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

Optimal. Leaf size=254 \[ -\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d) \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right )}{64 b^2 d^4}+\frac{(a+b x)^{3/2} \sqrt{c+d x} \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right )}{96 b^2 d^3}+\frac{(b c-a d)^2 \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{5/2} d^{9/2}}-\frac{(a+b x)^{5/2} \sqrt{c+d x} (3 a d+7 b c)}{24 b^2 d^2}+\frac{x (a+b x)^{5/2} \sqrt{c+d x}}{4 b d} \]

[Out]

-((b*c - a*d)*(35*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])
/(64*b^2*d^4) + ((35*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*(a + b*x)^(3/2)*Sqrt[c +
d*x])/(96*b^2*d^3) - ((7*b*c + 3*a*d)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(24*b^2*d^2
) + (x*(a + b*x)^(5/2)*Sqrt[c + d*x])/(4*b*d) + ((b*c - a*d)^2*(35*b^2*c^2 + 10*
a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(
64*b^(5/2)*d^(9/2))

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Rubi [A]  time = 0.537182, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d) \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right )}{64 b^2 d^4}+\frac{(a+b x)^{3/2} \sqrt{c+d x} \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right )}{96 b^2 d^3}+\frac{(b c-a d)^2 \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{5/2} d^{9/2}}-\frac{(a+b x)^{5/2} \sqrt{c+d x} (3 a d+7 b c)}{24 b^2 d^2}+\frac{x (a+b x)^{5/2} \sqrt{c+d x}}{4 b d} \]

Antiderivative was successfully verified.

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

[Out]

-((b*c - a*d)*(35*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])
/(64*b^2*d^4) + ((35*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*(a + b*x)^(3/2)*Sqrt[c +
d*x])/(96*b^2*d^3) - ((7*b*c + 3*a*d)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(24*b^2*d^2
) + (x*(a + b*x)^(5/2)*Sqrt[c + d*x])/(4*b*d) + ((b*c - a*d)^2*(35*b^2*c^2 + 10*
a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(
64*b^(5/2)*d^(9/2))

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Rubi in Sympy [A]  time = 40.3499, size = 241, normalized size = 0.95 \[ \frac{x \left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x}}{4 b d} - \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (3 a d + 7 b c\right )}{24 b^{2} d^{2}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (3 a^{2} d^{2} + 10 a b c d + 35 b^{2} c^{2}\right )}{96 b^{2} d^{3}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right ) \left (3 a^{2} d^{2} + 10 a b c d + 35 b^{2} c^{2}\right )}{64 b^{2} d^{4}} + \frac{\left (a d - b c\right )^{2} \left (3 a^{2} d^{2} + 10 a b c d + 35 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{64 b^{\frac{5}{2}} d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(a + b*x)**(5/2)*sqrt(c + d*x)/(4*b*d) - (a + b*x)**(5/2)*sqrt(c + d*x)*(3*a*d
+ 7*b*c)/(24*b**2*d**2) + (a + b*x)**(3/2)*sqrt(c + d*x)*(3*a**2*d**2 + 10*a*b*
c*d + 35*b**2*c**2)/(96*b**2*d**3) + sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)*(3*
a**2*d**2 + 10*a*b*c*d + 35*b**2*c**2)/(64*b**2*d**4) + (a*d - b*c)**2*(3*a**2*d
**2 + 10*a*b*c*d + 35*b**2*c**2)*atanh(sqrt(b)*sqrt(c + d*x)/(sqrt(d)*sqrt(a + b
*x)))/(64*b**(5/2)*d**(9/2))

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Mathematica [A]  time = 0.216728, size = 208, normalized size = 0.82 \[ \frac{3 (b c-a d)^2 \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )-2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x} \left (9 a^3 d^3+3 a^2 b d^2 (5 c-2 d x)+a b^2 d \left (-145 c^2+92 c d x-72 d^2 x^2\right )+b^3 \left (105 c^3-70 c^2 d x+56 c d^2 x^2-48 d^3 x^3\right )\right )}{384 b^{5/2} d^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]*(9*a^3*d^3 + 3*a^2*b*d^2*(5*c -
2*d*x) + a*b^2*d*(-145*c^2 + 92*c*d*x - 72*d^2*x^2) + b^3*(105*c^3 - 70*c^2*d*x
+ 56*c*d^2*x^2 - 48*d^3*x^3)) + 3*(b*c - a*d)^2*(35*b^2*c^2 + 10*a*b*c*d + 3*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]])/
(384*b^(5/2)*d^(9/2))

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Maple [B]  time = 0.032, size = 574, normalized size = 2.3 \[{\frac{1}{384\,{b}^{2}{d}^{4}}\sqrt{bx+a}\sqrt{dx+c} \left ( 96\,{x}^{3}{b}^{3}{d}^{3}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+144\,{x}^{2}a{b}^{2}{d}^{3}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}-112\,{x}^{2}{b}^{3}c{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+9\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{4}{d}^{4}+12\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}bc{d}^{3}+54\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}{b}^{2}{c}^{2}{d}^{2}-180\,{c}^{3}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) a{b}^{3}d+105\,{c}^{4}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{4}+12\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }x{a}^{2}b{d}^{3}-184\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }xa{b}^{2}c{d}^{2}+140\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }x{b}^{3}{c}^{2}d-18\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{a}^{3}{d}^{3}-30\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{a}^{2}bc{d}^{2}+290\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }a{b}^{2}{c}^{2}d-210\,{c}^{3}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{b}^{3}\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(96*x^3*b^3*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^
(1/2)+144*x^2*a*b^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-112*x^2*b^3*c*d^2*((
b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+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^4*d^4+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))*a^3*b*c*d^3+54*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*d^2-180*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))*a*b^3*d+10
5*c^4*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^4+12*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x*a^2*b*d^3-184*(b*d)^(1/2)*((b*x+a
)*(d*x+c))^(1/2)*x*a*b^2*c*d^2+140*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x*b^3*c^2
*d-18*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*d^3-30*(b*d)^(1/2)*((b*x+a)*(d*x+c
))^(1/2)*a^2*b*c*d^2+290*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b^2*c^2*d-210*c^3
*((b*x+a)*(d*x+c))^(1/2)*b^3*(b*d)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/b^2/d^4/(b*d)^
(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((b*x + a)^(3/2)*x^2/sqrt(d*x + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.282904, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (48 \, b^{3} d^{3} x^{3} - 105 \, b^{3} c^{3} + 145 \, a b^{2} c^{2} d - 15 \, a^{2} b c d^{2} - 9 \, a^{3} d^{3} - 8 \,{\left (7 \, b^{3} c d^{2} - 9 \, a b^{2} d^{3}\right )} x^{2} + 2 \,{\left (35 \, b^{3} c^{2} d - 46 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (35 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 3 \, a^{4} d^{4}\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{768 \, \sqrt{b d} b^{2} d^{4}}, \frac{2 \,{\left (48 \, b^{3} d^{3} x^{3} - 105 \, b^{3} c^{3} + 145 \, a b^{2} c^{2} d - 15 \, a^{2} b c d^{2} - 9 \, a^{3} d^{3} - 8 \,{\left (7 \, b^{3} c d^{2} - 9 \, a b^{2} d^{3}\right )} x^{2} + 2 \,{\left (35 \, b^{3} c^{2} d - 46 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (35 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 3 \, a^{4} d^{4}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{384 \, \sqrt{-b d} b^{2} d^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/768*(4*(48*b^3*d^3*x^3 - 105*b^3*c^3 + 145*a*b^2*c^2*d - 15*a^2*b*c*d^2 - 9*a
^3*d^3 - 8*(7*b^3*c*d^2 - 9*a*b^2*d^3)*x^2 + 2*(35*b^3*c^2*d - 46*a*b^2*c*d^2 +
3*a^2*b*d^3)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 3*(35*b^4*c^4 - 60*a*b^3
*c^3*d + 18*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 + 3*a^4*d^4)*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^2*d^4), 1/384*
(2*(48*b^3*d^3*x^3 - 105*b^3*c^3 + 145*a*b^2*c^2*d - 15*a^2*b*c*d^2 - 9*a^3*d^3
- 8*(7*b^3*c*d^2 - 9*a*b^2*d^3)*x^2 + 2*(35*b^3*c^2*d - 46*a*b^2*c*d^2 + 3*a^2*b
*d^3)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 3*(35*b^4*c^4 - 60*a*b^3*c^3*d
+ 18*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 + 3*a^4*d^4)*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^4)]

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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)**(3/2)/(d*x+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.252783, size = 393, normalized size = 1.55 \[ \frac{{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{6 \,{\left (b x + a\right )}}{b^{3} d} - \frac{7 \, b^{7} c d^{5} + 9 \, a b^{6} d^{6}}{b^{9} d^{7}}\right )} + \frac{35 \, b^{8} c^{2} d^{4} + 10 \, a b^{7} c d^{5} + 3 \, a^{2} b^{6} d^{6}}{b^{9} d^{7}}\right )} - \frac{3 \,{\left (35 \, b^{9} c^{3} d^{3} - 25 \, a b^{8} c^{2} d^{4} - 7 \, a^{2} b^{7} c d^{5} - 3 \, a^{3} b^{6} d^{6}\right )}}{b^{9} d^{7}}\right )} \sqrt{b x + a} - \frac{3 \,{\left (35 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 3 \, a^{4} d^{4}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b^{2} d^{4}}\right )} b}{192 \,{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/192*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a
)/(b^3*d) - (7*b^7*c*d^5 + 9*a*b^6*d^6)/(b^9*d^7)) + (35*b^8*c^2*d^4 + 10*a*b^7*
c*d^5 + 3*a^2*b^6*d^6)/(b^9*d^7)) - 3*(35*b^9*c^3*d^3 - 25*a*b^8*c^2*d^4 - 7*a^2
*b^7*c*d^5 - 3*a^3*b^6*d^6)/(b^9*d^7))*sqrt(b*x + a) - 3*(35*b^4*c^4 - 60*a*b^3*
c^3*d + 18*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 + 3*a^4*d^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^2*d^4))*b/abs(b)