3.738 \(\int \frac{x^2}{\sqrt{a+b x} (c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=126 \[ \frac{2 c^2 \sqrt{a+b x}}{3 d^2 (c+d x)^{3/2} (b c-a d)}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b} d^{5/2}}-\frac{4 c \sqrt{a+b x} (2 b c-3 a d)}{3 d^2 \sqrt{c+d x} (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.25334, antiderivative size = 126, 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 \[ \frac{2 c^2 \sqrt{a+b x}}{3 d^2 (c+d x)^{3/2} (b c-a d)}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b} d^{5/2}}-\frac{4 c \sqrt{a+b x} (2 b c-3 a d)}{3 d^2 \sqrt{c+d x} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(Sqrt[a + b*x]*(c + d*x)^(5/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 22.5582, size = 117, normalized size = 0.93 \[ - \frac{2 c^{2} \sqrt{a + b x}}{3 d^{2} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{4 c \sqrt{a + b x} \left (3 a d - 2 b c\right )}{3 d^{2} \sqrt{c + d x} \left (a d - b c\right )^{2}} + \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{\sqrt{b} d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(d*x+c)**(5/2)/(b*x+a)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.229743, size = 115, normalized size = 0.91 \[ \frac{\log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{\sqrt{b} d^{5/2}}+\frac{2 c \sqrt{a+b x} (a d (5 c+6 d x)-b c (3 c+4 d x))}{3 d^2 (c+d x)^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(Sqrt[a + b*x]*(c + d*x)^(5/2)),x]

[Out]

(2*c*Sqrt[a + b*x]*(-(b*c*(3*c + 4*d*x)) + a*d*(5*c + 6*d*x)))/(3*d^2*(b*c - a*d
)^2*(c + d*x)^(3/2)) + Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]
*Sqrt[c + d*x]]/(Sqrt[b]*d^(5/2))

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Maple [B]  time = 0.035, size = 604, normalized size = 4.8 \[{\frac{1}{3\, \left ( ad-bc \right ) ^{2}{d}^{2}}\sqrt{bx+a} \left ( 3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{a}^{2}{d}^{4}-6\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}abc{d}^{3}+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{b}^{2}{c}^{2}{d}^{2}+6\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{a}^{2}c{d}^{3}-12\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) xab{c}^{2}{d}^{2}+6\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{b}^{2}{c}^{3}d+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}{c}^{2}{d}^{2}-6\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) ab{c}^{3}d+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{2}{c}^{4}+12\,xac{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}-8\,xb{c}^{2}d\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+10\,a{c}^{2}d\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}-6\,b{c}^{3}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(d*x+c)^(5/2)/(b*x+a)^(1/2),x)

[Out]

1/3*(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))*x^2*a^2*d^4-6*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^2*a*b*c*d^3+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^2*b^2*c^2*d^2+6*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*c*d^3-12*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^2*d^2+6*
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^3*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))*a^2*c^2*d^2-6*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^3*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^4+12*x*a*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)
-8*x*b*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+10*a*c^2*d*((b*x+a)*(d*x+c))^(1
/2)*(b*d)^(1/2)-6*b*c^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2))/(b*d)^(1/2)/(a*d-b*
c)^2/((b*x+a)*(d*x+c))^(1/2)/d^2/(d*x+c)^(3/2)

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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)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.386807, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left (3 \, b c^{3} - 5 \, a c^{2} d + 2 \,{\left (2 \, b c^{2} d - 3 \, a c d^{2}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} - 3 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{6 \,{\left (b^{2} c^{4} d^{2} - 2 \, a b c^{3} d^{3} + a^{2} c^{2} d^{4} +{\left (b^{2} c^{2} d^{4} - 2 \, a b c d^{5} + a^{2} d^{6}\right )} x^{2} + 2 \,{\left (b^{2} c^{3} d^{3} - 2 \, a b c^{2} d^{4} + a^{2} c d^{5}\right )} x\right )} \sqrt{b d}}, -\frac{2 \,{\left (3 \, b c^{3} - 5 \, a c^{2} d + 2 \,{\left (2 \, b c^{2} d - 3 \, a c d^{2}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} - 3 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{3 \,{\left (b^{2} c^{4} d^{2} - 2 \, a b c^{3} d^{3} + a^{2} c^{2} d^{4} +{\left (b^{2} c^{2} d^{4} - 2 \, a b c d^{5} + a^{2} d^{6}\right )} x^{2} + 2 \,{\left (b^{2} c^{3} d^{3} - 2 \, a b c^{2} d^{4} + a^{2} c d^{5}\right )} x\right )} \sqrt{-b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(b*x + a)*(d*x + c)^(5/2)),x, algorithm="fricas")

[Out]

[-1/6*(4*(3*b*c^3 - 5*a*c^2*d + 2*(2*b*c^2*d - 3*a*c*d^2)*x)*sqrt(b*d)*sqrt(b*x
+ a)*sqrt(d*x + c) - 3*(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2 + (b^2*c^2*d^2 - 2*a
*b*c*d^3 + a^2*d^4)*x^2 + 2*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*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^4*d^2
- 2*a*b*c^3*d^3 + a^2*c^2*d^4 + (b^2*c^2*d^4 - 2*a*b*c*d^5 + a^2*d^6)*x^2 + 2*(
b^2*c^3*d^3 - 2*a*b*c^2*d^4 + a^2*c*d^5)*x)*sqrt(b*d)), -1/3*(2*(3*b*c^3 - 5*a*c
^2*d + 2*(2*b*c^2*d - 3*a*c*d^2)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 3*(
b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2 + (b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*x^2
+ 2*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*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^4*d^2 - 2*a*b*c^3*d^3 + a
^2*c^2*d^4 + (b^2*c^2*d^4 - 2*a*b*c*d^5 + a^2*d^6)*x^2 + 2*(b^2*c^3*d^3 - 2*a*b*
c^2*d^4 + a^2*c*d^5)*x)*sqrt(-b*d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(d*x+c)**(5/2)/(b*x+a)**(1/2),x)

[Out]

Integral(x**2/(sqrt(a + b*x)*(c + d*x)**(5/2)), x)

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GIAC/XCAS [A]  time = 0.248457, size = 288, normalized size = 2.29 \[ \frac{\sqrt{b x + a}{\left (\frac{2 \,{\left (2 \, b^{6} c^{2} d^{2} - 3 \, a b^{5} c d^{3}\right )}{\left (b x + a\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}} + \frac{3 \,{\left (b^{7} c^{3} d - 3 \, a b^{6} c^{2} d^{2} + 2 \, a^{2} b^{5} c d^{3}\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}}\right )}}{12 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} + \frac{{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{4 \, \sqrt{b d} b^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(b*x + a)*(d*x + c)^(5/2)),x, algorithm="giac")

[Out]

1/12*sqrt(b*x + a)*(2*(2*b^6*c^2*d^2 - 3*a*b^5*c*d^3)*(b*x + a)/(b^8*c^2*d^4 - 2
*a*b^7*c*d^5 + a^2*b^6*d^6) + 3*(b^7*c^3*d - 3*a*b^6*c^2*d^2 + 2*a^2*b^5*c*d^3)/
(b^8*c^2*d^4 - 2*a*b^7*c*d^5 + a^2*b^6*d^6))/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/
2) + 1/4*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)