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

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.235581, 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 \[ -\frac{2 a^2 \sqrt{c+d x}}{3 b^2 (a+b x)^{3/2} (b c-a d)}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{5/2} \sqrt{d}}+\frac{4 a \sqrt{c+d x} (3 b c-2 a d)}{3 b^2 \sqrt{a+b x} (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 22.7368, size = 117, normalized size = 0.93 \[ \frac{2 a^{2} \sqrt{c + d x}}{3 b^{2} \left (a + b x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{4 a \sqrt{c + d x} \left (2 a d - 3 b c\right )}{3 b^{2} \sqrt{a + b x} \left (a d - b c\right )^{2}} + \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{b^{\frac{5}{2}} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.240396, size = 116, normalized size = 0.92 \[ \frac{2 a \sqrt{c+d x} \left (-3 a^2 d+a b (5 c-4 d x)+6 b^2 c x\right )}{3 b^2 (a+b x)^{3/2} (b c-a d)^2}+\frac{\log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{b^{5/2} \sqrt{d}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.036, size = 604, normalized size = 4.8 \[{\frac{1}{3\, \left ( ad-bc \right ) ^{2}{b}^{2}}\sqrt{dx+c} \left ( 3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{a}^{2}{b}^{2}{d}^{2}-6\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}a{b}^{3}cd+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{b}^{4}{c}^{2}+6\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{a}^{3}b{d}^{2}-12\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{a}^{2}{b}^{2}cd+6\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) xa{b}^{3}{c}^{2}+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{4}{d}^{2}-6\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}bcd+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}{b}^{2}{c}^{2}-8\,x{a}^{2}bd\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+12\,xa{b}^{2}c\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}-6\,{a}^{3}d\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+10\,{a}^{2}bc\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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/((b*x + a)^(5/2)*sqrt(d*x + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.411298, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (5 \, a^{2} b c - 3 \, a^{3} d + 2 \,{\left (3 \, a b^{2} c - 2 \, a^{2} b d\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{6 \,{\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2} +{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{2} + 2 \,{\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x\right )} \sqrt{b d}}, \frac{2 \,{\left (5 \, a^{2} b c - 3 \, a^{3} d + 2 \,{\left (3 \, a b^{2} c - 2 \, a^{2} b d\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{3 \,{\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2} +{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{2} + 2 \,{\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x\right )} \sqrt{-b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.571745, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x