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

Optimal. Leaf size=112 \[ -\frac{2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{7/2}}+\frac{2 \sqrt{c+d x} (b c-a d)^2}{b^3}+\frac{2 (c+d x)^{3/2} (b c-a d)}{3 b^2}+\frac{2 (c+d x)^{5/2}}{5 b} \]

[Out]

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

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Rubi [A]  time = 0.145944, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{7/2}}+\frac{2 \sqrt{c+d x} (b c-a d)^2}{b^3}+\frac{2 (c+d x)^{3/2} (b c-a d)}{3 b^2}+\frac{2 (c+d x)^{5/2}}{5 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.3646, size = 99, normalized size = 0.88 \[ \frac{2 \left (c + d x\right )^{\frac{5}{2}}}{5 b} - \frac{2 \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )}{3 b^{2}} + \frac{2 \sqrt{c + d x} \left (a d - b c\right )^{2}}{b^{3}} - \frac{2 \left (a d - b c\right )^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.124098, size = 108, normalized size = 0.96 \[ \frac{2 \sqrt{c+d x} \left (15 a^2 d^2-5 a b d (7 c+d x)+b^2 \left (23 c^2+11 c d x+3 d^2 x^2\right )\right )}{15 b^3}-\frac{2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.007, size = 263, normalized size = 2.4 \[{\frac{2}{5\,b} \left ( dx+c \right ) ^{{\frac{5}{2}}}}-{\frac{2\,ad}{3\,{b}^{2}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+{\frac{2\,c}{3\,b} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+2\,{\frac{{a}^{2}{d}^{2}\sqrt{dx+c}}{{b}^{3}}}-4\,{\frac{acd\sqrt{dx+c}}{{b}^{2}}}+2\,{\frac{{c}^{2}\sqrt{dx+c}}{b}}-2\,{\frac{{a}^{3}{d}^{3}}{{b}^{3}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+6\,{\frac{{a}^{2}c{d}^{2}}{{b}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-6\,{\frac{a{c}^{2}d}{b\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+2\,{\frac{{c}^{3}}{\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/2)/(b*x+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238921, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x + 2 \, b c - a d - 2 \, \sqrt{d x + c} b \sqrt{\frac{b c - a d}{b}}}{b x + a}\right ) + 2 \,{\left (3 \, b^{2} d^{2} x^{2} + 23 \, b^{2} c^{2} - 35 \, a b c d + 15 \, a^{2} d^{2} +{\left (11 \, b^{2} c d - 5 \, a b d^{2}\right )} x\right )} \sqrt{d x + c}}{15 \, b^{3}}, -\frac{2 \,{\left (15 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) -{\left (3 \, b^{2} d^{2} x^{2} + 23 \, b^{2} c^{2} - 35 \, a b c d + 15 \, a^{2} d^{2} +{\left (11 \, b^{2} c d - 5 \, a b d^{2}\right )} x\right )} \sqrt{d x + c}\right )}}{15 \, b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/15*(15*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*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*(3*b^2*d^2*x^2 +
23*b^2*c^2 - 35*a*b*c*d + 15*a^2*d^2 + (11*b^2*c*d - 5*a*b*d^2)*x)*sqrt(d*x + c)
)/b^3, -2/15*(15*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(-(b*c - a*d)/b)*arctan(sqr
t(d*x + c)/sqrt(-(b*c - a*d)/b)) - (3*b^2*d^2*x^2 + 23*b^2*c^2 - 35*a*b*c*d + 15
*a^2*d^2 + (11*b^2*c*d - 5*a*b*d^2)*x)*sqrt(d*x + c))/b^3]

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Sympy [A]  time = 28.5752, size = 240, normalized size = 2.14 \[ \frac{2 \left (c + d x\right )^{\frac{5}{2}}}{5 b} + \frac{\left (c + d x\right )^{\frac{3}{2}} \left (- 2 a d + 2 b c\right )}{3 b^{2}} + \frac{\sqrt{c + d x} \left (2 a^{2} d^{2} - 4 a b c d + 2 b^{2} c^{2}\right )}{b^{3}} - \frac{2 \left (a d - b c\right )^{3} \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{a d - b c}{b}}} \right )}}{b \sqrt{\frac{a d - b c}{b}}} & \text{for}\: \frac{a d - b c}{b} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{- a d + b c}{b}}} \right )}}{b \sqrt{\frac{- a d + b c}{b}}} & \text{for}\: c + d x > \frac{- a d + b c}{b} \wedge \frac{a d - b c}{b} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{- a d + b c}{b}}} \right )}}{b \sqrt{\frac{- a d + b c}{b}}} & \text{for}\: \frac{a d - b c}{b} < 0 \wedge c + d x < \frac{- a d + b c}{b} \end{cases}\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.220888, size = 231, normalized size = 2.06 \[ \frac{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b^{3}} + \frac{2 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{4} + 5 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{4} c + 15 \, \sqrt{d x + c} b^{4} c^{2} - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{3} d - 30 \, \sqrt{d x + c} a b^{3} c d + 15 \, \sqrt{d x + c} a^{2} b^{2} d^{2}\right )}}{15 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*arctan(sqrt(d*x + c)*b/sqr
t(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*b^3) + 2/15*(3*(d*x + c)^(5/2)*b^4 + 5*
(d*x + c)^(3/2)*b^4*c + 15*sqrt(d*x + c)*b^4*c^2 - 5*(d*x + c)^(3/2)*a*b^3*d - 3
0*sqrt(d*x + c)*a*b^3*c*d + 15*sqrt(d*x + c)*a^2*b^2*d^2)/b^5