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

Optimal. Leaf size=220 \[ -\frac{a^2 (6 b c-11 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{13/2}}+\frac{a^2 \sqrt{c+d x} (6 b c-11 a d) (b c-a d)}{b^6}+\frac{a^2 (c+d x)^{3/2} (6 b c-11 a d)}{3 b^5}-\frac{(c+d x)^{5/2} \left (-693 a^2 d^2-5 b d x (10 b c-99 a d)+180 a b c d+20 b^2 c^2\right )}{315 b^4 d^2}-\frac{x^3 (c+d x)^{5/2}}{b (a+b x)}+\frac{11 x^2 (c+d x)^{5/2}}{9 b^2} \]

[Out]

(a^2*(6*b*c - 11*a*d)*(b*c - a*d)*Sqrt[c + d*x])/b^6 + (a^2*(6*b*c - 11*a*d)*(c
+ d*x)^(3/2))/(3*b^5) + (11*x^2*(c + d*x)^(5/2))/(9*b^2) - (x^3*(c + d*x)^(5/2))
/(b*(a + b*x)) - ((c + d*x)^(5/2)*(20*b^2*c^2 + 180*a*b*c*d - 693*a^2*d^2 - 5*b*
d*(10*b*c - 99*a*d)*x))/(315*b^4*d^2) - (a^2*(6*b*c - 11*a*d)*(b*c - a*d)^(3/2)*
ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/b^(13/2)

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Rubi [A]  time = 0.608583, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{a^2 (6 b c-11 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{13/2}}+\frac{a^2 \sqrt{c+d x} (6 b c-11 a d) (b c-a d)}{b^6}+\frac{a^2 (c+d x)^{3/2} (6 b c-11 a d)}{3 b^5}-\frac{(c+d x)^{5/2} \left (-693 a^2 d^2-5 b d x (10 b c-99 a d)+180 a b c d+20 b^2 c^2\right )}{315 b^4 d^2}-\frac{x^3 (c+d x)^{5/2}}{b (a+b x)}+\frac{11 x^2 (c+d x)^{5/2}}{9 b^2} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*(6*b*c - 11*a*d)*(b*c - a*d)*Sqrt[c + d*x])/b^6 + (a^2*(6*b*c - 11*a*d)*(c
+ d*x)^(3/2))/(3*b^5) + (11*x^2*(c + d*x)^(5/2))/(9*b^2) - (x^3*(c + d*x)^(5/2))
/(b*(a + b*x)) - ((c + d*x)^(5/2)*(20*b^2*c^2 + 180*a*b*c*d - 693*a^2*d^2 - 5*b*
d*(10*b*c - 99*a*d)*x))/(315*b^4*d^2) - (a^2*(6*b*c - 11*a*d)*(b*c - a*d)^(3/2)*
ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/b^(13/2)

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Rubi in Sympy [A]  time = 61.1699, size = 218, normalized size = 0.99 \[ - \frac{a^{2} \left (c + d x\right )^{\frac{3}{2}} \left (11 a d - 6 b c\right )}{3 b^{5}} + \frac{a^{2} \sqrt{c + d x} \left (a d - b c\right ) \left (11 a d - 6 b c\right )}{b^{6}} - \frac{a^{2} \left (a d - b c\right )^{\frac{3}{2}} \left (11 a d - 6 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{b^{\frac{13}{2}}} - \frac{x^{3} \left (c + d x\right )^{\frac{5}{2}}}{b \left (a + b x\right )} + \frac{11 x^{2} \left (c + d x\right )^{\frac{5}{2}}}{9 b^{2}} + \frac{8 \left (c + d x\right )^{\frac{5}{2}} \left (\frac{693 a^{2} d^{2}}{8} - \frac{45 a b c d}{2} - \frac{5 b^{2} c^{2}}{2} - \frac{5 b d x \left (99 a d - 10 b c\right )}{8}\right )}{315 b^{4} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**2*(c + d*x)**(3/2)*(11*a*d - 6*b*c)/(3*b**5) + a**2*sqrt(c + d*x)*(a*d - b*c
)*(11*a*d - 6*b*c)/b**6 - a**2*(a*d - b*c)**(3/2)*(11*a*d - 6*b*c)*atan(sqrt(b)*
sqrt(c + d*x)/sqrt(a*d - b*c))/b**(13/2) - x**3*(c + d*x)**(5/2)/(b*(a + b*x)) +
11*x**2*(c + d*x)**(5/2)/(9*b**2) + 8*(c + d*x)**(5/2)*(693*a**2*d**2/8 - 45*a*
b*c*d/2 - 5*b**2*c**2/2 - 5*b*d*x*(99*a*d - 10*b*c)/8)/(315*b**4*d**2)

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Mathematica [A]  time = 0.418519, size = 223, normalized size = 1.01 \[ \frac{\sqrt{c+d x} \left (3465 a^5 d^4+210 a^4 b d^3 (11 d x-31 c)-21 a^3 b^2 d^2 \left (-153 c^2+214 c d x+22 d^2 x^2\right )+18 a^2 b^3 d \left (-10 c^3+131 c^2 d x+47 c d^2 x^2+11 d^3 x^3\right )-10 a b^4 (c+d x)^3 (2 c+11 d x)+10 b^5 x (c+d x)^3 (7 d x-2 c)\right )}{315 b^6 d^2 (a+b x)}-\frac{a^2 (6 b c-11 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{13/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x]*(3465*a^5*d^4 + 10*b^5*x*(c + d*x)^3*(-2*c + 7*d*x) + 210*a^4*b*d
^3*(-31*c + 11*d*x) - 10*a*b^4*(c + d*x)^3*(2*c + 11*d*x) - 21*a^3*b^2*d^2*(-153
*c^2 + 214*c*d*x + 22*d^2*x^2) + 18*a^2*b^3*d*(-10*c^3 + 131*c^2*d*x + 47*c*d^2*
x^2 + 11*d^3*x^3)))/(315*b^6*d^2*(a + b*x)) - (a^2*(6*b*c - 11*a*d)*(b*c - a*d)^
(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/b^(13/2)

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Maple [B]  time = 0.028, size = 415, normalized size = 1.9 \[{\frac{2}{9\,{b}^{2}{d}^{2}} \left ( dx+c \right ) ^{{\frac{9}{2}}}}-{\frac{4\,a}{7\,d{b}^{3}} \left ( dx+c \right ) ^{{\frac{7}{2}}}}-{\frac{2\,c}{7\,{b}^{2}{d}^{2}} \left ( dx+c \right ) ^{{\frac{7}{2}}}}+{\frac{6\,{a}^{2}}{5\,{b}^{4}} \left ( dx+c \right ) ^{{\frac{5}{2}}}}-{\frac{8\,{a}^{3}d}{3\,{b}^{5}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+2\,{\frac{ \left ( dx+c \right ) ^{3/2}{a}^{2}c}{{b}^{4}}}+10\,{\frac{{d}^{2}{a}^{4}\sqrt{dx+c}}{{b}^{6}}}-16\,{\frac{{a}^{3}cd\sqrt{dx+c}}{{b}^{5}}}+6\,{\frac{{a}^{2}{c}^{2}\sqrt{dx+c}}{{b}^{4}}}+{\frac{{d}^{3}{a}^{5}}{{b}^{6} \left ( bdx+ad \right ) }\sqrt{dx+c}}-2\,{\frac{{d}^{2}{a}^{4}\sqrt{dx+c}c}{{b}^{5} \left ( bdx+ad \right ) }}+{\frac{{a}^{3}{c}^{2}d}{{b}^{4} \left ( bdx+ad \right ) }\sqrt{dx+c}}-11\,{\frac{{d}^{3}{a}^{5}}{{b}^{6}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+28\,{\frac{{d}^{2}{a}^{4}c}{{b}^{5}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-23\,{\frac{{a}^{3}{c}^{2}d}{{b}^{4}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+6\,{\frac{{a}^{2}{c}^{3}}{{b}^{3}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9/d^2/b^2*(d*x+c)^(9/2)-4/7/d/b^3*(d*x+c)^(7/2)*a-2/7/d^2/b^2*(d*x+c)^(7/2)*c+
6/5/b^4*a^2*(d*x+c)^(5/2)-8/3*d/b^5*(d*x+c)^(3/2)*a^3+2/b^4*(d*x+c)^(3/2)*a^2*c+
10*d^2/b^6*a^4*(d*x+c)^(1/2)-16*d/b^5*a^3*c*(d*x+c)^(1/2)+6/b^4*a^2*c^2*(d*x+c)^
(1/2)+d^3*a^5/b^6*(d*x+c)^(1/2)/(b*d*x+a*d)-2*d^2*a^4/b^5*(d*x+c)^(1/2)/(b*d*x+a
*d)*c+d*a^3/b^4*(d*x+c)^(1/2)/(b*d*x+a*d)*c^2-11*d^3*a^5/b^6/((a*d-b*c)*b)^(1/2)
*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))+28*d^2*a^4/b^5/((a*d-b*c)*b)^(1/2)*
arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*c-23*d*a^3/b^4/((a*d-b*c)*b)^(1/2)*a
rctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*c^2+6*a^2/b^3/((a*d-b*c)*b)^(1/2)*arc
tan((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^3/(b*x + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.303882, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/630*(315*(6*a^3*b^2*c^2*d^2 - 17*a^4*b*c*d^3 + 11*a^5*d^4 + (6*a^2*b^3*c^2*d^
2 - 17*a^3*b^2*c*d^3 + 11*a^4*b*d^4)*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*(70*b^5*d^4*x^5 - 2
0*a*b^4*c^4 - 180*a^2*b^3*c^3*d + 3213*a^3*b^2*c^2*d^2 - 6510*a^4*b*c*d^3 + 3465
*a^5*d^4 + 10*(19*b^5*c*d^3 - 11*a*b^4*d^4)*x^4 + 2*(75*b^5*c^2*d^2 - 175*a*b^4*
c*d^3 + 99*a^2*b^3*d^4)*x^3 + 2*(5*b^5*c^3*d - 195*a*b^4*c^2*d^2 + 423*a^2*b^3*c
*d^3 - 231*a^3*b^2*d^4)*x^2 - 2*(10*b^5*c^4 + 85*a*b^4*c^3*d - 1179*a^2*b^3*c^2*
d^2 + 2247*a^3*b^2*c*d^3 - 1155*a^4*b*d^4)*x)*sqrt(d*x + c))/(b^7*d^2*x + a*b^6*
d^2), -1/315*(315*(6*a^3*b^2*c^2*d^2 - 17*a^4*b*c*d^3 + 11*a^5*d^4 + (6*a^2*b^3*
c^2*d^2 - 17*a^3*b^2*c*d^3 + 11*a^4*b*d^4)*x)*sqrt(-(b*c - a*d)/b)*arctan(sqrt(d
*x + c)/sqrt(-(b*c - a*d)/b)) - (70*b^5*d^4*x^5 - 20*a*b^4*c^4 - 180*a^2*b^3*c^3
*d + 3213*a^3*b^2*c^2*d^2 - 6510*a^4*b*c*d^3 + 3465*a^5*d^4 + 10*(19*b^5*c*d^3 -
11*a*b^4*d^4)*x^4 + 2*(75*b^5*c^2*d^2 - 175*a*b^4*c*d^3 + 99*a^2*b^3*d^4)*x^3 +
2*(5*b^5*c^3*d - 195*a*b^4*c^2*d^2 + 423*a^2*b^3*c*d^3 - 231*a^3*b^2*d^4)*x^2 -
2*(10*b^5*c^4 + 85*a*b^4*c^3*d - 1179*a^2*b^3*c^2*d^2 + 2247*a^3*b^2*c*d^3 - 11
55*a^4*b*d^4)*x)*sqrt(d*x + c))/(b^7*d^2*x + a*b^6*d^2)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.233016, size = 436, normalized size = 1.98 \[ \frac{{\left (6 \, a^{2} b^{3} c^{3} - 23 \, a^{3} b^{2} c^{2} d + 28 \, a^{4} b c d^{2} - 11 \, a^{5} d^{3}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b^{6}} + \frac{\sqrt{d x + c} a^{3} b^{2} c^{2} d - 2 \, \sqrt{d x + c} a^{4} b c d^{2} + \sqrt{d x + c} a^{5} d^{3}}{{\left ({\left (d x + c\right )} b - b c + a d\right )} b^{6}} + \frac{2 \,{\left (35 \,{\left (d x + c\right )}^{\frac{9}{2}} b^{16} d^{16} - 45 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{16} c d^{16} - 90 \,{\left (d x + c\right )}^{\frac{7}{2}} a b^{15} d^{17} + 189 \,{\left (d x + c\right )}^{\frac{5}{2}} a^{2} b^{14} d^{18} + 315 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} b^{14} c d^{18} + 945 \, \sqrt{d x + c} a^{2} b^{14} c^{2} d^{18} - 420 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{3} b^{13} d^{19} - 2520 \, \sqrt{d x + c} a^{3} b^{13} c d^{19} + 1575 \, \sqrt{d x + c} a^{4} b^{12} d^{20}\right )}}{315 \, b^{18} d^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(6*a^2*b^3*c^3 - 23*a^3*b^2*c^2*d + 28*a^4*b*c*d^2 - 11*a^5*d^3)*arctan(sqrt(d*x
+ c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*b^6) + (sqrt(d*x + c)*a^3*b^
2*c^2*d - 2*sqrt(d*x + c)*a^4*b*c*d^2 + sqrt(d*x + c)*a^5*d^3)/(((d*x + c)*b - b
*c + a*d)*b^6) + 2/315*(35*(d*x + c)^(9/2)*b^16*d^16 - 45*(d*x + c)^(7/2)*b^16*c
*d^16 - 90*(d*x + c)^(7/2)*a*b^15*d^17 + 189*(d*x + c)^(5/2)*a^2*b^14*d^18 + 315
*(d*x + c)^(3/2)*a^2*b^14*c*d^18 + 945*sqrt(d*x + c)*a^2*b^14*c^2*d^18 - 420*(d*
x + c)^(3/2)*a^3*b^13*d^19 - 2520*sqrt(d*x + c)*a^3*b^13*c*d^19 + 1575*sqrt(d*x
+ c)*a^4*b^12*d^20)/(b^18*d^18)