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

Optimal. Leaf size=154 \[ -\frac{15 \sqrt{a} (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 c^{7/2}}+\frac{15 \sqrt{a+b x} (b c-a d)^2}{4 c^3 \sqrt{c+d x}}-\frac{5 (a+b x)^{3/2} (b c-a d)}{4 c^2 x \sqrt{c+d x}}-\frac{(a+b x)^{5/2}}{2 c x^2 \sqrt{c+d x}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.267505, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{15 \sqrt{a} (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 c^{7/2}}+\frac{15 \sqrt{a+b x} (b c-a d)^2}{4 c^3 \sqrt{c+d x}}-\frac{5 (a+b x)^{3/2} (b c-a d)}{4 c^2 x \sqrt{c+d x}}-\frac{(a+b x)^{5/2}}{2 c x^2 \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 23.0412, size = 136, normalized size = 0.88 \[ - \frac{15 \sqrt{a} \left (a d - b c\right )^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 c^{\frac{7}{2}}} - \frac{5 a \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x}}{2 c^{2} x^{2}} + \frac{15 a \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )}{4 c^{3} x} + \frac{2 \left (a + b x\right )^{\frac{5}{2}}}{c x^{2} \sqrt{c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15*sqrt(a)*(a*d - b*c)**2*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/
(4*c**(7/2)) - 5*a*(a + b*x)**(3/2)*sqrt(c + d*x)/(2*c**2*x**2) + 15*a*sqrt(a +
b*x)*sqrt(c + d*x)*(a*d - b*c)/(4*c**3*x) + 2*(a + b*x)**(5/2)/(c*x**2*sqrt(c +
d*x))

_______________________________________________________________________________________

Mathematica [A]  time = 0.306612, size = 158, normalized size = 1.03 \[ \frac{2 \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (-\frac{2 a^2 c}{x^2}+\frac{a (7 a d-9 b c)}{x}+\frac{8 (b c-a d)^2}{c+d x}\right )+15 \sqrt{a} \log (x) (b c-a d)^2-15 \sqrt{a} (b c-a d)^2 \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{8 c^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*((-2*a^2*c)/x^2 + (a*(-9*b*c + 7*a*d))/x
+ (8*(b*c - a*d)^2)/(c + d*x)) + 15*Sqrt[a]*(b*c - a*d)^2*Log[x] - 15*Sqrt[a]*(b
*c - a*d)^2*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c +
d*x]])/(8*c^(7/2))

_______________________________________________________________________________________

Maple [B]  time = 0.042, size = 507, normalized size = 3.3 \[ -{\frac{1}{8\,{c}^{3}{x}^{2}}\sqrt{bx+a} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{3}{d}^{3}-30\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{2}bc{d}^{2}+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}a{b}^{2}{c}^{2}d+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{3}c{d}^{2}-30\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{2}b{c}^{2}d+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}a{b}^{2}{c}^{3}-30\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{d}^{2}{a}^{2}{x}^{2}\sqrt{ac}+50\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dbca{x}^{2}\sqrt{ac}-16\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{b}^{2}{c}^{2}{x}^{2}\sqrt{ac}-10\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dc{a}^{2}x\sqrt{ac}+18\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }b{c}^{2}ax\sqrt{ac}+4\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{c}^{2}{a}^{2}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{dx+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(b*x+a)^(1/2)*(15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a
*c)/x)*x^3*a^3*d^3-30*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*
c)/x)*x^3*a^2*b*c*d^2+15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2
*a*c)/x)*x^3*a*b^2*c^2*d+15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2
)+2*a*c)/x)*x^2*a^3*c*d^2-30*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/
2)+2*a*c)/x)*x^2*a^2*b*c^2*d+15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^
(1/2)+2*a*c)/x)*x^2*a*b^2*c^3-30*((b*x+a)*(d*x+c))^(1/2)*d^2*a^2*x^2*(a*c)^(1/2)
+50*((b*x+a)*(d*x+c))^(1/2)*d*b*c*a*x^2*(a*c)^(1/2)-16*((b*x+a)*(d*x+c))^(1/2)*b
^2*c^2*x^2*(a*c)^(1/2)-10*((b*x+a)*(d*x+c))^(1/2)*d*c*a^2*x*(a*c)^(1/2)+18*((b*x
+a)*(d*x+c))^(1/2)*b*c^2*a*x*(a*c)^(1/2)+4*((b*x+a)*(d*x+c))^(1/2)*c^2*a^2*(a*c)
^(1/2))/c^3/((b*x+a)*(d*x+c))^(1/2)/(a*c)^(1/2)/x^2/(d*x+c)^(1/2)

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.43755, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left ({\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3} +{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} x^{2}\right )} \sqrt{\frac{a}{c}} \log \left (\frac{8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \,{\left (2 \, a c^{2} +{\left (b c^{2} + a c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{a}{c}} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \,{\left (2 \, a^{2} c^{2} -{\left (8 \, b^{2} c^{2} - 25 \, a b c d + 15 \, a^{2} d^{2}\right )} x^{2} +{\left (9 \, a b c^{2} - 5 \, a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{16 \,{\left (c^{3} d x^{3} + c^{4} x^{2}\right )}}, -\frac{15 \,{\left ({\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3} +{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} x^{2}\right )} \sqrt{-\frac{a}{c}} \arctan \left (\frac{2 \, a c +{\left (b c + a d\right )} x}{2 \, \sqrt{b x + a} \sqrt{d x + c} c \sqrt{-\frac{a}{c}}}\right ) + 2 \,{\left (2 \, a^{2} c^{2} -{\left (8 \, b^{2} c^{2} - 25 \, a b c d + 15 \, a^{2} d^{2}\right )} x^{2} +{\left (9 \, a b c^{2} - 5 \, a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{8 \,{\left (c^{3} d x^{3} + c^{4} x^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(15*((b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*x^3 + (b^2*c^3 - 2*a*b*c^2*d + a^
2*c*d^2)*x^2)*sqrt(a/c)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4
*(2*a*c^2 + (b*c^2 + a*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(a/c) + 8*(a*b*c^
2 + a^2*c*d)*x)/x^2) - 4*(2*a^2*c^2 - (8*b^2*c^2 - 25*a*b*c*d + 15*a^2*d^2)*x^2
+ (9*a*b*c^2 - 5*a^2*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(c^3*d*x^3 + c^4*x^2),
-1/8*(15*((b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*x^3 + (b^2*c^3 - 2*a*b*c^2*d + a^
2*c*d^2)*x^2)*sqrt(-a/c)*arctan(1/2*(2*a*c + (b*c + a*d)*x)/(sqrt(b*x + a)*sqrt(
d*x + c)*c*sqrt(-a/c))) + 2*(2*a^2*c^2 - (8*b^2*c^2 - 25*a*b*c*d + 15*a^2*d^2)*x
^2 + (9*a*b*c^2 - 5*a^2*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(c^3*d*x^3 + c^4*x^
2)]

_______________________________________________________________________________________

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((b*x+a)**(5/2)/x**3/(d*x+c)**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError