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

Optimal. Leaf size=135 \[ \frac{2 a (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{9/2}}-\frac{2 a \sqrt{c+d x} (b c-a d)^2}{b^4}-\frac{2 a (c+d x)^{3/2} (b c-a d)}{3 b^3}-\frac{2 a (c+d x)^{5/2}}{5 b^2}+\frac{2 (c+d x)^{7/2}}{7 b d} \]

[Out]

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

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Rubi [A]  time = 0.2202, antiderivative size = 135, 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 \[ \frac{2 a (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{9/2}}-\frac{2 a \sqrt{c+d x} (b c-a d)^2}{b^4}-\frac{2 a (c+d x)^{3/2} (b c-a d)}{3 b^3}-\frac{2 a (c+d x)^{5/2}}{5 b^2}+\frac{2 (c+d x)^{7/2}}{7 b d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 30.3947, size = 122, normalized size = 0.9 \[ - \frac{2 a \left (c + d x\right )^{\frac{5}{2}}}{5 b^{2}} + \frac{2 a \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )}{3 b^{3}} - \frac{2 a \sqrt{c + d x} \left (a d - b c\right )^{2}}{b^{4}} + \frac{2 a \left (a d - b c\right )^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{b^{\frac{9}{2}}} + \frac{2 \left (c + d x\right )^{\frac{7}{2}}}{7 b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.171546, size = 131, normalized size = 0.97 \[ \frac{2 \sqrt{c+d x} \left (-105 a^3 d^3+35 a^2 b d^2 (7 c+d x)-7 a b^2 d \left (23 c^2+11 c d x+3 d^2 x^2\right )+15 b^3 (c+d x)^3\right )}{105 b^4 d}+\frac{2 a (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c + d*x]*(-105*a^3*d^3 + 15*b^3*(c + d*x)^3 + 35*a^2*b*d^2*(7*c + d*x) -
7*a*b^2*d*(23*c^2 + 11*c*d*x + 3*d^2*x^2)))/(105*b^4*d) + (2*a*(b*c - a*d)^(5/2
)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/b^(9/2)

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Maple [B]  time = 0.013, size = 291, normalized size = 2.2 \[{\frac{2}{7\,bd} \left ( dx+c \right ) ^{{\frac{7}{2}}}}-{\frac{2\,a}{5\,{b}^{2}} \left ( dx+c \right ) ^{{\frac{5}{2}}}}+{\frac{2\,d{a}^{2}}{3\,{b}^{3}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{2\,ac}{3\,{b}^{2}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-2\,{\frac{{d}^{2}{a}^{3}\sqrt{dx+c}}{{b}^{4}}}+4\,{\frac{d{a}^{2}c\sqrt{dx+c}}{{b}^{3}}}-2\,{\frac{a{c}^{2}\sqrt{dx+c}}{{b}^{2}}}+2\,{\frac{{d}^{3}{a}^{4}}{{b}^{4}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-6\,{\frac{{d}^{2}{a}^{3}c}{{b}^{3}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+6\,{\frac{{a}^{2}{c}^{2}d}{{b}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-2\,{\frac{a{c}^{3}}{b\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(d*x+c)^(5/2)/(b*x+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.248069, size = 1, normalized size = 0.01 \[ \left [\frac{105 \,{\left (a b^{2} c^{2} d - 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x + 2 \, b c - a d + 2 \, \sqrt{d x + c} b \sqrt{\frac{b c - a d}{b}}}{b x + a}\right ) + 2 \,{\left (15 \, b^{3} d^{3} x^{3} + 15 \, b^{3} c^{3} - 161 \, a b^{2} c^{2} d + 245 \, a^{2} b c d^{2} - 105 \, a^{3} d^{3} + 3 \,{\left (15 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} +{\left (45 \, b^{3} c^{2} d - 77 \, a b^{2} c d^{2} + 35 \, a^{2} b d^{3}\right )} x\right )} \sqrt{d x + c}}{105 \, b^{4} d}, \frac{2 \,{\left (105 \,{\left (a b^{2} c^{2} d - 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) +{\left (15 \, b^{3} d^{3} x^{3} + 15 \, b^{3} c^{3} - 161 \, a b^{2} c^{2} d + 245 \, a^{2} b c d^{2} - 105 \, a^{3} d^{3} + 3 \,{\left (15 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} +{\left (45 \, b^{3} c^{2} d - 77 \, a b^{2} c d^{2} + 35 \, a^{2} b d^{3}\right )} x\right )} \sqrt{d x + c}\right )}}{105 \, b^{4} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/105*(105*(a*b^2*c^2*d - 2*a^2*b*c*d^2 + a^3*d^3)*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*(15*b^3
*d^3*x^3 + 15*b^3*c^3 - 161*a*b^2*c^2*d + 245*a^2*b*c*d^2 - 105*a^3*d^3 + 3*(15*
b^3*c*d^2 - 7*a*b^2*d^3)*x^2 + (45*b^3*c^2*d - 77*a*b^2*c*d^2 + 35*a^2*b*d^3)*x)
*sqrt(d*x + c))/(b^4*d), 2/105*(105*(a*b^2*c^2*d - 2*a^2*b*c*d^2 + a^3*d^3)*sqrt
(-(b*c - a*d)/b)*arctan(sqrt(d*x + c)/sqrt(-(b*c - a*d)/b)) + (15*b^3*d^3*x^3 +
15*b^3*c^3 - 161*a*b^2*c^2*d + 245*a^2*b*c*d^2 - 105*a^3*d^3 + 3*(15*b^3*c*d^2 -
7*a*b^2*d^3)*x^2 + (45*b^3*c^2*d - 77*a*b^2*c*d^2 + 35*a^2*b*d^3)*x)*sqrt(d*x +
c))/(b^4*d)]

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Sympy [A]  time = 43.5803, size = 267, normalized size = 1.98 \[ - \frac{2 a \left (c + d x\right )^{\frac{5}{2}}}{5 b^{2}} + \frac{2 a \left (a d - b c\right )^{3} \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{a d - b c}{b}}} \right )}}{b \sqrt{\frac{a d - b c}{b}}} & \text{for}\: \frac{a d - b c}{b} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{- a d + b c}{b}}} \right )}}{b \sqrt{\frac{- a d + b c}{b}}} & \text{for}\: c + d x > \frac{- a d + b c}{b} \wedge \frac{a d - b c}{b} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{- a d + b c}{b}}} \right )}}{b \sqrt{\frac{- a d + b c}{b}}} & \text{for}\: \frac{a d - b c}{b} < 0 \wedge c + d x < \frac{- a d + b c}{b} \end{cases}\right )}{b^{4}} + \frac{2 \left (c + d x\right )^{\frac{7}{2}}}{7 b d} + \frac{\left (c + d x\right )^{\frac{3}{2}} \left (2 a^{2} d - 2 a b c\right )}{3 b^{3}} + \frac{\sqrt{c + d x} \left (- 2 a^{3} d^{2} + 4 a^{2} b c d - 2 a b^{2} c^{2}\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.256521, size = 286, normalized size = 2.12 \[ -\frac{2 \,{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b^{4}} + \frac{2 \,{\left (15 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{6} d^{6} - 21 \,{\left (d x + c\right )}^{\frac{5}{2}} a b^{5} d^{7} - 35 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{5} c d^{7} - 105 \, \sqrt{d x + c} a b^{5} c^{2} d^{7} + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} b^{4} d^{8} + 210 \, \sqrt{d x + c} a^{2} b^{4} c d^{8} - 105 \, \sqrt{d x + c} a^{3} b^{3} d^{9}\right )}}{105 \, b^{7} d^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - 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^4) + 2/105*(15*(d*x + c)^(7/2)*b
^6*d^6 - 21*(d*x + c)^(5/2)*a*b^5*d^7 - 35*(d*x + c)^(3/2)*a*b^5*c*d^7 - 105*sqr
t(d*x + c)*a*b^5*c^2*d^7 + 35*(d*x + c)^(3/2)*a^2*b^4*d^8 + 210*sqrt(d*x + c)*a^
2*b^4*c*d^8 - 105*sqrt(d*x + c)*a^3*b^3*d^9)/(b^7*d^7)