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

Optimal. Leaf size=116 \[ -\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{c}}-\frac{\sqrt{b} (b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{3/2}}+\frac{b \sqrt{a+b x} \sqrt{c+d x}}{d} \]

[Out]

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

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Rubi [A]  time = 0.295036, antiderivative size = 116, 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 \[ -\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{c}}-\frac{\sqrt{b} (b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{3/2}}+\frac{b \sqrt{a+b x} \sqrt{c+d x}}{d} \]

Antiderivative was successfully verified.

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

[Out]

(b*Sqrt[a + b*x]*Sqrt[c + d*x])/d - (2*a^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(
Sqrt[a]*Sqrt[c + d*x])])/Sqrt[c] - (Sqrt[b]*(b*c - 3*a*d)*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 = 26.9738, size = 107, normalized size = 0.92 \[ - \frac{2 a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{\sqrt{c}} + \frac{\sqrt{b} \left (3 a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{d^{\frac{3}{2}}} + \frac{b \sqrt{a + b x} \sqrt{c + d x}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)/x/(d*x+c)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.30225, size = 159, normalized size = 1.37 \[ -\frac{a^{3/2} \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{\sqrt{c}}+\frac{a^{3/2} \log (x)}{\sqrt{c}}+\frac{\sqrt{b} (3 a d-b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2 d^{3/2}}+\frac{b \sqrt{a+b x} \sqrt{c+d x}}{d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.027, size = 220, normalized size = 1.9 \[{\frac{1}{2\,d}\sqrt{bx+a}\sqrt{dx+c} \left ( 3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) abd\sqrt{ac}-\ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ){b}^{2}c\sqrt{ac}-2\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){a}^{2}d\sqrt{bd}+2\,b\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)/x/(d*x+c)^(1/2),x)

[Out]

1/2*(b*x+a)^(1/2)*(d*x+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))*a*b*d*(a*c)^(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*(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^2*d*(b*d)^(1/2)+2*b*(a*c)^(1/
2)*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/(b*d)^(1/2)/(a*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)/(sqrt(d*x + c)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.865033, 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)/(sqrt(d*x + c)*x),x, algorithm="fricas")

[Out]

[1/4*(2*a*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) - (b*c - 3*a*d)*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*sqrt(b*x + a)*sqrt(d*x + c)*b)/d, 1/2*(a*d*s
qrt(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) - (b*c - 3*a*d)*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*sqrt(b*x + a)*sqrt(d*x + c)*b)/d, -1/4*(4*a*d*
sqrt(-a/c)*arctan(1/2*(2*a*c + (b*c + a*d)*x)/(sqrt(b*x + a)*sqrt(d*x + c)*c*sqr
t(-a/c))) + (b*c - 3*a*d)*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*sqrt(b*x + a)*sqrt(d*x + c)*b)/d, -1/2*(2*a*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 - 3*a*d)*sqrt(-b/d)*arctan(1/2*(2*b*d*x + b*c + a*d)/(sqrt(b*x + a)*sq
rt(d*x + c)*d*sqrt(-b/d))) - 2*sqrt(b*x + a)*sqrt(d*x + c)*b)/d]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(3/2)/x/(d*x+c)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.234515, size = 267, normalized size = 2.3 \[ -\frac{{\left (\frac{4 \, \sqrt{b d} a^{2} \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} b} - \frac{2 \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}}{b d} - \frac{{\left (\sqrt{b d} b c - 3 \, \sqrt{b d} a d\right )}{\rm ln}\left ({\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{b d^{2}}\right )} b^{2}}{2 \,{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(4*sqrt(b*d)*a^2*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sq
rt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*b) - 2
*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)/(b*d) - (sqrt(b*d)*b*c - 3*sq
rt(b*d)*a*d)*ln((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^
2)/(b*d^2))*b^2/abs(b)