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

Optimal. Leaf size=119 \[ -\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{3/2}}+\frac{2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{3/2}}-\frac{2 \sqrt{a+b x} (b c-a d)}{c d \sqrt{c+d x}} \]

[Out]

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

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Rubi [A]  time = 0.269298, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{3/2}}+\frac{2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{3/2}}-\frac{2 \sqrt{a+b x} (b c-a d)}{c d \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*c - a*d)*Sqrt[a + b*x])/(c*d*Sqrt[c + d*x]) - (2*a^(3/2)*ArcTanh[(Sqrt[c]
*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/c^(3/2) + (2*b^(3/2)*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 = 28.843, size = 109, normalized size = 0.92 \[ - \frac{2 a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{c^{\frac{3}{2}}} + \frac{2 b^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{d^{\frac{3}{2}}} + \frac{2 \sqrt{a + b x} \left (a d - b c\right )}{c d \sqrt{c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.305515, size = 158, normalized size = 1.33 \[ -\frac{a^{3/2} \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{c^{3/2}}+\frac{a^{3/2} \log (x)}{c^{3/2}}+\frac{b^{3/2} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{d^{3/2}}+\frac{2 \sqrt{a+b x} (a d-b c)}{c d \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-(b*c) + a*d)*Sqrt[a + b*x])/(c*d*Sqrt[c + d*x]) + (a^(3/2)*Log[x])/c^(3/2)
- (a^(3/2)*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)*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 = 0.033, size = 306, normalized size = 2.6 \[{\frac{1}{dc}\sqrt{bx+a} \left ( \ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) x{b}^{2}cd\sqrt{ac}-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) x{a}^{2}{d}^{2}\sqrt{bd}+\ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ){b}^{2}{c}^{2}\sqrt{ac}-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ){a}^{2}cd\sqrt{bd}+2\,ad\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac}-2\,bc\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{dx+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*x+a)^(1/2)*(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^2*c*d*(a*c)^(1/2)-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^2*d^2*(b*d)^(1/2)+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))*b^2*c^2*(a*c)^(1/2)-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^2*c*d*(b*d)^(1/2)+2*a*d*((b*x+a)*(d*x+c
))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)-2*b*c*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*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

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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)^(3/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.725198, 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)^(3/2)/((d*x + c)^(3/2)*x),x, algorithm="fricas")

[Out]

[1/2*((b*c*d*x + b*c^2)*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) + (a*d^2*x + a*c*d)*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*c - a*d)*sqrt(b*x + a)*sq
rt(d*x + c))/(c*d^2*x + c^2*d), 1/2*(2*(b*c*d*x + b*c^2)*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*d^2*x + a*
c*d)*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*c - a*d)*sqrt(b*x + a)*sqrt(d*x + c))/(c*d^2*x + c^2*d), -1/2
*(2*(a*d^2*x + a*c*d)*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*c*d*x + b*c^2)*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) + 4*(b*c - a*d)*sqrt(b*x + a)*s
qrt(d*x + c))/(c*d^2*x + c^2*d), -((a*d^2*x + a*c*d)*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*c*d*x + b*c^
2)*sqrt(-b/d)*arctan(1/2*(2*b*d*x + b*c + a*d)/(sqrt(b*x + a)*sqrt(d*x + c)*d*sq
rt(-b/d))) + 2*(b*c - a*d)*sqrt(b*x + a)*sqrt(d*x + c))/(c*d^2*x + c^2*d)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.261141, size = 277, normalized size = 2.33 \[ -\frac{2 \, \sqrt{b d} a^{2} b \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} c{\left | b \right |}} - \frac{\sqrt{b d} b^{2}{\rm ln}\left ({\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{d^{2}{\left | b \right |}} - \frac{2 \,{\left (b^{3} c{\left | b \right |} - a b^{2} d{\left | b \right |}\right )} \sqrt{b x + a}}{\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} b^{2} c d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(b*d)*a^2*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))
- sqrt(b*d)*b^2*ln((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*
d))^2)/(d^2*abs(b)) - 2*(b^3*c*abs(b) - a*b^2*d*abs(b))*sqrt(b*x + a)/(sqrt(b^2*
c + (b*x + a)*b*d - a*b*d)*b^2*c*d)