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

Optimal. Leaf size=164 \[ -\frac{a^{3/2} (5 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{5/2}}+\frac{2 b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{3/2}}-\frac{\sqrt{a+b x} (2 b c-3 a d) (b c-a d)}{c^2 d \sqrt{c+d x}}-\frac{a (a+b x)^{3/2}}{c x \sqrt{c+d x}} \]

[Out]

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

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Rubi [A]  time = 0.513821, antiderivative size = 164, 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 \[ -\frac{a^{3/2} (5 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{5/2}}+\frac{2 b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{3/2}}-\frac{\sqrt{a+b x} (2 b c-3 a d) (b c-a d)}{c^2 d \sqrt{c+d x}}-\frac{a (a+b x)^{3/2}}{c x \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 48.1153, size = 150, normalized size = 0.91 \[ \frac{a^{\frac{3}{2}} \left (3 a d - 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{c^{\frac{5}{2}}} - \frac{a \left (a + b x\right )^{\frac{3}{2}}}{c x \sqrt{c + d x}} + \frac{2 b^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{d^{\frac{3}{2}}} - \frac{\sqrt{a + b x} \left (a d - b c\right ) \left (3 a d - 2 b c\right )}{c^{2} d \sqrt{c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.759201, size = 203, normalized size = 1.24 \[ -\frac{a^{3/2} \log (x) (3 a d-5 b c)}{2 c^{5/2}}+\frac{a^{3/2} (3 a d-5 b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{2 c^{5/2}}+\sqrt{a+b x} \sqrt{c+d x} \left (-\frac{a^2}{c^2 x}-\frac{2 (b c-a d)^2}{c^2 d (c+d x)}\right )+\frac{b^{5/2} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{d^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.037, size = 502, normalized size = 3.1 \[{\frac{1}{2\,{c}^{2}xd}\sqrt{bx+a} \left ( 3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{3}{d}^{3}\sqrt{bd}-5\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{2}bc{d}^{2}\sqrt{bd}+2\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{b}^{3}{c}^{2}d\sqrt{ac}+3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) x{a}^{3}c{d}^{2}\sqrt{bd}-5\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) x{a}^{2}b{c}^{2}d\sqrt{bd}+2\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{b}^{3}{c}^{3}\sqrt{ac}-6\,x{a}^{2}{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac}+8\,xabcd\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac}-4\,x{b}^{2}{c}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac}-2\,{a}^{2}cd\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{dx+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*x+a)^(1/2)*(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^3*d^3*(b*d)^(1/2)-5*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*d^2*(b*d)^(1/2)+2*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))
^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^2*b^3*c^2*d*(a*c)^(1/2)+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*a^3*c*d^2*(b*d)^(1/2)-5*l
n((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*b*c^2*d*(b*
d)^(1/2)+2*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^
(1/2))*x*b^3*c^3*(a*c)^(1/2)-6*x*a^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*
c)^(1/2)+8*x*a*b*c*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)-4*x*b^2*c^2
*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)-2*a^2*c*d*((b*x+a)*(d*x+c))^(1/
2)*(b*d)^(1/2)*(a*c)^(1/2))/c^2/((b*x+a)*(d*x+c))^(1/2)/x/(b*d)^(1/2)/(a*c)^(1/2
)/(d*x+c)^(1/2)/d

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.81857, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(2*(b^2*c^2*d*x^2 + b^2*c^3*x)*sqrt(b/d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*
b*c*d + a^2*d^2 + 4*(2*b*d^2*x + b*c*d + a*d^2)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt
(b/d) + 8*(b^2*c*d + a*b*d^2)*x) - ((5*a*b*c*d^2 - 3*a^2*d^3)*x^2 + (5*a*b*c^2*d
- 3*a^2*c*d^2)*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*(a^2*c*d + (2*b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2)*x)*
sqrt(b*x + a)*sqrt(d*x + c))/(c^2*d^2*x^2 + c^3*d*x), 1/4*(4*(b^2*c^2*d*x^2 + b^
2*c^3*x)*sqrt(-b/d)*arctan(1/2*(2*b*d*x + b*c + a*d)/(sqrt(b*x + a)*sqrt(d*x + c
)*d*sqrt(-b/d))) - ((5*a*b*c*d^2 - 3*a^2*d^3)*x^2 + (5*a*b*c^2*d - 3*a^2*c*d^2)*
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*(a^2*c*d + (2*b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2)*x)*sqrt(b*x + a)*sq
rt(d*x + c))/(c^2*d^2*x^2 + c^3*d*x), -1/2*(((5*a*b*c*d^2 - 3*a^2*d^3)*x^2 + (5*
a*b*c^2*d - 3*a^2*c*d^2)*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))) - (b^2*c^2*d*x^2 + b^2*c^3*x)*sqrt(b/d)*lo
g(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d^2*x + b*c*d + a*d^2)*
sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b/d) + 8*(b^2*c*d + a*b*d^2)*x) + 2*(a^2*c*d +
(2*b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(c^2*d^2*x^2
+ c^3*d*x), -1/2*(((5*a*b*c*d^2 - 3*a^2*d^3)*x^2 + (5*a*b*c^2*d - 3*a^2*c*d^2)*
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*(b^2*c^2*d*x^2 + b^2*c^3*x)*sqrt(-b/d)*arctan(1/2*(2*b*d*x + b*
c + a*d)/(sqrt(b*x + a)*sqrt(d*x + c)*d*sqrt(-b/d))) + 2*(a^2*c*d + (2*b^2*c^2 -
4*a*b*c*d + 3*a^2*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(c^2*d^2*x^2 + c^3*d*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)**(5/2)/x**2/(d*x+c)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 1.33111, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x