3.735 \(\int \frac{1}{x^3 \sqrt{a+b x} (c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=194 \[ \frac{d \sqrt{a+b x} (3 b c-5 a d) (3 a d+b c)}{4 a^2 c^3 \sqrt{c+d x} (b c-a d)}+\frac{\sqrt{a+b x} (5 a d+3 b c)}{4 a^2 c^2 x \sqrt{c+d x}}-\frac{3 \left (5 a^2 d^2+2 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{5/2} c^{7/2}}-\frac{\sqrt{a+b x}}{2 a c x^2 \sqrt{c+d x}} \]

[Out]

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

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Rubi [A]  time = 0.577584, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{d \sqrt{a+b x} (3 b c-5 a d) (3 a d+b c)}{4 a^2 c^3 \sqrt{c+d x} (b c-a d)}+\frac{\sqrt{a+b x} (5 a d+3 b c)}{4 a^2 c^2 x \sqrt{c+d x}}-\frac{3 \left (5 a^2 d^2+2 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{5/2} c^{7/2}}-\frac{\sqrt{a+b x}}{2 a c x^2 \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*Sqrt[a + b*x]*(c + d*x)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 75.9479, size = 180, normalized size = 0.93 \[ - \frac{\sqrt{a + b x}}{2 a c x^{2} \sqrt{c + d x}} + \frac{\sqrt{a + b x} \left (5 a d + 3 b c\right )}{4 a^{2} c^{2} x \sqrt{c + d x}} + \frac{d \sqrt{a + b x} \left (3 a d + b c\right ) \left (5 a d - 3 b c\right )}{4 a^{2} c^{3} \sqrt{c + d x} \left (a d - b c\right )} - \frac{3 \left (5 a^{2} d^{2} + 2 a b c d + b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 a^{\frac{5}{2}} c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(d*x+c)**(3/2)/(b*x+a)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.630155, size = 187, normalized size = 0.96 \[ \frac{2 \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (\frac{3 b c}{a^2 x}+\frac{8 d^3}{(c+d x) (a d-b c)}+\frac{7 d x-2 c}{a x^2}\right )+\frac{3 \log (x) \left (5 a^2 d^2+2 a b c d+b^2 c^2\right )}{a^{5/2}}-\frac{3 \left (5 a^2 d^2+2 a b c d+b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{a^{5/2}}}{8 c^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*Sqrt[a + b*x]*(c + d*x)^(3/2)),x]

[Out]

(2*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*((3*b*c)/(a^2*x) + (8*d^3)/((-(b*c) + a*d
)*(c + d*x)) + (-2*c + 7*d*x)/(a*x^2)) + (3*(b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*Lo
g[x])/a^(5/2) - (3*(b^2*c^2 + 2*a*b*c*d + 5*a^2*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]])/a^(5/2))/(8*c^(7/2))

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Maple [B]  time = 0.051, size = 683, normalized size = 3.5 \[ -{\frac{1}{8\,{a}^{2}{c}^{3}{x}^{2} \left ( ad-bc \right ) }\sqrt{bx+a} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{3}{d}^{4}-9\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{2}bc{d}^{3}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}a{b}^{2}{c}^{2}{d}^{2}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{b}^{3}{c}^{3}d+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{3}c{d}^{3}-9\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{2}b{c}^{2}{d}^{2}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}a{b}^{2}{c}^{3}d-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{b}^{3}{c}^{4}-30\,{x}^{2}{a}^{2}{d}^{3}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+8\,{x}^{2}abc{d}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+6\,{x}^{2}{b}^{2}{c}^{2}d\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }-10\,x{a}^{2}c{d}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+4\,xab{c}^{2}d\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+6\,x{b}^{2}{c}^{3}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+4\,{a}^{2}{c}^{2}d\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }-4\,ab{c}^{3}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) } \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{dx+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(d*x+c)^(3/2)/(b*x+a)^(1/2),x)

[Out]

-1/8*(b*x+a)^(1/2)/a^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^3*d^4-9*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^2*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*a*b^2*c^2*d^2-3*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^3*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^2*a^3*c*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^2*a^2*b*c^2*d^2-3*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^2*a*b^2*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^2*b^3*c^4-30*x^2*a^2*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c
))^(1/2)+8*x^2*a*b*c*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+6*x^2*b^2*c^2*d*(a*
c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-10*x*a^2*c*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1
/2)+4*x*a*b*c^2*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+6*x*b^2*c^3*(a*c)^(1/2)*((
b*x+a)*(d*x+c))^(1/2)+4*a^2*c^2*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-4*a*b*c^3*
(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x^2/(a*c)^(1/2)/(a*d-b*c)/((b*x+a)*(d*x+c))
^(1/2)/(d*x+c)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{3}{2}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*(d*x + c)^(3/2)*x^3),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*x + a)*(d*x + c)^(3/2)*x^3), x)

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Fricas [A]  time = 0.419839, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left (2 \, a b c^{3} - 2 \, a^{2} c^{2} d -{\left (3 \, b^{2} c^{2} d + 4 \, a b c d^{2} - 15 \, a^{2} d^{3}\right )} x^{2} -{\left (3 \, b^{2} c^{3} + 2 \, a b c^{2} d - 5 \, a^{2} c d^{2}\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c} - 3 \,{\left ({\left (b^{3} c^{3} d + a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - 5 \, a^{3} d^{4}\right )} x^{3} +{\left (b^{3} c^{4} + a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - 5 \, a^{3} c d^{3}\right )} x^{2}\right )} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right )}{16 \,{\left ({\left (a^{2} b c^{4} d - a^{3} c^{3} d^{2}\right )} x^{3} +{\left (a^{2} b c^{5} - a^{3} c^{4} d\right )} x^{2}\right )} \sqrt{a c}}, -\frac{2 \,{\left (2 \, a b c^{3} - 2 \, a^{2} c^{2} d -{\left (3 \, b^{2} c^{2} d + 4 \, a b c d^{2} - 15 \, a^{2} d^{3}\right )} x^{2} -{\left (3 \, b^{2} c^{3} + 2 \, a b c^{2} d - 5 \, a^{2} c d^{2}\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left ({\left (b^{3} c^{3} d + a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - 5 \, a^{3} d^{4}\right )} x^{3} +{\left (b^{3} c^{4} + a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - 5 \, a^{3} c d^{3}\right )} x^{2}\right )} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right )}{8 \,{\left ({\left (a^{2} b c^{4} d - a^{3} c^{3} d^{2}\right )} x^{3} +{\left (a^{2} b c^{5} - a^{3} c^{4} d\right )} x^{2}\right )} \sqrt{-a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*(d*x + c)^(3/2)*x^3),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(d*x+c)**(3/2)/(b*x+a)**(1/2),x)

[Out]

Integral(1/(x**3*sqrt(a + b*x)*(c + d*x)**(3/2)), x)

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

[Out]

Exception raised: TypeError