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

Optimal. Leaf size=204 \[ -\frac{\left (35 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 \sqrt{a} c^{9/2}}-\frac{5 d \sqrt{a+b x} (11 b c-21 a d)}{12 c^4 \sqrt{c+d x}}-\frac{d \sqrt{a+b x} (23 b c-35 a d)}{12 c^3 (c+d x)^{3/2}}-\frac{\sqrt{a+b x} (5 b c-7 a d)}{4 c^2 x (c+d x)^{3/2}}-\frac{a \sqrt{a+b x}}{2 c x^2 (c+d x)^{3/2}} \]

[Out]

-(d*(23*b*c - 35*a*d)*Sqrt[a + b*x])/(12*c^3*(c + d*x)^(3/2)) - (a*Sqrt[a + b*x]
)/(2*c*x^2*(c + d*x)^(3/2)) - ((5*b*c - 7*a*d)*Sqrt[a + b*x])/(4*c^2*x*(c + d*x)
^(3/2)) - (5*d*(11*b*c - 21*a*d)*Sqrt[a + b*x])/(12*c^4*Sqrt[c + d*x]) - ((3*b^2
*c^2 - 30*a*b*c*d + 35*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c
+ d*x])])/(4*Sqrt[a]*c^(9/2))

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Rubi [A]  time = 0.797931, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{\left (35 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 \sqrt{a} c^{9/2}}-\frac{5 d \sqrt{a+b x} (11 b c-21 a d)}{12 c^4 \sqrt{c+d x}}-\frac{d \sqrt{a+b x} (23 b c-35 a d)}{12 c^3 (c+d x)^{3/2}}-\frac{\sqrt{a+b x} (5 b c-7 a d)}{4 c^2 x (c+d x)^{3/2}}-\frac{a \sqrt{a+b x}}{2 c x^2 (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(d*(23*b*c - 35*a*d)*Sqrt[a + b*x])/(12*c^3*(c + d*x)^(3/2)) - (a*Sqrt[a + b*x]
)/(2*c*x^2*(c + d*x)^(3/2)) - ((5*b*c - 7*a*d)*Sqrt[a + b*x])/(4*c^2*x*(c + d*x)
^(3/2)) - (5*d*(11*b*c - 21*a*d)*Sqrt[a + b*x])/(12*c^4*Sqrt[c + d*x]) - ((3*b^2
*c^2 - 30*a*b*c*d + 35*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c
+ d*x])])/(4*Sqrt[a]*c^(9/2))

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Rubi in Sympy [A]  time = 106.523, size = 194, normalized size = 0.95 \[ - \frac{a \sqrt{a + b x}}{2 c x^{2} \left (c + d x\right )^{\frac{3}{2}}} + \frac{\sqrt{a + b x} \left (7 a d - 5 b c\right )}{4 c^{2} x \left (c + d x\right )^{\frac{3}{2}}} + \frac{d \sqrt{a + b x} \left (35 a d - 23 b c\right )}{12 c^{3} \left (c + d x\right )^{\frac{3}{2}}} + \frac{5 d \sqrt{a + b x} \left (21 a d - 11 b c\right )}{12 c^{4} \sqrt{c + d x}} - \frac{\left (35 a^{2} d^{2} - 30 a b c d + 3 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 \sqrt{a} c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*sqrt(a + b*x)/(2*c*x**2*(c + d*x)**(3/2)) + sqrt(a + b*x)*(7*a*d - 5*b*c)/(4*
c**2*x*(c + d*x)**(3/2)) + d*sqrt(a + b*x)*(35*a*d - 23*b*c)/(12*c**3*(c + d*x)*
*(3/2)) + 5*d*sqrt(a + b*x)*(21*a*d - 11*b*c)/(12*c**4*sqrt(c + d*x)) - (35*a**2
*d**2 - 30*a*b*c*d + 3*b**2*c**2)*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c +
d*x)))/(4*sqrt(a)*c**(9/2))

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Mathematica [A]  time = 0.32687, size = 201, normalized size = 0.99 \[ \frac{\frac{3 \log (x) \left (35 a^2 d^2-30 a b c d+3 b^2 c^2\right )}{\sqrt{a}}-\frac{3 \left (35 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{\sqrt{a}}+\frac{2 \sqrt{c} \sqrt{a+b x} \left (a \left (-6 c^3+21 c^2 d x+140 c d^2 x^2+105 d^3 x^3\right )-b c x \left (15 c^2+78 c d x+55 d^2 x^2\right )\right )}{x^2 (c+d x)^{3/2}}}{24 c^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[c]*Sqrt[a + b*x]*(-(b*c*x*(15*c^2 + 78*c*d*x + 55*d^2*x^2)) + a*(-6*c^3
+ 21*c^2*d*x + 140*c*d^2*x^2 + 105*d^3*x^3)))/(x^2*(c + d*x)^(3/2)) + (3*(3*b^2
*c^2 - 30*a*b*c*d + 35*a^2*d^2)*Log[x])/Sqrt[a] - (3*(3*b^2*c^2 - 30*a*b*c*d + 3
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]])/Sqrt[a])/(24*c^(9/2))

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Maple [B]  time = 0.047, size = 679, normalized size = 3.3 \[ -{\frac{1}{24\,{c}^{4}{x}^{2}}\sqrt{bx+a} \left ( 105\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}{a}^{2}{d}^{4}-90\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}abc{d}^{3}+9\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}{b}^{2}{c}^{2}{d}^{2}+210\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{2}c{d}^{3}-180\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}ab{c}^{2}{d}^{2}+18\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{b}^{2}{c}^{3}d+105\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{2}{c}^{2}{d}^{2}-90\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}ab{c}^{3}d+9\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{b}^{2}{c}^{4}-210\,{x}^{3}a{d}^{3}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+110\,{x}^{3}bc{d}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }-280\,{x}^{2}ac{d}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+156\,{x}^{2}b{c}^{2}d\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }-42\,xa{c}^{2}d\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+30\,xb{c}^{3}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+12\,a{c}^{3}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) } \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*(b*x+a)^(1/2)/c^4*(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*a^2*d^4-90*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*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^4*b^2*c^2*d^2+210*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*c*d^3-180*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^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^3*b^2*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^2*a^2*c^2*d^2-90*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^3*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^2*b^2*c^4-210*x^3*a*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c)
)^(1/2)+110*x^3*b*c*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-280*x^2*a*c*d^2*(a*c
)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+156*x^2*b*c^2*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1
/2)-42*x*a*c^2*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+30*x*b*c^3*(a*c)^(1/2)*((b*
x+a)*(d*x+c))^(1/2)+12*a*c^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/(a*c)^(1/2)/x^
2/((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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.802259, size = 1, normalized size = 0. \[ \left [-\frac{4 \,{\left (6 \, a c^{3} + 5 \,{\left (11 \, b c d^{2} - 21 \, a d^{3}\right )} x^{3} + 2 \,{\left (39 \, b c^{2} d - 70 \, a c d^{2}\right )} x^{2} + 3 \,{\left (5 \, b c^{3} - 7 \, a c^{2} d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c} - 3 \,{\left ({\left (3 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{4} + 2 \,{\left (3 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{3} +{\left (3 \, b^{2} c^{4} - 30 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{2}\right )} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right )}{48 \,{\left (c^{4} d^{2} x^{4} + 2 \, c^{5} d x^{3} + c^{6} x^{2}\right )} \sqrt{a c}}, -\frac{2 \,{\left (6 \, a c^{3} + 5 \,{\left (11 \, b c d^{2} - 21 \, a d^{3}\right )} x^{3} + 2 \,{\left (39 \, b c^{2} d - 70 \, a c d^{2}\right )} x^{2} + 3 \,{\left (5 \, b c^{3} - 7 \, a c^{2} d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left ({\left (3 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{4} + 2 \,{\left (3 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{3} +{\left (3 \, b^{2} c^{4} - 30 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{2}\right )} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right )}{24 \,{\left (c^{4} d^{2} x^{4} + 2 \, c^{5} d x^{3} + c^{6} x^{2}\right )} \sqrt{-a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/48*(4*(6*a*c^3 + 5*(11*b*c*d^2 - 21*a*d^3)*x^3 + 2*(39*b*c^2*d - 70*a*c*d^2)
*x^2 + 3*(5*b*c^3 - 7*a*c^2*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) - 3*((3*
b^2*c^2*d^2 - 30*a*b*c*d^3 + 35*a^2*d^4)*x^4 + 2*(3*b^2*c^3*d - 30*a*b*c^2*d^2 +
35*a^2*c*d^3)*x^3 + (3*b^2*c^4 - 30*a*b*c^3*d + 35*a^2*c^2*d^2)*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))/((c
^4*d^2*x^4 + 2*c^5*d*x^3 + c^6*x^2)*sqrt(a*c)), -1/24*(2*(6*a*c^3 + 5*(11*b*c*d^
2 - 21*a*d^3)*x^3 + 2*(39*b*c^2*d - 70*a*c*d^2)*x^2 + 3*(5*b*c^3 - 7*a*c^2*d)*x)
*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 3*((3*b^2*c^2*d^2 - 30*a*b*c*d^3 + 35*
a^2*d^4)*x^4 + 2*(3*b^2*c^3*d - 30*a*b*c^2*d^2 + 35*a^2*c*d^3)*x^3 + (3*b^2*c^4
- 30*a*b*c^3*d + 35*a^2*c^2*d^2)*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)))/((c^4*d^2*x^4 + 2*c^5*d*x^3 + c^6*x^2)*s
qrt(-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)**(3/2)/x**3/(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^3),x, algorithm="giac")

[Out]

Exception raised: TypeError