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

Optimal. Leaf size=191 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (a^2 d^2+2 a b c d+5 b^2 c^2\right )}{8 b^2 d^3}-\frac{(b c-a d) \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{5/2} d^{7/2}}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (3 a d+5 b c)}{12 b^2 d^2}+\frac{x (a+b x)^{3/2} \sqrt{c+d x}}{3 b d} \]

[Out]

((5*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*b^2*d^3) - ((
5*b*c + 3*a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(12*b^2*d^2) + (x*(a + b*x)^(3/2)*
Sqrt[c + d*x])/(3*b*d) - ((b*c - a*d)*(5*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*ArcTanh[
(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(8*b^(5/2)*d^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.398857, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (a^2 d^2+2 a b c d+5 b^2 c^2\right )}{8 b^2 d^3}-\frac{(b c-a d) \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{5/2} d^{7/2}}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (3 a d+5 b c)}{12 b^2 d^2}+\frac{x (a+b x)^{3/2} \sqrt{c+d x}}{3 b d} \]

Antiderivative was successfully verified.

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

[Out]

((5*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*b^2*d^3) - ((
5*b*c + 3*a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(12*b^2*d^2) + (x*(a + b*x)^(3/2)*
Sqrt[c + d*x])/(3*b*d) - ((b*c - a*d)*(5*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*ArcTanh[
(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(8*b^(5/2)*d^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 26.7646, size = 178, normalized size = 0.93 \[ \frac{x \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x}}{3 b d} - \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (3 a d + 5 b c\right )}{12 b^{2} d^{2}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a^{2} d^{2} + 2 a b c d + 5 b^{2} c^{2}\right )}{8 b^{2} d^{3}} + \frac{\left (a d - b c\right ) \left (a^{2} d^{2} + 2 a b c d + 5 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{8 b^{\frac{5}{2}} d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(a + b*x)**(3/2)*sqrt(c + d*x)/(3*b*d) - (a + b*x)**(3/2)*sqrt(c + d*x)*(3*a*d
+ 5*b*c)/(12*b**2*d**2) + sqrt(a + b*x)*sqrt(c + d*x)*(a**2*d**2 + 2*a*b*c*d +
5*b**2*c**2)/(8*b**2*d**3) + (a*d - b*c)*(a**2*d**2 + 2*a*b*c*d + 5*b**2*c**2)*a
tanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x)))/(8*b**(5/2)*d**(7/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.131271, size = 160, normalized size = 0.84 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (-3 a^2 d^2+2 a b d (d x-2 c)+b^2 \left (15 c^2-10 c d x+8 d^2 x^2\right )\right )}{24 b^2 d^3}-\frac{(b c-a d) \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{16 b^{5/2} d^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-3*a^2*d^2 + 2*a*b*d*(-2*c + d*x) + b^2*(15*c^2 -
10*c*d*x + 8*d^2*x^2)))/(24*b^2*d^3) - ((b*c - a*d)*(5*b^2*c^2 + 2*a*b*c*d + a^2
*d^2)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/
(16*b^(5/2)*d^(7/2))

_______________________________________________________________________________________

Maple [B]  time = 0.031, size = 395, normalized size = 2.1 \[{\frac{1}{48\,{b}^{2}{d}^{3}}\sqrt{bx+a}\sqrt{dx+c} \left ( 16\,{x}^{2}{b}^{2}{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}{d}^{3}+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}bc{d}^{2}+9\,{c}^{2}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) a{b}^{2}d-15\,{c}^{3}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{3}+4\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }xab{d}^{2}-20\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }x{b}^{2}cd-6\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{a}^{2}{d}^{2}-8\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }abcd+30\,{c}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{b}^{2}\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(16*x^2*b^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(
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^3*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))*a^2*b*c*d^2+9*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))*a*b^2*d-15*c^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^3+4*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x*a*
b*d^2-20*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x*b^2*c*d-6*(b*d)^(1/2)*((b*x+a)*(d
*x+c))^(1/2)*a^2*d^2-8*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b*c*d+30*c^2*((b*x+
a)*(d*x+c))^(1/2)*b^2*(b*d)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/b^2/d^3/(b*d)^(1/2)

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.251883, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (8 \, b^{2} d^{2} x^{2} + 15 \, b^{2} c^{2} - 4 \, a b c d - 3 \, a^{2} d^{2} - 2 \,{\left (5 \, b^{2} c d - a b d^{2}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} - 3 \,{\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{96 \, \sqrt{b d} b^{2} d^{3}}, \frac{2 \,{\left (8 \, b^{2} d^{2} x^{2} + 15 \, b^{2} c^{2} - 4 \, a b c d - 3 \, a^{2} d^{2} - 2 \,{\left (5 \, b^{2} c d - a b d^{2}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} - 3 \,{\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{48 \, \sqrt{-b d} b^{2} d^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(4*(8*b^2*d^2*x^2 + 15*b^2*c^2 - 4*a*b*c*d - 3*a^2*d^2 - 2*(5*b^2*c*d - a*
b*d^2)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 3*(5*b^3*c^3 - 3*a*b^2*c^2*d -
a^2*b*c*d^2 - a^3*d^3)*log(4*(2*b^2*d^2*x + b^2*c*d + a*b*d^2)*sqrt(b*x + a)*sq
rt(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^3), 1/48*(2*(8*b^2*d^2*x^2 + 15*b^2*c^2 - 4
*a*b*c*d - 3*a^2*d^2 - 2*(5*b^2*c*d - a*b*d^2)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(
d*x + c) - 3*(5*b^3*c^3 - 3*a*b^2*c^2*d - a^2*b*c*d^2 - a^3*d^3)*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^3)]

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.243134, size = 289, normalized size = 1.51 \[ \frac{{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b x + a\right )}}{b^{3} d} - \frac{5 \, b^{7} c d^{3} + 7 \, a b^{6} d^{4}}{b^{9} d^{5}}\right )} + \frac{3 \,{\left (5 \, b^{8} c^{2} d^{2} + 2 \, a b^{7} c d^{3} + a^{2} b^{6} d^{4}\right )}}{b^{9} d^{5}}\right )} + \frac{3 \,{\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b^{2} d^{3}}\right )} b}{24 \,{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a
)/(b^3*d) - (5*b^7*c*d^3 + 7*a*b^6*d^4)/(b^9*d^5)) + 3*(5*b^8*c^2*d^2 + 2*a*b^7*
c*d^3 + a^2*b^6*d^4)/(b^9*d^5)) + 3*(5*b^3*c^3 - 3*a*b^2*c^2*d - a^2*b*c*d^2 - a
^3*d^3)*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^2*d^3))*b/abs(b)