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

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

[Out]

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

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Rubi [A]  time = 0.17056, 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 \[ -\frac{(b c-a d) (3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{5/2} d^{3/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (3 a d+b c)}{4 b^2 d}+\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 b d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.7173, size = 109, normalized size = 0.87 \[ \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}}}{2 b d} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (3 a d + b c\right )}{4 b^{2} d} + \frac{\left (a d - b c\right ) \left (3 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{4 b^{\frac{5}{2}} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0915731, size = 115, normalized size = 0.92 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} (b (c+2 d x)-3 a d)}{4 b^2 d}-\frac{(b c-a d) (3 a d+b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{8 b^{5/2} d^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.023, size = 251, normalized size = 2. \[{\frac{1}{8\,{b}^{2}d}\sqrt{bx+a}\sqrt{dx+c} \left ( 3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}{d}^{2}-2\,c\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) adb-{c}^{2}\ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ){b}^{2}+4\,x\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }db\sqrt{bd}-6\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }ad\sqrt{bd}+2\,c\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }b\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.248564, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (2 \, b d x + b c - 3 \, a d\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} -{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{16 \, \sqrt{b d} b^{2} d}, \frac{2 \,{\left (2 \, b d x + b c - 3 \, a d\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} -{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{8 \, \sqrt{-b d} b^{2} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.23052, size = 190, normalized size = 1.52 \[ \frac{{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (\frac{2 \,{\left (b x + a\right )}}{b^{4} d^{2}} + \frac{b c d - 5 \, a d^{2}}{b^{4} d^{4}}\right )} + \frac{{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b^{3} d^{3}}\right )}{\left | b \right |}}{48 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)/(b^4*d^2) +
(b*c*d - 5*a*d^2)/(b^4*d^4)) + (b^2*c^2 + 2*a*b*c*d - 3*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)*b^3*d^3))*a
bs(b)/b^4