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

Optimal. Leaf size=169 \[ -\frac{(a d+b c) \left (5 a^2 d^2-2 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{7/2} d^{7/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (15 a^2 d^2-10 b d x (a d+b c)+14 a b c d+15 b^2 c^2\right )}{24 b^3 d^3}+\frac{x^2 \sqrt{a+b x} \sqrt{c+d x}}{3 b d} \]

[Out]

(x^2*Sqrt[a + b*x]*Sqrt[c + d*x])/(3*b*d) + (Sqrt[a + b*x]*Sqrt[c + d*x]*(15*b^2
*c^2 + 14*a*b*c*d + 15*a^2*d^2 - 10*b*d*(b*c + a*d)*x))/(24*b^3*d^3) - ((b*c + a
*d)*(5*b^2*c^2 - 2*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]
*Sqrt[c + d*x])])/(8*b^(7/2)*d^(7/2))

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Rubi [A]  time = 0.316136, antiderivative size = 169, 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 \[ -\frac{(a d+b c) \left (5 a^2 d^2-2 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{7/2} d^{7/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (15 a^2 d^2-10 b d x (a d+b c)+14 a b c d+15 b^2 c^2\right )}{24 b^3 d^3}+\frac{x^2 \sqrt{a+b x} \sqrt{c+d x}}{3 b d} \]

Antiderivative was successfully verified.

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

[Out]

(x^2*Sqrt[a + b*x]*Sqrt[c + d*x])/(3*b*d) + (Sqrt[a + b*x]*Sqrt[c + d*x]*(15*b^2
*c^2 + 14*a*b*c*d + 15*a^2*d^2 - 10*b*d*(b*c + a*d)*x))/(24*b^3*d^3) - ((b*c + a
*d)*(5*b^2*c^2 - 2*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]
*Sqrt[c + d*x])])/(8*b^(7/2)*d^(7/2))

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Rubi in Sympy [A]  time = 23.5251, size = 168, normalized size = 0.99 \[ \frac{x^{2} \sqrt{a + b x} \sqrt{c + d x}}{3 b d} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (\frac{15 a^{2} d^{2}}{4} + \frac{7 a b c d}{2} + \frac{15 b^{2} c^{2}}{4} - \frac{5 b d x \left (a d + b c\right )}{2}\right )}{6 b^{3} d^{3}} - \frac{\left (a d + b c\right ) \left (5 a^{2} d^{2} - 2 a b c d + 5 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{8 b^{\frac{7}{2}} d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*sqrt(a + b*x)*sqrt(c + d*x)/(3*b*d) + sqrt(a + b*x)*sqrt(c + d*x)*(15*a**2*
d**2/4 + 7*a*b*c*d/2 + 15*b**2*c**2/4 - 5*b*d*x*(a*d + b*c)/2)/(6*b**3*d**3) - (
a*d + b*c)*(5*a**2*d**2 - 2*a*b*c*d + 5*b**2*c**2)*atanh(sqrt(d)*sqrt(a + b*x)/(
sqrt(b)*sqrt(c + d*x)))/(8*b**(7/2)*d**(7/2))

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Mathematica [A]  time = 0.148937, size = 161, normalized size = 0.95 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (15 a^2 d^2+2 a b d (7 c-5 d x)+b^2 \left (15 c^2-10 c d x+8 d^2 x^2\right )\right )}{24 b^3 d^3}-\frac{(a d+b c) \left (5 a^2 d^2-2 a b c d+5 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{16 b^{7/2} d^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.038, size = 395, normalized size = 2.3 \[ -{\frac{1}{48\,{b}^{3}{d}^{3}} \left ( -16\,{x}^{2}{b}^{2}{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}{d}^{3}+9\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}bc{d}^{2}+9\,{c}^{2}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) a{b}^{2}d+15\,{c}^{3}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{3}+20\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }xab{d}^{2}+20\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }x{b}^{2}cd-30\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{a}^{2}{d}^{2}-28\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }abcd-30\,{c}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{b}^{2}\sqrt{bd} \right ) \sqrt{bx+a}\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*(-16*x^2*b^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+15*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+9*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+20*(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-30*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*d^2-28
*(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)^(1/2)*(d*x+c)^(1/2)/(b*d)^(1/2)/d^3/b^3/((b*x+a)*(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^3/(sqrt(b*x + a)*sqrt(d*x + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.300168, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (8 \, b^{2} d^{2} x^{2} + 15 \, b^{2} c^{2} + 14 \, a b c d + 15 \, a^{2} d^{2} - 10 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (5 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} \log \left (-4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{96 \, \sqrt{b d} b^{3} d^{3}}, \frac{2 \,{\left (8 \, b^{2} d^{2} x^{2} + 15 \, b^{2} c^{2} + 14 \, a b c d + 15 \, a^{2} d^{2} - 10 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} - 3 \,{\left (5 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{48 \, \sqrt{-b d} b^{3} d^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(4*(8*b^2*d^2*x^2 + 15*b^2*c^2 + 14*a*b*c*d + 15*a^2*d^2 - 10*(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
+ 3*a^2*b*c*d^2 + 5*a^3*d^3)*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^3*d^3), 1/48*(2*(8*b^2*d^2*x^2 + 15*b^2*c
^2 + 14*a*b*c*d + 15*a^2*d^2 - 10*(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 + 3*a^2*b*c*d^2 + 5*a^3*d^3)*arct
an(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*d)))/(sqr
t(-b*d)*b^3*d^3)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.241673, size = 290, normalized size = 1.72 \[ \frac{{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b x + a\right )}}{b^{4} d} - \frac{5 \, b^{12} c d^{3} + 13 \, a b^{11} d^{4}}{b^{15} d^{5}}\right )} + \frac{3 \,{\left (5 \, b^{13} c^{2} d^{2} + 8 \, a b^{12} c d^{3} + 11 \, a^{2} b^{11} d^{4}\right )}}{b^{15} d^{5}}\right )} + \frac{3 \,{\left (5 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b^{3} d^{3}}\right )} b}{24 \,{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(sqrt(b*x + a)*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^4*d) - (5*b^12*c*d^3 + 13*a*b^11*d^4)/(b^15*d^5)) + 3*(5*b^13*c^2*d^2 + 8*a
*b^12*c*d^3 + 11*a^2*b^11*d^4)/(b^15*d^5)) + 3*(5*b^3*c^3 + 3*a*b^2*c^2*d + 3*a^
2*b*c*d^2 + 5*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^3*d^3))*b/abs(b)