+ad \right ) }\sqrt{dx+c}}-7\,{\frac{{a}^{3}{d}^{3}}{{b}^{4}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+16\,{\frac{c{a}^{2}{d}^{2}}{{b}^{3}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-11\,{\frac{a{c}^{2}d}{{b}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+2\,{\frac{{c}^{3}}{b\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5/b^2*(d*x+c)^(5/2)-4/3/b^3*(d*x+c)^(3/2)*a*d+2/3/b^2*(d*x+c)^(3/2)*c+6/b^4*a^
2*d^2*(d*x+c)^(1/2)-8/b^3*a*c*d*(d*x+c)^(1/2)+2/b^2*c^2*(d*x+c)^(1/2)+1/b^4*(d*x
+c)^(1/2)/(b*d*x+a*d)*a^3*d^3-2/b^3*(d*x+c)^(1/2)/(b*d*x+a*d)*a^2*c*d^2+1/b^2*(d
*x+c)^(1/2)/(b*d*x+a*d)*a*c^2*d-7/b^4/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b
/((a*d-b*c)*b)^(1/2))*a^3*d^3+16/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*c*d^2-11/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^2*d+2/b/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a
*d-b*c)*b)^(1/2))*c^3

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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)*x/(b*x + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.272732, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (2 \, a b^{2} c^{2} - 9 \, a^{2} b c d + 7 \, a^{3} d^{2} +{\left (2 \, b^{3} c^{2} - 9 \, a b^{2} c d + 7 \, a^{2} b d^{2}\right )} x\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x + 2 \, b c - a d - 2 \, \sqrt{d x + c} b \sqrt{\frac{b c - a d}{b}}}{b x + a}\right ) + 2 \,{\left (6 \, b^{3} d^{2} x^{3} + 61 \, a b^{2} c^{2} - 170 \, a^{2} b c d + 105 \, a^{3} d^{2} + 2 \,{\left (11 \, b^{3} c d - 7 \, a b^{2} d^{2}\right )} x^{2} + 2 \,{\left (23 \, b^{3} c^{2} - 59 \, a b^{2} c d + 35 \, a^{2} b d^{2}\right )} x\right )} \sqrt{d x + c}}{30 \,{\left (b^{5} x + a b^{4}\right )}}, -\frac{15 \,{\left (2 \, a b^{2} c^{2} - 9 \, a^{2} b c d + 7 \, a^{3} d^{2} +{\left (2 \, b^{3} c^{2} - 9 \, a b^{2} c d + 7 \, a^{2} b d^{2}\right )} x\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) -{\left (6 \, b^{3} d^{2} x^{3} + 61 \, a b^{2} c^{2} - 170 \, a^{2} b c d + 105 \, a^{3} d^{2} + 2 \,{\left (11 \, b^{3} c d - 7 \, a b^{2} d^{2}\right )} x^{2} + 2 \,{\left (23 \, b^{3} c^{2} - 59 \, a b^{2} c d + 35 \, a^{2} b d^{2}\right )} x\right )} \sqrt{d x + c}}{15 \,{\left (b^{5} x + a b^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/30*(15*(2*a*b^2*c^2 - 9*a^2*b*c*d + 7*a^3*d^2 + (2*b^3*c^2 - 9*a*b^2*c*d + 7*
a^2*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*s
qrt((b*c - a*d)/b))/(b*x + a)) + 2*(6*b^3*d^2*x^3 + 61*a*b^2*c^2 - 170*a^2*b*c*d
+ 105*a^3*d^2 + 2*(11*b^3*c*d - 7*a*b^2*d^2)*x^2 + 2*(23*b^3*c^2 - 59*a*b^2*c*d
+ 35*a^2*b*d^2)*x)*sqrt(d*x + c))/(b^5*x + a*b^4), -1/15*(15*(2*a*b^2*c^2 - 9*a
^2*b*c*d + 7*a^3*d^2 + (2*b^3*c^2 - 9*a*b^2*c*d + 7*a^2*b*d^2)*x)*sqrt(-(b*c - a
*d)/b)*arctan(sqrt(d*x + c)/sqrt(-(b*c - a*d)/b)) - (6*b^3*d^2*x^3 + 61*a*b^2*c^
2 - 170*a^2*b*c*d + 105*a^3*d^2 + 2*(11*b^3*c*d - 7*a*b^2*d^2)*x^2 + 2*(23*b^3*c
^2 - 59*a*b^2*c*d + 35*a^2*b*d^2)*x)*sqrt(d*x + c))/(b^5*x + a*b^4)]

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Sympy [A]  time = 160.4, size = 1846, normalized size = 10.37 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**4*d**4*sqrt(c + d*x)/(2*a**2*b**4*d**2 - 2*a*b**5*c*d + 2*a*b**5*d**2*x - 2
*b**6*c*d*x) - a**4*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**4) + a**4*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**4) - 6*a**3*c
*d**3*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) + 3*a**3*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**3) - 3*a**3*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**3) - 8*a**3
*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*sq
rt(-a*d/b + c)), (a*d/b - c < 0) & (c + d*x < -a*d/b + c)))/b**4 + 6*a**2*c**2*d
**2*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**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**2) + 3*a**2*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**2) + 18
*a**2*c*d**2*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*sqrt(-a*d/b + c)), (a*d/b - c < 0) & (c + d*x < -a*d/b + c)))/b**3 + 6*a**2
*d**2*sqrt(c + d*x)/b**4 - 2*a*c**3*d*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) + a*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*b) - a*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*b) - 12*a*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)/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 -
8*a*c*d*sqrt(c + d*x)/b**3 - 4*a*d*(c + d*x)**(3/2)/(3*b**3) + 2*c**3*Piecewise(
(atan(sqrt(c + d*x)/sqrt(a*d/b - c))/(b*sqrt(a*d/b - c)), a*d/b - c > 0), (-acot
h(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*c**2*sqrt(c + d*x)/b**2 + 2*c*
(c + d*x)**(3/2)/(3*b**2) + 2*(c + d*x)**(5/2)/(5*b**2)

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GIAC/XCAS [A]  time = 0.245332, size = 324, normalized size = 1.82 \[ \frac{{\left (2 \, b^{3} c^{3} - 11 \, a b^{2} c^{2} d + 16 \, a^{2} b c d^{2} - 7 \, a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b^{4}} + \frac{\sqrt{d x + c} a b^{2} c^{2} d - 2 \, \sqrt{d x + c} a^{2} b c d^{2} + \sqrt{d x + c} a^{3} d^{3}}{{\left ({\left (d x + c\right )} b - b c + a d\right )} b^{4}} + \frac{2 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{8} + 5 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{8} c + 15 \, \sqrt{d x + c} b^{8} c^{2} - 10 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{7} d - 60 \, \sqrt{d x + c} a b^{7} c d + 45 \, \sqrt{d x + c} a^{2} b^{6} d^{2}\right )}}{15 \, b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*b^3*c^3 - 11*a*b^2*c^2*d + 16*a^2*b*c*d^2 - 7*a^3*d^3)*arctan(sqrt(d*x + c)*b
/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*b^4) + (sqrt(d*x + c)*a*b^2*c^2*d -
2*sqrt(d*x + c)*a^2*b*c*d^2 + sqrt(d*x + c)*a^3*d^3)/(((d*x + c)*b - b*c + a*d)
*b^4) + 2/15*(3*(d*x + c)^(5/2)*b^8 + 5*(d*x + c)^(3/2)*b^8*c + 15*sqrt(d*x + c)
*b^8*c^2 - 10*(d*x + c)^(3/2)*a*b^7*d - 60*sqrt(d*x + c)*a*b^7*c*d + 45*sqrt(d*x
+ c)*a^2*b^6*d^2)/b^10