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

Optimal. Leaf size=229 \[ -\frac{5 (b c-a d)^3 (a d+7 b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{9/2} c^{3/2}}+\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2 (a d+7 b c)}{64 a^4 c x}-\frac{5 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d) (a d+7 b c)}{96 a^3 c x^2}+\frac{\sqrt{a+b x} (c+d x)^{5/2} (a d+7 b c)}{24 a^2 c x^3}-\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 a c x^4} \]

[Out]

(5*(b*c - a*d)^2*(7*b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*a^4*c*x) - (5*(b
*c - a*d)*(7*b*c + a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(96*a^3*c*x^2) + ((7*b*c
+ a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(24*a^2*c*x^3) - (Sqrt[a + b*x]*(c + d*x)^
(7/2))/(4*a*c*x^4) - (5*(b*c - a*d)^3*(7*b*c + a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*
x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(9/2)*c^(3/2))

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Rubi [A]  time = 0.427561, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{5 (b c-a d)^3 (a d+7 b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{9/2} c^{3/2}}+\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2 (a d+7 b c)}{64 a^4 c x}-\frac{5 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d) (a d+7 b c)}{96 a^3 c x^2}+\frac{\sqrt{a+b x} (c+d x)^{5/2} (a d+7 b c)}{24 a^2 c x^3}-\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 a c x^4} \]

Antiderivative was successfully verified.

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

[Out]

(5*(b*c - a*d)^2*(7*b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*a^4*c*x) - (5*(b
*c - a*d)*(7*b*c + a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(96*a^3*c*x^2) + ((7*b*c
+ a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(24*a^2*c*x^3) - (Sqrt[a + b*x]*(c + d*x)^
(7/2))/(4*a*c*x^4) - (5*(b*c - a*d)^3*(7*b*c + a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*
x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(9/2)*c^(3/2))

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Rubi in Sympy [A]  time = 36.2704, size = 207, normalized size = 0.9 \[ - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{7}{2}}}{4 a c x^{4}} + \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}} \left (a d + 7 b c\right )}{24 a^{2} c x^{3}} + \frac{5 \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right ) \left (a d + 7 b c\right )}{96 a^{3} c x^{2}} + \frac{5 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{2} \left (a d + 7 b c\right )}{64 a^{4} c x} + \frac{5 \left (a d - b c\right )^{3} \left (a d + 7 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{64 a^{\frac{9}{2}} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(5/2)/x**5/(b*x+a)**(1/2),x)

[Out]

-sqrt(a + b*x)*(c + d*x)**(7/2)/(4*a*c*x**4) + sqrt(a + b*x)*(c + d*x)**(5/2)*(a
*d + 7*b*c)/(24*a**2*c*x**3) + 5*sqrt(a + b*x)*(c + d*x)**(3/2)*(a*d - b*c)*(a*d
+ 7*b*c)/(96*a**3*c*x**2) + 5*sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)**2*(a*d +
7*b*c)/(64*a**4*c*x) + 5*(a*d - b*c)**3*(a*d + 7*b*c)*atanh(sqrt(c)*sqrt(a + b*
x)/(sqrt(a)*sqrt(c + d*x)))/(64*a**(9/2)*c**(3/2))

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Mathematica [A]  time = 0.274044, size = 233, normalized size = 1.02 \[ \frac{-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (a^3 \left (48 c^3+136 c^2 d x+118 c d^2 x^2+15 d^3 x^3\right )-a^2 b c x \left (56 c^2+172 c d x+191 d^2 x^2\right )+5 a b^2 c^2 x^2 (14 c+53 d x)-105 b^3 c^3 x^3\right )+15 x^4 \log (x) (b c-a d)^3 (a d+7 b c)-15 x^4 (b c-a d)^3 (a d+7 b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{384 a^{9/2} c^{3/2} x^4} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(-105*b^3*c^3*x^3 + 5*a*b^2*c^2*
x^2*(14*c + 53*d*x) - a^2*b*c*x*(56*c^2 + 172*c*d*x + 191*d^2*x^2) + a^3*(48*c^3
+ 136*c^2*d*x + 118*c*d^2*x^2 + 15*d^3*x^3)) + 15*(b*c - a*d)^3*(7*b*c + a*d)*x
^4*Log[x] - 15*(b*c - a*d)^3*(7*b*c + a*d)*x^4*Log[2*a*c + b*c*x + a*d*x + 2*Sqr
t[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(384*a^(9/2)*c^(3/2)*x^4)

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Maple [B]  time = 0.04, size = 593, normalized size = 2.6 \[{\frac{1}{384\,{a}^{4}c{x}^{4}}\sqrt{bx+a}\sqrt{dx+c} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}{a}^{4}{d}^{4}+60\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}{a}^{3}bc{d}^{3}-270\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}{a}^{2}{b}^{2}{c}^{2}{d}^{2}+300\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}a{b}^{3}{c}^{3}d-105\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}{b}^{4}{c}^{4}-30\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{d}^{3}{a}^{3}{x}^{3}\sqrt{ac}+382\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{d}^{2}bc{a}^{2}{x}^{3}\sqrt{ac}-530\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }d{b}^{2}{c}^{2}a{x}^{3}\sqrt{ac}+210\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{b}^{3}{c}^{3}{x}^{3}\sqrt{ac}-236\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{d}^{2}c{a}^{3}{x}^{2}\sqrt{ac}+344\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }db{c}^{2}{a}^{2}{x}^{2}\sqrt{ac}-140\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{b}^{2}{c}^{3}a{x}^{2}\sqrt{ac}-272\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }d{c}^{2}{a}^{3}x\sqrt{ac}+112\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }b{c}^{3}{a}^{2}x\sqrt{ac}-96\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{c}^{3}{a}^{3}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/2)/x^5/(b*x+a)^(1/2),x)

[Out]

1/384*(d*x+c)^(1/2)*(b*x+a)^(1/2)/a^4/c*(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^4*a^4*d^4+60*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^4*a^3*b*c*d^3-270*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^4*a^2*b^2*c^2*d^2+300*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^4*a*b^3*c^3*d-105*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^4*b^4*c^4-30*((b*x+a)*(d*x+c))^(1/2
)*d^3*a^3*x^3*(a*c)^(1/2)+382*((b*x+a)*(d*x+c))^(1/2)*d^2*b*c*a^2*x^3*(a*c)^(1/2
)-530*((b*x+a)*(d*x+c))^(1/2)*d*b^2*c^2*a*x^3*(a*c)^(1/2)+210*((b*x+a)*(d*x+c))^
(1/2)*b^3*c^3*x^3*(a*c)^(1/2)-236*((b*x+a)*(d*x+c))^(1/2)*d^2*c*a^3*x^2*(a*c)^(1
/2)+344*((b*x+a)*(d*x+c))^(1/2)*d*b*c^2*a^2*x^2*(a*c)^(1/2)-140*((b*x+a)*(d*x+c)
)^(1/2)*b^2*c^3*a*x^2*(a*c)^(1/2)-272*((b*x+a)*(d*x+c))^(1/2)*d*c^2*a^3*x*(a*c)^
(1/2)+112*((b*x+a)*(d*x+c))^(1/2)*b*c^3*a^2*x*(a*c)^(1/2)-96*((b*x+a)*(d*x+c))^(
1/2)*c^3*a^3*(a*c)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/(a*c)^(1/2)/x^4

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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)/(sqrt(b*x + a)*x^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.786231, size = 1, normalized size = 0. \[ \left [-\frac{15 \,{\left (7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x^{4} \log \left (\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) + 4 \,{\left (48 \, a^{3} c^{3} -{\left (105 \, b^{3} c^{3} - 265 \, a b^{2} c^{2} d + 191 \, a^{2} b c d^{2} - 15 \, a^{3} d^{3}\right )} x^{3} + 2 \,{\left (35 \, a b^{2} c^{3} - 86 \, a^{2} b c^{2} d + 59 \, a^{3} c d^{2}\right )} x^{2} - 8 \,{\left (7 \, a^{2} b c^{3} - 17 \, a^{3} c^{2} d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{768 \, \sqrt{a c} a^{4} c x^{4}}, -\frac{15 \,{\left (7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x^{4} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) + 2 \,{\left (48 \, a^{3} c^{3} -{\left (105 \, b^{3} c^{3} - 265 \, a b^{2} c^{2} d + 191 \, a^{2} b c d^{2} - 15 \, a^{3} d^{3}\right )} x^{3} + 2 \,{\left (35 \, a b^{2} c^{3} - 86 \, a^{2} b c^{2} d + 59 \, a^{3} c d^{2}\right )} x^{2} - 8 \,{\left (7 \, a^{2} b c^{3} - 17 \, a^{3} c^{2} d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{384 \, \sqrt{-a c} a^{4} c x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/768*(15*(7*b^4*c^4 - 20*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 - a
^4*d^4)*x^4*log((4*(2*a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)*sqrt(b*x + a)*sqrt(d*x +
c) + (8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 + 8*(a*b*c^2 + a^2*c*d)*x)
*sqrt(a*c))/x^2) + 4*(48*a^3*c^3 - (105*b^3*c^3 - 265*a*b^2*c^2*d + 191*a^2*b*c*
d^2 - 15*a^3*d^3)*x^3 + 2*(35*a*b^2*c^3 - 86*a^2*b*c^2*d + 59*a^3*c*d^2)*x^2 - 8
*(7*a^2*b*c^3 - 17*a^3*c^2*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*
c)*a^4*c*x^4), -1/384*(15*(7*b^4*c^4 - 20*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 4*a
^3*b*c*d^3 - a^4*d^4)*x^4*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)/(sqrt(b*
x + a)*sqrt(d*x + c)*a*c)) + 2*(48*a^3*c^3 - (105*b^3*c^3 - 265*a*b^2*c^2*d + 19
1*a^2*b*c*d^2 - 15*a^3*d^3)*x^3 + 2*(35*a*b^2*c^3 - 86*a^2*b*c^2*d + 59*a^3*c*d^
2)*x^2 - 8*(7*a^2*b*c^3 - 17*a^3*c^2*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c
))/(sqrt(-a*c)*a^4*c*x^4)]

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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**5/(b*x+a)**(1/2),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)/(sqrt(b*x + a)*x^5),x, algorithm="giac")

[Out]

Exception raised: TypeError