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

Optimal. Leaf size=163 \[ -\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{3/2}}-\frac{b^{3/2} (3 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{5/2}}+\frac{b \sqrt{a+b x} \sqrt{c+d x} (3 b c-2 a d)}{c d^2}-\frac{2 (a+b x)^{3/2} (b c-a d)}{c d \sqrt{c+d x}} \]

[Out]

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

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Rubi [A]  time = 0.494514, antiderivative size = 163, 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 \[ -\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{3/2}}-\frac{b^{3/2} (3 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{5/2}}+\frac{b \sqrt{a+b x} \sqrt{c+d x} (3 b c-2 a d)}{c d^2}-\frac{2 (a+b x)^{3/2} (b c-a d)}{c d \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*c - a*d)*(a + b*x)^(3/2))/(c*d*Sqrt[c + d*x]) + (b*(3*b*c - 2*a*d)*Sqrt[a
+ b*x]*Sqrt[c + d*x])/(c*d^2) - (2*a^(5/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqr
t[a]*Sqrt[c + d*x])])/c^(3/2) - (b^(3/2)*(3*b*c - 5*a*d)*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 = 49.0528, size = 151, normalized size = 0.93 \[ - \frac{2 a^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{c^{\frac{3}{2}}} + \frac{b^{\frac{3}{2}} \left (5 a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{d^{\frac{5}{2}}} - \frac{b \sqrt{a + b x} \sqrt{c + d x} \left (2 a d - 3 b c\right )}{c d^{2}} + \frac{2 \left (a + b x\right )^{\frac{3}{2}} \left (a d - b c\right )}{c d \sqrt{c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.598872, size = 188, normalized size = 1.15 \[ -\frac{a^{5/2} \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{c^{3/2}}+\frac{a^{5/2} \log (x)}{c^{3/2}}-\frac{b^{3/2} (3 b c-5 a d) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2 d^{5/2}}+\sqrt{a+b x} \sqrt{c+d x} \left (\frac{2 (b c-a d)^2}{c d^2 (c+d x)}+\frac{b^2}{d^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.036, size = 492, normalized size = 3. \[ -{\frac{1}{2\,{d}^{2}c}\sqrt{bx+a} \left ( 2\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) x{a}^{3}{d}^{3}\sqrt{bd}-5\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) xa{b}^{2}c{d}^{2}\sqrt{ac}+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{b}^{3}{c}^{2}d\sqrt{ac}+2\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){a}^{3}c{d}^{2}\sqrt{bd}-5\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) a{b}^{2}{c}^{2}d\sqrt{ac}+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{3}{c}^{3}\sqrt{ac}-2\,x{b}^{2}cd\sqrt{bd}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }-4\,{a}^{2}{d}^{2}\sqrt{bd}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+8\,abcd\sqrt{bd}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }-6\,{b}^{2}{c}^{2}\sqrt{bd}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) } \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{dx+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.42861, 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),x, algorithm="fricas")

[Out]

[-1/4*((3*b^2*c^3 - 5*a*b*c^2*d + (3*b^2*c^2*d - 5*a*b*c*d^2)*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)*sqr
t(b*x + a)*sqrt(d*x + c)*sqrt(b/d) + 8*(b^2*c*d + a*b*d^2)*x) - 2*(a^2*d^3*x + a
^2*c*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*(b^2*c*d*x + 3*b^2*c^2 - 4*a*b*c*d + 2*a^2*d^2)*sqrt(b*x +
a)*sqrt(d*x + c))/(c*d^3*x + c^2*d^2), -1/2*((3*b^2*c^3 - 5*a*b*c^2*d + (3*b^2*c
^2*d - 5*a*b*c*d^2)*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))) - (a^2*d^3*x + a^2*c*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) - 2*(b^2*c*d*x +
3*b^2*c^2 - 4*a*b*c*d + 2*a^2*d^2)*sqrt(b*x + a)*sqrt(d*x + c))/(c*d^3*x + c^2*
d^2), -1/4*(4*(a^2*d^3*x + a^2*c*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^3 - 5*a*b*c^2*d + (3*
b^2*c^2*d - 5*a*b*c*d^2)*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) - 4*(b^2*c*d*x + 3*b^2*c^2 - 4*a*b*c*d + 2*a^2*d^2)*sqr
t(b*x + a)*sqrt(d*x + c))/(c*d^3*x + c^2*d^2), -1/2*(2*(a^2*d^3*x + a^2*c*d^2)*s
qrt(-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^3 - 5*a*b*c^2*d + (3*b^2*c^2*d - 5*a*b*c*d^2)*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
*(b^2*c*d*x + 3*b^2*c^2 - 4*a*b*c*d + 2*a^2*d^2)*sqrt(b*x + a)*sqrt(d*x + c))/(c
*d^3*x + c^2*d^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{\frac{5}{2}}}{x \left (c + d x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*x)**(5/2)/(x*(c + d*x)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.283905, size = 392, normalized size = 2.4 \[ -\frac{2 \, \sqrt{b d} a^{3} b \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} c{\left | b \right |}} - \frac{{\left (\frac{{\left (b x + a\right )} b^{5} c d^{2}}{b^{6} c d^{4} - a b^{5} d^{5}} + \frac{3 \, b^{6} c^{2} d - 5 \, a b^{5} c d^{2} + 2 \, a^{2} b^{4} d^{3}}{b^{6} c d^{4} - a b^{5} d^{5}}\right )} \sqrt{b x + a}}{32 \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}} - \frac{{\left (3 \, \sqrt{b d} b c^{2} - 5 \, \sqrt{b d} a c d\right )}{\rm ln}\left ({\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{64 \,{\left (b^{2} c d^{4} - a b d^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-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*abs(b))
- 1/32*((b*x + a)*b^5*c*d^2/(b^6*c*d^4 - a*b^5*d^5) + (3*b^6*c^2*d - 5*a*b^5*c*
d^2 + 2*a^2*b^4*d^3)/(b^6*c*d^4 - a*b^5*d^5))*sqrt(b*x + a)/sqrt(b^2*c + (b*x +
a)*b*d - a*b*d) - 1/64*(3*sqrt(b*d)*b*c^2 - 5*sqrt(b*d)*a*c*d)*ln((sqrt(b*d)*sqr
t(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)