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

Optimal. Leaf size=169 \[ -\frac{2 a^2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{11/2}}+\frac{2 a^2 \sqrt{c+d x} (b c-a d)^2}{b^5}+\frac{2 a^2 (c+d x)^{3/2} (b c-a d)}{3 b^4}+\frac{2 a^2 (c+d x)^{5/2}}{5 b^3}-\frac{2 (c+d x)^{7/2} (a d+b c)}{7 b^2 d^2}+\frac{2 (c+d x)^{9/2}}{9 b d^2} \]

[Out]

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

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Rubi [A]  time = 0.314229, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{2 a^2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{11/2}}+\frac{2 a^2 \sqrt{c+d x} (b c-a d)^2}{b^5}+\frac{2 a^2 (c+d x)^{3/2} (b c-a d)}{3 b^4}+\frac{2 a^2 (c+d x)^{5/2}}{5 b^3}-\frac{2 (c+d x)^{7/2} (a d+b c)}{7 b^2 d^2}+\frac{2 (c+d x)^{9/2}}{9 b d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 45.7572, size = 156, normalized size = 0.92 \[ \frac{2 a^{2} \left (c + d x\right )^{\frac{5}{2}}}{5 b^{3}} - \frac{2 a^{2} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )}{3 b^{4}} + \frac{2 a^{2} \sqrt{c + d x} \left (a d - b c\right )^{2}}{b^{5}} - \frac{2 a^{2} \left (a d - b c\right )^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{b^{\frac{11}{2}}} + \frac{2 \left (c + d x\right )^{\frac{9}{2}}}{9 b d^{2}} - \frac{2 \left (c + d x\right )^{\frac{7}{2}} \left (a d + b c\right )}{7 b^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.231315, size = 159, normalized size = 0.94 \[ \frac{2 \sqrt{c+d x} \left (315 a^4 d^4-105 a^3 b d^3 (7 c+d x)+21 a^2 b^2 d^2 \left (23 c^2+11 c d x+3 d^2 x^2\right )-45 a b^3 d (c+d x)^3-5 b^4 (2 c-7 d x) (c+d x)^3\right )}{315 b^5 d^2}-\frac{2 a^2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c + d*x]*(315*a^4*d^4 - 45*a*b^3*d*(c + d*x)^3 - 5*b^4*(2*c - 7*d*x)*(c
+ d*x)^3 - 105*a^3*b*d^3*(7*c + d*x) + 21*a^2*b^2*d^2*(23*c^2 + 11*c*d*x + 3*d^2
*x^2)))/(315*b^5*d^2) - (2*a^2*(b*c - a*d)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])
/Sqrt[b*c - a*d]])/b^(11/2)

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Maple [B]  time = 0.015, size = 331, normalized size = 2. \[{\frac{2}{9\,b{d}^{2}} \left ( dx+c \right ) ^{{\frac{9}{2}}}}-{\frac{2\,a}{7\,{b}^{2}d} \left ( dx+c \right ) ^{{\frac{7}{2}}}}-{\frac{2\,c}{7\,b{d}^{2}} \left ( dx+c \right ) ^{{\frac{7}{2}}}}+{\frac{2\,{a}^{2}}{5\,{b}^{3}} \left ( dx+c \right ) ^{{\frac{5}{2}}}}-{\frac{2\,{a}^{3}d}{3\,{b}^{4}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+{\frac{2\,{a}^{2}c}{3\,{b}^{3}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+2\,{\frac{{d}^{2}{a}^{4}\sqrt{dx+c}}{{b}^{5}}}-4\,{\frac{{a}^{3}cd\sqrt{dx+c}}{{b}^{4}}}+2\,{\frac{{a}^{2}{c}^{2}\sqrt{dx+c}}{{b}^{3}}}-2\,{\frac{{d}^{3}{a}^{5}}{{b}^{5}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+6\,{\frac{{d}^{2}{a}^{4}c}{{b}^{4}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-6\,{\frac{{a}^{3}{c}^{2}d}{{b}^{3}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+2\,{\frac{{a}^{2}{c}^{3}}{{b}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251144, size = 1, normalized size = 0.01 \[ \left [\frac{315 \,{\left (a^{2} b^{2} c^{2} d^{2} - 2 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x + 2 \, b c - a d - 2 \, \sqrt{d x + c} b \sqrt{\frac{b c - a d}{b}}}{b x + a}\right ) + 2 \,{\left (35 \, b^{4} d^{4} x^{4} - 10 \, b^{4} c^{4} - 45 \, a b^{3} c^{3} d + 483 \, a^{2} b^{2} c^{2} d^{2} - 735 \, a^{3} b c d^{3} + 315 \, a^{4} d^{4} + 5 \,{\left (19 \, b^{4} c d^{3} - 9 \, a b^{3} d^{4}\right )} x^{3} + 3 \,{\left (25 \, b^{4} c^{2} d^{2} - 45 \, a b^{3} c d^{3} + 21 \, a^{2} b^{2} d^{4}\right )} x^{2} +{\left (5 \, b^{4} c^{3} d - 135 \, a b^{3} c^{2} d^{2} + 231 \, a^{2} b^{2} c d^{3} - 105 \, a^{3} b d^{4}\right )} x\right )} \sqrt{d x + c}}{315 \, b^{5} d^{2}}, -\frac{2 \,{\left (315 \,{\left (a^{2} b^{2} c^{2} d^{2} - 2 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) -{\left (35 \, b^{4} d^{4} x^{4} - 10 \, b^{4} c^{4} - 45 \, a b^{3} c^{3} d + 483 \, a^{2} b^{2} c^{2} d^{2} - 735 \, a^{3} b c d^{3} + 315 \, a^{4} d^{4} + 5 \,{\left (19 \, b^{4} c d^{3} - 9 \, a b^{3} d^{4}\right )} x^{3} + 3 \,{\left (25 \, b^{4} c^{2} d^{2} - 45 \, a b^{3} c d^{3} + 21 \, a^{2} b^{2} d^{4}\right )} x^{2} +{\left (5 \, b^{4} c^{3} d - 135 \, a b^{3} c^{2} d^{2} + 231 \, a^{2} b^{2} c d^{3} - 105 \, a^{3} b d^{4}\right )} x\right )} \sqrt{d x + c}\right )}}{315 \, b^{5} d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/315*(315*(a^2*b^2*c^2*d^2 - 2*a^3*b*c*d^3 + a^4*d^4)*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*(35
*b^4*d^4*x^4 - 10*b^4*c^4 - 45*a*b^3*c^3*d + 483*a^2*b^2*c^2*d^2 - 735*a^3*b*c*d
^3 + 315*a^4*d^4 + 5*(19*b^4*c*d^3 - 9*a*b^3*d^4)*x^3 + 3*(25*b^4*c^2*d^2 - 45*a
*b^3*c*d^3 + 21*a^2*b^2*d^4)*x^2 + (5*b^4*c^3*d - 135*a*b^3*c^2*d^2 + 231*a^2*b^
2*c*d^3 - 105*a^3*b*d^4)*x)*sqrt(d*x + c))/(b^5*d^2), -2/315*(315*(a^2*b^2*c^2*d
^2 - 2*a^3*b*c*d^3 + a^4*d^4)*sqrt(-(b*c - a*d)/b)*arctan(sqrt(d*x + c)/sqrt(-(b
*c - a*d)/b)) - (35*b^4*d^4*x^4 - 10*b^4*c^4 - 45*a*b^3*c^3*d + 483*a^2*b^2*c^2*
d^2 - 735*a^3*b*c*d^3 + 315*a^4*d^4 + 5*(19*b^4*c*d^3 - 9*a*b^3*d^4)*x^3 + 3*(25
*b^4*c^2*d^2 - 45*a*b^3*c*d^3 + 21*a^2*b^2*d^4)*x^2 + (5*b^4*c^3*d - 135*a*b^3*c
^2*d^2 + 231*a^2*b^2*c*d^3 - 105*a^3*b*d^4)*x)*sqrt(d*x + c))/(b^5*d^2)]

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Sympy [A]  time = 57.7372, size = 304, normalized size = 1.8 \[ \frac{2 a^{2} \left (c + d x\right )^{\frac{5}{2}}}{5 b^{3}} - \frac{2 a^{2} \left (a d - b c\right )^{3} \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{a d - b c}{b}}} \right )}}{b \sqrt{\frac{a d - b c}{b}}} & \text{for}\: \frac{a d - b c}{b} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{- a d + b c}{b}}} \right )}}{b \sqrt{\frac{- a d + b c}{b}}} & \text{for}\: c + d x > \frac{- a d + b c}{b} \wedge \frac{a d - b c}{b} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{- a d + b c}{b}}} \right )}}{b \sqrt{\frac{- a d + b c}{b}}} & \text{for}\: \frac{a d - b c}{b} < 0 \wedge c + d x < \frac{- a d + b c}{b} \end{cases}\right )}{b^{5}} + \frac{2 \left (c + d x\right )^{\frac{9}{2}}}{9 b d^{2}} + \frac{\left (c + d x\right )^{\frac{7}{2}} \left (- 2 a d - 2 b c\right )}{7 b^{2} d^{2}} + \frac{\left (c + d x\right )^{\frac{3}{2}} \left (- 2 a^{3} d + 2 a^{2} b c\right )}{3 b^{4}} + \frac{\sqrt{c + d x} \left (2 a^{4} d^{2} - 4 a^{3} b c d + 2 a^{2} b^{2} c^{2}\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*(c + d*x)**(5/2)/(5*b**3) - 2*a**2*(a*d - b*c)**3*Piecewise((atan(sqrt(c
+ d*x)/sqrt((a*d - b*c)/b))/(b*sqrt((a*d - b*c)/b)), (a*d - b*c)/b > 0), (-acoth
(sqrt(c + d*x)/sqrt((-a*d + b*c)/b))/(b*sqrt((-a*d + b*c)/b)), ((a*d - b*c)/b <
0) & (c + d*x > (-a*d + b*c)/b)), (-atanh(sqrt(c + d*x)/sqrt((-a*d + b*c)/b))/(b
*sqrt((-a*d + b*c)/b)), ((a*d - b*c)/b < 0) & (c + d*x < (-a*d + b*c)/b)))/b**5
+ 2*(c + d*x)**(9/2)/(9*b*d**2) + (c + d*x)**(7/2)*(-2*a*d - 2*b*c)/(7*b**2*d**2
) + (c + d*x)**(3/2)*(-2*a**3*d + 2*a**2*b*c)/(3*b**4) + sqrt(c + d*x)*(2*a**4*d
**2 - 4*a**3*b*c*d + 2*a**2*b**2*c**2)/b**5

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GIAC/XCAS [A]  time = 0.325361, size = 340, normalized size = 2.01 \[ \frac{2 \,{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b^{5}} + \frac{2 \,{\left (35 \,{\left (d x + c\right )}^{\frac{9}{2}} b^{8} d^{16} - 45 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{8} c d^{16} - 45 \,{\left (d x + c\right )}^{\frac{7}{2}} a b^{7} d^{17} + 63 \,{\left (d x + c\right )}^{\frac{5}{2}} a^{2} b^{6} d^{18} + 105 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} b^{6} c d^{18} + 315 \, \sqrt{d x + c} a^{2} b^{6} c^{2} d^{18} - 105 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{3} b^{5} d^{19} - 630 \, \sqrt{d x + c} a^{3} b^{5} c d^{19} + 315 \, \sqrt{d x + c} a^{4} b^{4} d^{20}\right )}}{315 \, b^{9} d^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - 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^5) + 2/315*(35*(d*x + c)^(9/2)*
b^8*d^16 - 45*(d*x + c)^(7/2)*b^8*c*d^16 - 45*(d*x + c)^(7/2)*a*b^7*d^17 + 63*(d
*x + c)^(5/2)*a^2*b^6*d^18 + 105*(d*x + c)^(3/2)*a^2*b^6*c*d^18 + 315*sqrt(d*x +
c)*a^2*b^6*c^2*d^18 - 105*(d*x + c)^(3/2)*a^3*b^5*d^19 - 630*sqrt(d*x + c)*a^3*
b^5*c*d^19 + 315*sqrt(d*x + c)*a^4*b^4*d^20)/(b^9*d^18)