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

Optimal. Leaf size=149 \[ -\frac{\sqrt{a} (3 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{7/2}}+\frac{\sqrt{a+b x} (3 b c-5 a d)}{c^3 \sqrt{c+d x}}+\frac{(a+b x)^{3/2} (3 b c-5 a d)}{3 a c^2 (c+d x)^{3/2}}-\frac{(a+b x)^{5/2}}{a c x (c+d x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.28288, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{\sqrt{a} (3 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{7/2}}+\frac{\sqrt{a+b x} (3 b c-5 a d)}{c^3 \sqrt{c+d x}}+\frac{(a+b x)^{3/2} (3 b c-5 a d)}{3 a c^2 (c+d x)^{3/2}}-\frac{(a+b x)^{5/2}}{a c x (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 25.6615, size = 151, normalized size = 1.01 \[ \frac{\sqrt{a} \left (5 a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{c^{\frac{7}{2}}} + \frac{2 d \left (a + b x\right )^{\frac{5}{2}}}{3 c x \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{\left (a + b x\right )^{\frac{3}{2}} \left (5 a d - 3 b c\right )}{3 c^{2} x \sqrt{c + d x} \left (a d - b c\right )} - \frac{\sqrt{a + b x} \left (5 a d - 3 b c\right )}{c^{3} \sqrt{c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.472769, size = 151, normalized size = 1.01 \[ \frac{-\frac{2 \sqrt{c} \sqrt{a+b x} \left (a \left (3 c^2+20 c d x+15 d^2 x^2\right )-2 b c x (3 c+2 d x)\right )}{x (c+d x)^{3/2}}+3 \sqrt{a} \log (x) (3 b c-5 a d)+3 \sqrt{a} (5 a d-3 b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{6 c^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

((-2*Sqrt[c]*Sqrt[a + b*x]*(-2*b*c*x*(3*c + 2*d*x) + a*(3*c^2 + 20*c*d*x + 15*d^
2*x^2)))/(x*(c + d*x)^(3/2)) + 3*Sqrt[a]*(3*b*c - 5*a*d)*Log[x] + 3*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]])/(6*c^(7/2))

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Maple [B]  time = 0.042, size = 459, normalized size = 3.1 \[{\frac{1}{6\,{c}^{3}x}\sqrt{bx+a} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{2}{d}^{3}-9\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}abc{d}^{2}+30\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{2}c{d}^{2}-18\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}ab{c}^{2}d+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) x{a}^{2}{c}^{2}d-9\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) xab{c}^{3}-30\,{x}^{2}a{d}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+8\,{x}^{2}bcd\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }-40\,xacd\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+12\,xb{c}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }-6\,a{c}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) } \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(b*x+a)^(1/2)/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^3*a^2*d^3-9*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^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^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^2*a*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*a^2*c^2*d-9*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^3-30*x^2*a*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+8*x^2*b*c*d*
(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-40*x*a*c*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/
2)+12*x*b*c^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-6*a*c^2*(a*c)^(1/2)*((b*x+a)*(
d*x+c))^(1/2))/(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((b*x + a)^(3/2)/((d*x + c)^(5/2)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.463321, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left ({\left (3 \, b c d^{2} - 5 \, a d^{3}\right )} x^{3} + 2 \,{\left (3 \, b c^{2} d - 5 \, a c d^{2}\right )} x^{2} +{\left (3 \, b c^{3} - 5 \, a c^{2} d\right )} x\right )} \sqrt{\frac{a}{c}} \log \left (\frac{8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \,{\left (2 \, a c^{2} +{\left (b c^{2} + a c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{a}{c}} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \,{\left (3 \, a c^{2} -{\left (4 \, b c d - 15 \, a d^{2}\right )} x^{2} - 2 \,{\left (3 \, b c^{2} - 10 \, a c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{12 \,{\left (c^{3} d^{2} x^{3} + 2 \, c^{4} d x^{2} + c^{5} x\right )}}, -\frac{3 \,{\left ({\left (3 \, b c d^{2} - 5 \, a d^{3}\right )} x^{3} + 2 \,{\left (3 \, b c^{2} d - 5 \, a c d^{2}\right )} x^{2} +{\left (3 \, b c^{3} - 5 \, a c^{2} d\right )} x\right )} \sqrt{-\frac{a}{c}} \arctan \left (\frac{2 \, a c +{\left (b c + a d\right )} x}{2 \, \sqrt{b x + a} \sqrt{d x + c} c \sqrt{-\frac{a}{c}}}\right ) + 2 \,{\left (3 \, a c^{2} -{\left (4 \, b c d - 15 \, a d^{2}\right )} x^{2} - 2 \,{\left (3 \, b c^{2} - 10 \, a c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{6 \,{\left (c^{3} d^{2} x^{3} + 2 \, c^{4} d x^{2} + c^{5} x\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: TypeError