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

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

[Out]

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

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Rubi [A]  time = 0.295989, 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 \[ -\frac{15 \sqrt{c} (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{7/2}}+\frac{15 \sqrt{c+d x} (b c-a d)^2}{4 a^3 \sqrt{a+b x}}+\frac{5 (c+d x)^{3/2} (b c-a d)}{4 a^2 x \sqrt{a+b x}}-\frac{(c+d x)^{5/2}}{2 a x^2 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.2219, size = 136, normalized size = 0.88 \[ \frac{2 \left (c + d x\right )^{\frac{5}{2}}}{a x^{2} \sqrt{a + b x}} - \frac{5 c \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}}}{2 a^{2} x^{2}} - \frac{15 c \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )}{4 a^{3} x} - \frac{15 \sqrt{c} \left (a d - b c\right )^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.24275, size = 165, normalized size = 1.07 \[ \frac{\frac{2 \sqrt{a} \sqrt{c+d x} \left (a^2 \left (-2 c^2-9 c d x+8 d^2 x^2\right )+5 a b c x (c-5 d x)+15 b^2 c^2 x^2\right )}{x^2 \sqrt{a+b x}}+15 \sqrt{c} \log (x) (b c-a d)^2-15 \sqrt{c} (b c-a d)^2 \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{8 a^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[a]*Sqrt[c + d*x]*(15*b^2*c^2*x^2 + 5*a*b*c*x*(c - 5*d*x) + a^2*(-2*c^2
- 9*c*d*x + 8*d^2*x^2)))/(x^2*Sqrt[a + b*x]) + 15*Sqrt[c]*(b*c - a*d)^2*Log[x] -
15*Sqrt[c]*(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*a^(7/2))

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Maple [B]  time = 0.04, size = 507, normalized size = 3.3 \[ -{\frac{1}{8\,{a}^{3}{x}^{2}}\sqrt{dx+c} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{2}bc{d}^{2}-30\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}a{b}^{2}{c}^{2}d+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{b}^{3}{c}^{3}+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{3}c{d}^{2}-30\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{2}b{c}^{2}d+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}a{b}^{2}{c}^{3}-16\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{d}^{2}{a}^{2}{x}^{2}\sqrt{ac}+50\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dbca{x}^{2}\sqrt{ac}-30\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{b}^{2}{c}^{2}{x}^{2}\sqrt{ac}+18\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dc{a}^{2}x\sqrt{ac}-10\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }b{c}^{2}ax\sqrt{ac}+4\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{c}^{2}{a}^{2}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)^(3/2)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.423604, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left ({\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} +{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2}\right )} \sqrt{\frac{c}{a}} \log \left (\frac{8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \,{\left (2 \, a^{2} c +{\left (a b c + a^{2} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{c}{a}} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \,{\left (2 \, a^{2} c^{2} -{\left (15 \, b^{2} c^{2} - 25 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{2} -{\left (5 \, a b c^{2} - 9 \, a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{16 \,{\left (a^{3} b x^{3} + a^{4} x^{2}\right )}}, -\frac{15 \,{\left ({\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} +{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2}\right )} \sqrt{-\frac{c}{a}} \arctan \left (\frac{2 \, a c +{\left (b c + a d\right )} x}{2 \, \sqrt{b x + a} \sqrt{d x + c} a \sqrt{-\frac{c}{a}}}\right ) + 2 \,{\left (2 \, a^{2} c^{2} -{\left (15 \, b^{2} c^{2} - 25 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{2} -{\left (5 \, a b c^{2} - 9 \, a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{8 \,{\left (a^{3} b x^{3} + a^{4} x^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Timed out

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

[Out]

Exception raised: TypeError