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

Optimal. Leaf size=204 \[ -\frac{a^2 (c+d x)^{7/2}}{b^2 (a+b x) (b c-a d)}+\frac{a (4 b c-9 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{11/2}}-\frac{a \sqrt{c+d x} (4 b c-9 a d) (b c-a d)}{b^5}-\frac{a (c+d x)^{3/2} (4 b c-9 a d)}{3 b^4}-\frac{a (c+d x)^{5/2} (4 b c-9 a d)}{5 b^3 (b c-a d)}+\frac{2 (c+d x)^{7/2}}{7 b^2 d} \]

[Out]

-((a*(4*b*c - 9*a*d)*(b*c - a*d)*Sqrt[c + d*x])/b^5) - (a*(4*b*c - 9*a*d)*(c + d
*x)^(3/2))/(3*b^4) - (a*(4*b*c - 9*a*d)*(c + d*x)^(5/2))/(5*b^3*(b*c - a*d)) + (
2*(c + d*x)^(7/2))/(7*b^2*d) - (a^2*(c + d*x)^(7/2))/(b^2*(b*c - a*d)*(a + b*x))
+ (a*(4*b*c - 9*a*d)*(b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c
- a*d]])/b^(11/2)

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Rubi [A]  time = 0.551751, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{a^2 (c+d x)^{7/2}}{b^2 (a+b x) (b c-a d)}+\frac{a (4 b c-9 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{11/2}}-\frac{a \sqrt{c+d x} (4 b c-9 a d) (b c-a d)}{b^5}-\frac{a (c+d x)^{3/2} (4 b c-9 a d)}{3 b^4}-\frac{a (c+d x)^{5/2} (4 b c-9 a d)}{5 b^3 (b c-a d)}+\frac{2 (c+d x)^{7/2}}{7 b^2 d} \]

Antiderivative was successfully verified.

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

[Out]

-((a*(4*b*c - 9*a*d)*(b*c - a*d)*Sqrt[c + d*x])/b^5) - (a*(4*b*c - 9*a*d)*(c + d
*x)^(3/2))/(3*b^4) - (a*(4*b*c - 9*a*d)*(c + d*x)^(5/2))/(5*b^3*(b*c - a*d)) + (
2*(c + d*x)^(7/2))/(7*b^2*d) - (a^2*(c + d*x)^(7/2))/(b^2*(b*c - a*d)*(a + b*x))
+ (a*(4*b*c - 9*a*d)*(b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c
- a*d]])/b^(11/2)

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Rubi in Sympy [A]  time = 50.3252, size = 184, normalized size = 0.9 \[ \frac{a^{2} \left (c + d x\right )^{\frac{7}{2}}}{b^{2} \left (a + b x\right ) \left (a d - b c\right )} - \frac{a \left (c + d x\right )^{\frac{5}{2}} \left (9 a d - 4 b c\right )}{5 b^{3} \left (a d - b c\right )} + \frac{a \left (c + d x\right )^{\frac{3}{2}} \left (9 a d - 4 b c\right )}{3 b^{4}} - \frac{a \sqrt{c + d x} \left (a d - b c\right ) \left (9 a d - 4 b c\right )}{b^{5}} + \frac{a \left (a d - b c\right )^{\frac{3}{2}} \left (9 a d - 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{b^{\frac{11}{2}}} + \frac{2 \left (c + d x\right )^{\frac{7}{2}}}{7 b^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*(c + d*x)**(7/2)/(b**2*(a + b*x)*(a*d - b*c)) - a*(c + d*x)**(5/2)*(9*a*d -
4*b*c)/(5*b**3*(a*d - b*c)) + a*(c + d*x)**(3/2)*(9*a*d - 4*b*c)/(3*b**4) - a*s
qrt(c + d*x)*(a*d - b*c)*(9*a*d - 4*b*c)/b**5 + a*(a*d - b*c)**(3/2)*(9*a*d - 4*
b*c)*atan(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c))/b**(11/2) + 2*(c + d*x)**(7/2)/
(7*b**2*d)

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Mathematica [A]  time = 0.289946, size = 186, normalized size = 0.91 \[ \frac{\sqrt{c+d x} \left (-945 a^4 d^3+210 a^3 b d^2 (8 c-3 d x)+7 a^2 b^2 d \left (-107 c^2+166 c d x+18 d^2 x^2\right )+2 a b^3 \left (15 c^3-277 c^2 d x-109 c d^2 x^2-27 d^3 x^3\right )+30 b^4 x (c+d x)^3\right )}{105 b^5 d (a+b x)}+\frac{a (4 b c-9 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x]*(-945*a^4*d^3 + 210*a^3*b*d^2*(8*c - 3*d*x) + 30*b^4*x*(c + d*x)^
3 + 7*a^2*b^2*d*(-107*c^2 + 166*c*d*x + 18*d^2*x^2) + 2*a*b^3*(15*c^3 - 277*c^2*
d*x - 109*c*d^2*x^2 - 27*d^3*x^3)))/(105*b^5*d*(a + b*x)) + (a*(4*b*c - 9*a*d)*(
b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/b^(11/2)

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Maple [B]  time = 0.026, size = 377, normalized size = 1.9 \[{\frac{2}{7\,{b}^{2}d} \left ( dx+c \right ) ^{{\frac{7}{2}}}}-{\frac{4\,a}{5\,{b}^{3}} \left ( dx+c \right ) ^{{\frac{5}{2}}}}+2\,{\frac{d \left ( dx+c \right ) ^{3/2}{a}^{2}}{{b}^{4}}}-{\frac{4\,ac}{3\,{b}^{3}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-8\,{\frac{{d}^{2}{a}^{3}\sqrt{dx+c}}{{b}^{5}}}+12\,{\frac{{a}^{2}dc\sqrt{dx+c}}{{b}^{4}}}-4\,{\frac{a{c}^{2}\sqrt{dx+c}}{{b}^{3}}}-{\frac{{d}^{3}{a}^{4}}{{b}^{5} \left ( bdx+ad \right ) }\sqrt{dx+c}}+2\,{\frac{{d}^{2}{a}^{3}\sqrt{dx+c}c}{{b}^{4} \left ( bdx+ad \right ) }}-{\frac{{a}^{2}{c}^{2}d}{{b}^{3} \left ( bdx+ad \right ) }\sqrt{dx+c}}+9\,{\frac{{d}^{3}{a}^{4}}{{b}^{5}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-22\,{\frac{{d}^{2}{a}^{3}c}{{b}^{4}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+17\,{\frac{{a}^{2}{c}^{2}d}{{b}^{3}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-4\,{\frac{a{c}^{3}}{{b}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*(d*x+c)^(7/2)/b^2/d-4/5/b^3*a*(d*x+c)^(5/2)+2*d/b^4*(d*x+c)^(3/2)*a^2-4/3/b^
3*(d*x+c)^(3/2)*a*c-8*d^2/b^5*a^3*(d*x+c)^(1/2)+12*d/b^4*a^2*c*(d*x+c)^(1/2)-4/b
^3*a*c^2*(d*x+c)^(1/2)-d^3*a^4/b^5*(d*x+c)^(1/2)/(b*d*x+a*d)+2*d^2*a^3/b^4*(d*x+
c)^(1/2)/(b*d*x+a*d)*c-d*a^2/b^3*(d*x+c)^(1/2)/(b*d*x+a*d)*c^2+9*d^3*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))-22*d^2*a^3/b^4/((a*
d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*c+17*d*a^2/b^3/((a*d
-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*c^2-4*a/b^2/((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^2/(b*x + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.300533, size = 1, normalized size = 0. \[ \left [\frac{105 \,{\left (4 \, a^{2} b^{2} c^{2} d - 13 \, a^{3} b c d^{2} + 9 \, a^{4} d^{3} +{\left (4 \, a b^{3} c^{2} d - 13 \, a^{2} b^{2} c d^{2} + 9 \, a^{3} b d^{3}\right )} x\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x + 2 \, b c - a d + 2 \, \sqrt{d x + c} b \sqrt{\frac{b c - a d}{b}}}{b x + a}\right ) + 2 \,{\left (30 \, b^{4} d^{3} x^{4} + 30 \, a b^{3} c^{3} - 749 \, a^{2} b^{2} c^{2} d + 1680 \, a^{3} b c d^{2} - 945 \, a^{4} d^{3} + 18 \,{\left (5 \, b^{4} c d^{2} - 3 \, a b^{3} d^{3}\right )} x^{3} + 2 \,{\left (45 \, b^{4} c^{2} d - 109 \, a b^{3} c d^{2} + 63 \, a^{2} b^{2} d^{3}\right )} x^{2} + 2 \,{\left (15 \, b^{4} c^{3} - 277 \, a b^{3} c^{2} d + 581 \, a^{2} b^{2} c d^{2} - 315 \, a^{3} b d^{3}\right )} x\right )} \sqrt{d x + c}}{210 \,{\left (b^{6} d x + a b^{5} d\right )}}, \frac{105 \,{\left (4 \, a^{2} b^{2} c^{2} d - 13 \, a^{3} b c d^{2} + 9 \, a^{4} d^{3} +{\left (4 \, a b^{3} c^{2} d - 13 \, a^{2} b^{2} c d^{2} + 9 \, a^{3} b d^{3}\right )} x\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) +{\left (30 \, b^{4} d^{3} x^{4} + 30 \, a b^{3} c^{3} - 749 \, a^{2} b^{2} c^{2} d + 1680 \, a^{3} b c d^{2} - 945 \, a^{4} d^{3} + 18 \,{\left (5 \, b^{4} c d^{2} - 3 \, a b^{3} d^{3}\right )} x^{3} + 2 \,{\left (45 \, b^{4} c^{2} d - 109 \, a b^{3} c d^{2} + 63 \, a^{2} b^{2} d^{3}\right )} x^{2} + 2 \,{\left (15 \, b^{4} c^{3} - 277 \, a b^{3} c^{2} d + 581 \, a^{2} b^{2} c d^{2} - 315 \, a^{3} b d^{3}\right )} x\right )} \sqrt{d x + c}}{105 \,{\left (b^{6} d x + a b^{5} d\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/210*(105*(4*a^2*b^2*c^2*d - 13*a^3*b*c*d^2 + 9*a^4*d^3 + (4*a*b^3*c^2*d - 13*
a^2*b^2*c*d^2 + 9*a^3*b*d^3)*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*(30*b^4*d^3*x^4 + 30*a*b^3*
c^3 - 749*a^2*b^2*c^2*d + 1680*a^3*b*c*d^2 - 945*a^4*d^3 + 18*(5*b^4*c*d^2 - 3*a
*b^3*d^3)*x^3 + 2*(45*b^4*c^2*d - 109*a*b^3*c*d^2 + 63*a^2*b^2*d^3)*x^2 + 2*(15*
b^4*c^3 - 277*a*b^3*c^2*d + 581*a^2*b^2*c*d^2 - 315*a^3*b*d^3)*x)*sqrt(d*x + c))
/(b^6*d*x + a*b^5*d), 1/105*(105*(4*a^2*b^2*c^2*d - 13*a^3*b*c*d^2 + 9*a^4*d^3 +
(4*a*b^3*c^2*d - 13*a^2*b^2*c*d^2 + 9*a^3*b*d^3)*x)*sqrt(-(b*c - a*d)/b)*arctan
(sqrt(d*x + c)/sqrt(-(b*c - a*d)/b)) + (30*b^4*d^3*x^4 + 30*a*b^3*c^3 - 749*a^2*
b^2*c^2*d + 1680*a^3*b*c*d^2 - 945*a^4*d^3 + 18*(5*b^4*c*d^2 - 3*a*b^3*d^3)*x^3
+ 2*(45*b^4*c^2*d - 109*a*b^3*c*d^2 + 63*a^2*b^2*d^3)*x^2 + 2*(15*b^4*c^3 - 277*
a*b^3*c^2*d + 581*a^2*b^2*c*d^2 - 315*a^3*b*d^3)*x)*sqrt(d*x + c))/(b^6*d*x + a*
b^5*d)]

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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**2*(d*x+c)**(5/2)/(b*x+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.242542, size = 385, normalized size = 1.89 \[ -\frac{{\left (4 \, a b^{3} c^{3} - 17 \, a^{2} b^{2} c^{2} d + 22 \, a^{3} b c d^{2} - 9 \, a^{4} d^{3}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b^{5}} - \frac{\sqrt{d x + c} a^{2} b^{2} c^{2} d - 2 \, \sqrt{d x + c} a^{3} b c d^{2} + \sqrt{d x + c} a^{4} d^{3}}{{\left ({\left (d x + c\right )} b - b c + a d\right )} b^{5}} + \frac{2 \,{\left (15 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{12} d^{6} - 42 \,{\left (d x + c\right )}^{\frac{5}{2}} a b^{11} d^{7} - 70 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{11} c d^{7} - 210 \, \sqrt{d x + c} a b^{11} c^{2} d^{7} + 105 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} b^{10} d^{8} + 630 \, \sqrt{d x + c} a^{2} b^{10} c d^{8} - 420 \, \sqrt{d x + c} a^{3} b^{9} d^{9}\right )}}{105 \, b^{14} d^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(4*a*b^3*c^3 - 17*a^2*b^2*c^2*d + 22*a^3*b*c*d^2 - 9*a^4*d^3)*arctan(sqrt(d*x +
c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*b^5) - (sqrt(d*x + c)*a^2*b^2*
c^2*d - 2*sqrt(d*x + c)*a^3*b*c*d^2 + sqrt(d*x + c)*a^4*d^3)/(((d*x + c)*b - b*c
+ a*d)*b^5) + 2/105*(15*(d*x + c)^(7/2)*b^12*d^6 - 42*(d*x + c)^(5/2)*a*b^11*d^
7 - 70*(d*x + c)^(3/2)*a*b^11*c*d^7 - 210*sqrt(d*x + c)*a*b^11*c^2*d^7 + 105*(d*
x + c)^(3/2)*a^2*b^10*d^8 + 630*sqrt(d*x + c)*a^2*b^10*c*d^8 - 420*sqrt(d*x + c)
*a^3*b^9*d^9)/(b^14*d^7)