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

Optimal. Leaf size=112 \[ -\frac{(a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2} d^{5/2}}+\frac{2 c^2 \sqrt{a+b x}}{d^2 \sqrt{c+d x} (b c-a d)}+\frac{\sqrt{a+b x} \sqrt{c+d x}}{b d^2} \]

[Out]

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

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Rubi [A]  time = 0.262226, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{(a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2} d^{5/2}}+\frac{2 c^2 \sqrt{a+b x}}{d^2 \sqrt{c+d x} (b c-a d)}+\frac{\sqrt{a+b x} \sqrt{c+d x}}{b d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.0102, size = 100, normalized size = 0.89 \[ - \frac{2 c^{2} \sqrt{a + b x}}{d^{2} \sqrt{c + d x} \left (a d - b c\right )} + \frac{\sqrt{a + b x} \sqrt{c + d x}}{b d^{2}} - \frac{\left (a d + 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{b^{\frac{3}{2}} d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.211043, size = 114, normalized size = 1.02 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (\frac{2 c^2}{(c+d x) (b c-a d)}+\frac{1}{b}\right )}{d^2}-\frac{(a d+3 b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2 b^{3/2} d^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(b^(-1) + (2*c^2)/((b*c - a*d)*(c + d*x))))/d^2 - (
(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]])/(2*b^(3/2)*d^(5/2))

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Maple [B]  time = 0.035, size = 439, normalized size = 3.9 \[ -{\frac{1}{2\, \left ( ad-bc \right ) b{d}^{2}}\sqrt{bx+a} \left ( \ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) x{a}^{2}{d}^{3}+2\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) xabc{d}^{2}-3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{b}^{2}{c}^{2}d+\ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ){a}^{2}c{d}^{2}+2\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) ab{c}^{2}d-3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{2}{c}^{3}-2\,xa{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+2\,xbcd\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}-2\,acd\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+6\,b{c}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{dx+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.323048, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (3 \, b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} +{\left (3 \, b^{2} c^{3} - 2 \, a b c^{2} d - a^{2} c d^{2} +{\left (3 \, b^{2} c^{2} d - 2 \, a b c d^{2} - a^{2} d^{3}\right )} x\right )} \log \left (-4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{4 \,{\left (b^{2} c^{2} d^{2} - a b c d^{3} +{\left (b^{2} c d^{3} - a b d^{4}\right )} x\right )} \sqrt{b d}}, \frac{2 \,{\left (3 \, b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} -{\left (3 \, b^{2} c^{3} - 2 \, a b c^{2} d - a^{2} c d^{2} +{\left (3 \, b^{2} c^{2} d - 2 \, a b c d^{2} - a^{2} d^{3}\right )} x\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{2 \,{\left (b^{2} c^{2} d^{2} - a b c d^{3} +{\left (b^{2} c d^{3} - a b d^{4}\right )} x\right )} \sqrt{-b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.250993, size = 251, normalized size = 2.24 \[ \frac{\sqrt{b x + a}{\left (\frac{{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )}{\left (b x + a\right )}}{b^{6} c d^{4} - a b^{5} d^{5}} + \frac{3 \, b^{4} c^{2} d - 2 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}}{b^{6} c d^{4} - a b^{5} d^{5}}\right )}}{8 \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}} + \frac{{\left (3 \, b c + a d\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{8 \, \sqrt{b d} b^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(b*x + a)*((b^3*c*d^2 - a*b^2*d^3)*(b*x + a)/(b^6*c*d^4 - a*b^5*d^5) + (
3*b^4*c^2*d - 2*a*b^3*c*d^2 + a^2*b^2*d^3)/(b^6*c*d^4 - a*b^5*d^5))/sqrt(b^2*c +
(b*x + a)*b*d - a*b*d) + 1/8*(3*b*c + a*d)*ln(abs(-sqrt(b*d)*sqrt(b*x + a) + sq
rt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^3*d^3)