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

Optimal. Leaf size=148 \[ -\frac{(b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{a} c^{7/2}}-\frac{d \sqrt{a+b x} (13 b c-15 a d)}{3 c^3 \sqrt{c+d x} (b c-a d)}-\frac{5 d \sqrt{a+b x}}{3 c^2 (c+d x)^{3/2}}-\frac{\sqrt{a+b x}}{c x (c+d x)^{3/2}} \]

[Out]

(-5*d*Sqrt[a + b*x])/(3*c^2*(c + d*x)^(3/2)) - Sqrt[a + b*x]/(c*x*(c + d*x)^(3/2
)) - (d*(13*b*c - 15*a*d)*Sqrt[a + b*x])/(3*c^3*(b*c - a*d)*Sqrt[c + d*x]) - ((b
*c - 5*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(Sqrt[a]*c
^(7/2))

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Rubi [A]  time = 0.498362, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{(b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{a} c^{7/2}}-\frac{d \sqrt{a+b x} (13 b c-15 a d)}{3 c^3 \sqrt{c+d x} (b c-a d)}-\frac{5 d \sqrt{a+b x}}{3 c^2 (c+d x)^{3/2}}-\frac{\sqrt{a+b x}}{c x (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-5*d*Sqrt[a + b*x])/(3*c^2*(c + d*x)^(3/2)) - Sqrt[a + b*x]/(c*x*(c + d*x)^(3/2
)) - (d*(13*b*c - 15*a*d)*Sqrt[a + b*x])/(3*c^3*(b*c - a*d)*Sqrt[c + d*x]) - ((b
*c - 5*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(Sqrt[a]*c
^(7/2))

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Rubi in Sympy [A]  time = 58.8884, size = 133, normalized size = 0.9 \[ - \frac{\sqrt{a + b x}}{c x \left (c + d x\right )^{\frac{3}{2}}} - \frac{5 d \sqrt{a + b x}}{3 c^{2} \left (c + d x\right )^{\frac{3}{2}}} - \frac{d \sqrt{a + b x} \left (15 a d - 13 b c\right )}{3 c^{3} \sqrt{c + d x} \left (a d - b c\right )} + \frac{\left (5 a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{\sqrt{a} c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + b*x)/(c*x*(c + d*x)**(3/2)) - 5*d*sqrt(a + b*x)/(3*c**2*(c + d*x)**(3/
2)) - d*sqrt(a + b*x)*(15*a*d - 13*b*c)/(3*c**3*sqrt(c + d*x)*(a*d - b*c)) + (5*
a*d - b*c)*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/(sqrt(a)*c**(7/2
))

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Mathematica [A]  time = 0.528113, size = 158, normalized size = 1.07 \[ \frac{2 \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (\frac{2 d (6 a d-5 b c)}{(c+d x) (b c-a d)}-\frac{2 c d}{(c+d x)^2}-\frac{3}{x}\right )+\frac{3 \log (x) (b c-5 a d)}{\sqrt{a}}+\frac{3 (5 a d-b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{\sqrt{a}}}{6 c^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(-3/x - (2*c*d)/(c + d*x)^2 + (2*d*(-5*b*
c + 6*a*d))/((b*c - a*d)*(c + d*x))) + (3*(b*c - 5*a*d)*Log[x])/Sqrt[a] + (3*(-(
b*c) + 5*a*d)*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c
+ d*x]])/Sqrt[a])/(6*c^(7/2))

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Maple [B]  time = 0.041, size = 653, normalized size = 4.4 \[{\frac{1}{6\,{c}^{3} \left ( ad-bc \right ) x} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{2}{d}^{4}-18\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}abc{d}^{3}+3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{b}^{2}{c}^{2}{d}^{2}+30\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{2}c{d}^{3}-36\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}ab{c}^{2}{d}^{2}+6\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{b}^{2}{c}^{3}d+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) x{a}^{2}{c}^{2}{d}^{2}-18\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) xab{c}^{3}d+3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) x{b}^{2}{c}^{4}-30\,{x}^{2}a{d}^{3}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+26\,{x}^{2}bc{d}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }-40\,xac{d}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+36\,xb{c}^{2}d\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }-6\,a{c}^{2}d\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+6\,b{c}^{3}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) } \right ) \sqrt{bx+a}{\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(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*
d^4-18*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*c
*d^3+3*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^2*c
^2*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*c*d^3-36*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*c^2*d^2+6*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
*b^2*c^3*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*
a^2*c^2*d^2-18*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
*a*b*c^3*d+3*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*b
^2*c^4-30*x^2*a*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+26*x^2*b*c*d^2*(a*c)^(1/
2)*((b*x+a)*(d*x+c))^(1/2)-40*x*a*c*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+36*x
*b*c^2*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-6*a*c^2*d*(a*c)^(1/2)*((b*x+a)*(d*x
+c))^(1/2)+6*b*c^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/c^3*(b*x+a)^(1/2)/(a*d-b
*c)/(a*c)^(1/2)/x/((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(3/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.446685, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left (3 \, b c^{3} - 3 \, a c^{2} d +{\left (13 \, b c d^{2} - 15 \, a d^{3}\right )} x^{2} + 2 \,{\left (9 \, b c^{2} d - 10 \, a c d^{2}\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left ({\left (b^{2} c^{2} d^{2} - 6 \, a b c d^{3} + 5 \, a^{2} d^{4}\right )} x^{3} + 2 \,{\left (b^{2} c^{3} d - 6 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} x^{2} +{\left (b^{2} c^{4} - 6 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2}\right )} x\right )} \log \left (\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right )}{12 \,{\left ({\left (b c^{4} d^{2} - a c^{3} d^{3}\right )} x^{3} + 2 \,{\left (b c^{5} d - a c^{4} d^{2}\right )} x^{2} +{\left (b c^{6} - a c^{5} d\right )} x\right )} \sqrt{a c}}, -\frac{2 \,{\left (3 \, b c^{3} - 3 \, a c^{2} d +{\left (13 \, b c d^{2} - 15 \, a d^{3}\right )} x^{2} + 2 \,{\left (9 \, b c^{2} d - 10 \, a c d^{2}\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left ({\left (b^{2} c^{2} d^{2} - 6 \, a b c d^{3} + 5 \, a^{2} d^{4}\right )} x^{3} + 2 \,{\left (b^{2} c^{3} d - 6 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} x^{2} +{\left (b^{2} c^{4} - 6 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2}\right )} x\right )} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right )}{6 \,{\left ({\left (b c^{4} d^{2} - a c^{3} d^{3}\right )} x^{3} + 2 \,{\left (b c^{5} d - a c^{4} d^{2}\right )} x^{2} +{\left (b c^{6} - a c^{5} d\right )} x\right )} \sqrt{-a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/12*(4*(3*b*c^3 - 3*a*c^2*d + (13*b*c*d^2 - 15*a*d^3)*x^2 + 2*(9*b*c^2*d - 10
*a*c*d^2)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 3*((b^2*c^2*d^2 - 6*a*b*c*d
^3 + 5*a^2*d^4)*x^3 + 2*(b^2*c^3*d - 6*a*b*c^2*d^2 + 5*a^2*c*d^3)*x^2 + (b^2*c^4
- 6*a*b*c^3*d + 5*a^2*c^2*d^2)*x)*log((4*(2*a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)*sq
rt(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))/(((b*c^4*d^2 - a*c^3*d^3)*x^3 + 2*(b*c^
5*d - a*c^4*d^2)*x^2 + (b*c^6 - a*c^5*d)*x)*sqrt(a*c)), -1/6*(2*(3*b*c^3 - 3*a*c
^2*d + (13*b*c*d^2 - 15*a*d^3)*x^2 + 2*(9*b*c^2*d - 10*a*c*d^2)*x)*sqrt(-a*c)*sq
rt(b*x + a)*sqrt(d*x + c) + 3*((b^2*c^2*d^2 - 6*a*b*c*d^3 + 5*a^2*d^4)*x^3 + 2*(
b^2*c^3*d - 6*a*b*c^2*d^2 + 5*a^2*c*d^3)*x^2 + (b^2*c^4 - 6*a*b*c^3*d + 5*a^2*c^
2*d^2)*x)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)/(sqrt(b*x + a)*sqrt(d*x
+ c)*a*c)))/(((b*c^4*d^2 - a*c^3*d^3)*x^3 + 2*(b*c^5*d - a*c^4*d^2)*x^2 + (b*c^6
- a*c^5*d)*x)*sqrt(-a*c))]

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

[Out]

Exception raised: TypeError