3.566 \(\int \frac{x \sqrt{a+b x}}{\sqrt{c+d x}} \, dx\)

Optimal. Leaf size=125 \[ \frac{(b c-a d) (a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{3/2} d^{5/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (a d+3 b c)}{4 b d^2}+\frac{(a+b x)^{3/2} \sqrt{c+d x}}{2 b d} \]

[Out]

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

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Rubi [A]  time = 0.165917, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{(b c-a d) (a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{3/2} d^{5/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (a d+3 b c)}{4 b d^2}+\frac{(a+b x)^{3/2} \sqrt{c+d x}}{2 b d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.552, size = 109, normalized size = 0.87 \[ \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x}}{2 b d} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d + 3 b c\right )}{4 b d^{2}} - \frac{\left (a d - b c\right ) \left (a d + 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{4 b^{\frac{3}{2}} d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x)**(3/2)*sqrt(c + d*x)/(2*b*d) - sqrt(a + b*x)*sqrt(c + d*x)*(a*d + 3*b*
c)/(4*b*d**2) - (a*d - b*c)*(a*d + 3*b*c)*atanh(sqrt(b)*sqrt(c + d*x)/(sqrt(d)*s
qrt(a + b*x)))/(4*b**(3/2)*d**(5/2))

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Mathematica [A]  time = 0.0887706, size = 115, normalized size = 0.92 \[ \frac{(b c-a d) (a d+3 b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{8 b^{3/2} d^{5/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (a d-3 b c+2 b d x)}{4 b d^2} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-3*b*c + a*d + 2*b*d*x))/(4*b*d^2) + ((b*c - a*d)*
(3*b*c + a*d)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c +
d*x]])/(8*b^(3/2)*d^(5/2))

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Maple [B]  time = 0.021, size = 250, normalized size = 2. \[ -{\frac{1}{8\,{d}^{2}b}\sqrt{bx+a}\sqrt{dx+c} \left ( \ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ){a}^{2}{d}^{2}+2\,c\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) adb-3\,{c}^{2}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{2}-4\,x\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }db\sqrt{bd}+6\,c\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }b\sqrt{bd}-2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }ad\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242395, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (2 \, b d x - 3 \, b c + a d\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} -{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \log \left (-4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{16 \, \sqrt{b d} b d^{2}}, \frac{2 \,{\left (2 \, b d x - 3 \, b c + a d\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} +{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{8 \, \sqrt{-b d} b d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)*x/sqrt(d*x + c),x, algorithm="fricas")

[Out]

[1/16*(4*(2*b*d*x - 3*b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) - (3*b^2*
c^2 - 2*a*b*c*d - a^2*d^2)*log(-4*(2*b^2*d^2*x + b^2*c*d + a*b*d^2)*sqrt(b*x + a
)*sqrt(d*x + c) + (8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 8*(b^2*c*d +
a*b*d^2)*x)*sqrt(b*d)))/(sqrt(b*d)*b*d^2), 1/8*(2*(2*b*d*x - 3*b*c + a*d)*sqrt(-
b*d)*sqrt(b*x + a)*sqrt(d*x + c) + (3*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*arctan(1/2*
(2*b*d*x + b*c + a*d)*sqrt(-b*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*d)))/(sqrt(-b*d)
*b*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(x*(b*x+a)**(1/2)/(d*x+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.232762, size = 193, normalized size = 1.54 \[ \frac{\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (\frac{2 \,{\left (b x + a\right )}}{b d} - \frac{3 \, b^{2} c d + a b d^{2}}{b^{2} d^{3}}\right )} - \frac{{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} d^{2}}}{4 \,{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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