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

Optimal. Leaf size=157 \[ -\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{5/2}}+\frac{2 \sqrt{a+b x} \left (\frac{a^2}{c^2}-\frac{b^2}{d^2}\right )}{\sqrt{c+d x}}+\frac{2 b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{5/2}}-\frac{2 (a+b x)^{3/2} (b c-a d)}{3 c d (c+d x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.442132, antiderivative size = 157, 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 \[ -\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{5/2}}+\frac{2 \sqrt{a+b x} \left (\frac{a^2}{c^2}-\frac{b^2}{d^2}\right )}{\sqrt{c+d x}}+\frac{2 b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{5/2}}-\frac{2 (a+b x)^{3/2} (b c-a d)}{3 c d (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 46.6948, size = 144, normalized size = 0.92 \[ - \frac{2 a^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{c^{\frac{5}{2}}} + \frac{2 b^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{d^{\frac{5}{2}}} + \frac{\sqrt{a + b x} \left (\frac{2 a^{2}}{c^{2}} - \frac{2 b^{2}}{d^{2}}\right )}{\sqrt{c + d x}} + \frac{2 \left (a + b x\right )^{\frac{3}{2}} \left (a d - b c\right )}{3 c d \left (c + d x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.670164, size = 183, normalized size = 1.17 \[ -\frac{a^{5/2} \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{c^{5/2}}+\frac{a^{5/2} \log (x)}{c^{5/2}}+\frac{b^{5/2} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{d^{5/2}}-\frac{2 \sqrt{a+b x} (b c-a d) (a d (4 c+3 d x)+b c (3 c+4 d x))}{3 c^2 d^2 (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*c - a*d)*Sqrt[a + b*x]*(a*d*(4*c + 3*d*x) + b*c*(3*c + 4*d*x)))/(3*c^2*d^
2*(c + d*x)^(3/2)) + (a^(5/2)*Log[x])/c^(5/2) - (a^(5/2)*Log[2*a*c + b*c*x + a*d
*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]])/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^(5/2)

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Maple [B]  time = 0.036, size = 566, normalized size = 3.6 \[{\frac{1}{3\,{c}^{2}{d}^{2}}\sqrt{bx+a} \left ( 3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{b}^{3}{c}^{2}{d}^{2}\sqrt{ac}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{3}{d}^{4}\sqrt{bd}+6\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{b}^{3}{c}^{3}d\sqrt{ac}-6\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) x{a}^{3}c{d}^{3}\sqrt{bd}+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{3}{c}^{4}\sqrt{ac}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){a}^{3}{c}^{2}{d}^{2}\sqrt{bd}+6\,x{a}^{2}{d}^{3}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac}+2\,xabc{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac}-8\,x{b}^{2}{c}^{2}d\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac}+8\,{a}^{2}c{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac}-2\,ab{c}^{2}d\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac}-6\,{b}^{2}{c}^{3}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(b*x+a)^(1/2)*(3*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^2*(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^2*a^3*d^4*(b*d)^(1/2)+6*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*d*(a*c)^(1/2)-6*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^3*c*d^3*(b*d)
^(1/2)+3*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))*b^3*c^4*(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)*a^3*c^2*d^2*(b*d)^(1/2)+6*x*a^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/
2)*(a*c)^(1/2)+2*x*a*b*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)-8*x
*b^2*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)+8*a^2*c*d^2*((b*x+a)*
(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)-2*a*b*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)
^(1/2)*(a*c)^(1/2)-6*b^2*c^3*((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)/(b*d)^(1/2)/(a*c)^(1/2)/(d*x+c)^(3/2)/d^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)^(5/2)/((d*x + c)^(5/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.75527, 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)^(5/2)*x),x, algorithm="fricas")

[Out]

[1/6*(3*(b^2*c^2*d^2*x^2 + 2*b^2*c^3*d*x + b^2*c^4)*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)*sq
rt(d*x + c)*sqrt(b/d) + 8*(b^2*c*d + a*b*d^2)*x) + 3*(a^2*d^4*x^2 + 2*a^2*c*d^3*
x + a^2*c^2*d^2)*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*b^2*c^3 + a*b*c^2*d - 4*a^2*c*d^2 + (4*b^2*c^2*d
- a*b*c*d^2 - 3*a^2*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(c^2*d^4*x^2 + 2*c^3*d^
3*x + c^4*d^2), 1/6*(6*(b^2*c^2*d^2*x^2 + 2*b^2*c^3*d*x + b^2*c^4)*sqrt(-b/d)*ar
ctan(1/2*(2*b*d*x + b*c + a*d)/(sqrt(b*x + a)*sqrt(d*x + c)*d*sqrt(-b/d))) + 3*(
a^2*d^4*x^2 + 2*a^2*c*d^3*x + a^2*c^2*d^2)*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*b^2*c^3 + a*b*c^2*d - 4
*a^2*c*d^2 + (4*b^2*c^2*d - a*b*c*d^2 - 3*a^2*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c
))/(c^2*d^4*x^2 + 2*c^3*d^3*x + c^4*d^2), -1/6*(6*(a^2*d^4*x^2 + 2*a^2*c*d^3*x +
a^2*c^2*d^2)*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))) - 3*(b^2*c^2*d^2*x^2 + 2*b^2*c^3*d*x + b^2*c^4)*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) + 4*(3*b^2*
c^3 + a*b*c^2*d - 4*a^2*c*d^2 + (4*b^2*c^2*d - a*b*c*d^2 - 3*a^2*d^3)*x)*sqrt(b*
x + a)*sqrt(d*x + c))/(c^2*d^4*x^2 + 2*c^3*d^3*x + c^4*d^2), -1/3*(3*(a^2*d^4*x^
2 + 2*a^2*c*d^3*x + a^2*c^2*d^2)*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))) - 3*(b^2*c^2*d^2*x^2 + 2*b^2*c^3*d*x
+ b^2*c^4)*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*(3*b^2*c^3 + a*b*c^2*d - 4*a^2*c*d^2 + (4*b^2*c^2*d - a*b
*c*d^2 - 3*a^2*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(c^2*d^4*x^2 + 2*c^3*d^3*x +
c^4*d^2)]

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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/(d*x+c)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.304313, size = 494, normalized size = 3.15 \[ \frac{\sqrt{b d} c^{4}{\rm ln}\left ({\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{32 \,{\left (b^{2} c d^{4} - a b d^{5}\right )}} - \frac{2 \, \sqrt{b d} a^{3} b \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} c^{2}{\left | b \right |}} + \frac{\sqrt{b x + a}{\left (\frac{{\left (4 \, b^{8} c^{5} d^{2} - 5 \, a b^{7} c^{4} d^{3} - 2 \, a^{2} b^{6} c^{3} d^{4} + 3 \, a^{3} b^{5} c^{2} d^{5}\right )}{\left (b x + a\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}} + \frac{3 \,{\left (b^{9} c^{6} d - 2 \, a b^{8} c^{5} d^{2} + 2 \, a^{3} b^{6} c^{3} d^{4} - a^{4} b^{5} c^{2} d^{5}\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}}\right )}}{48 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/32*sqrt(b*d)*c^4*ln((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*
b*d))^2)/(b^2*c*d^4 - a*b*d^5) - 2*sqrt(b*d)*a^3*b*arctan(-1/2*(b^2*c + a*b*d -
(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*
d)*b))/(sqrt(-a*b*c*d)*c^2*abs(b)) + 1/48*sqrt(b*x + a)*((4*b^8*c^5*d^2 - 5*a*b^
7*c^4*d^3 - 2*a^2*b^6*c^3*d^4 + 3*a^3*b^5*c^2*d^5)*(b*x + a)/(b^8*c^2*d^4 - 2*a*
b^7*c*d^5 + a^2*b^6*d^6) + 3*(b^9*c^6*d - 2*a*b^8*c^5*d^2 + 2*a^3*b^6*c^3*d^4 -
a^4*b^5*c^2*d^5)/(b^8*c^2*d^4 - 2*a*b^7*c*d^5 + a^2*b^6*d^6))/(b^2*c + (b*x + a)
*b*d - a*b*d)^(3/2)