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

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

[Out]

((b*c - a*d)*(3*b^2*c^2 + 10*a*b*c*d + 35*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/
(64*b^4*d^2) + ((3*b^2*c^2 + 10*a*b*c*d + 35*a^2*d^2)*Sqrt[a + b*x]*(c + d*x)^(3
/2))/(96*b^3*d^2) - ((3*b*c + 7*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(24*b^2*d^2)
+ (x*Sqrt[a + b*x]*(c + d*x)^(5/2))/(4*b*d) + ((b*c - a*d)^2*(3*b^2*c^2 + 10*a*
b*c*d + 35*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(6
4*b^(9/2)*d^(5/2))

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Rubi [A]  time = 0.536165, 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 \[ \frac{(b c-a d)^2 \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{9/2} d^{5/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d) \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right )}{64 b^4 d^2}+\frac{\sqrt{a+b x} (c+d x)^{3/2} \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right )}{96 b^3 d^2}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (7 a d+3 b c)}{24 b^2 d^2}+\frac{x \sqrt{a+b x} (c+d x)^{5/2}}{4 b d} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)*(3*b^2*c^2 + 10*a*b*c*d + 35*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/
(64*b^4*d^2) + ((3*b^2*c^2 + 10*a*b*c*d + 35*a^2*d^2)*Sqrt[a + b*x]*(c + d*x)^(3
/2))/(96*b^3*d^2) - ((3*b*c + 7*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(24*b^2*d^2)
+ (x*Sqrt[a + b*x]*(c + d*x)^(5/2))/(4*b*d) + ((b*c - a*d)^2*(3*b^2*c^2 + 10*a*
b*c*d + 35*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(6
4*b^(9/2)*d^(5/2))

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Rubi in Sympy [A]  time = 40.3931, size = 241, normalized size = 0.95 \[ \frac{x \sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}}}{4 b d} - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}} \left (7 a d + 3 b c\right )}{24 b^{2} d^{2}} + \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (35 a^{2} d^{2} + 10 a b c d + 3 b^{2} c^{2}\right )}{96 b^{3} d^{2}} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right ) \left (35 a^{2} d^{2} + 10 a b c d + 3 b^{2} c^{2}\right )}{64 b^{4} d^{2}} + \frac{\left (a d - b c\right )^{2} \left (35 a^{2} d^{2} + 10 a b c d + 3 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{64 b^{\frac{9}{2}} d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.215226, size = 208, normalized size = 0.82 \[ \frac{3 (b c-a d)^2 \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )-2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x} \left (105 a^3 d^3-5 a^2 b d^2 (29 c+14 d x)+a b^2 d \left (15 c^2+92 c d x+56 d^2 x^2\right )+b^3 \left (9 c^3-6 c^2 d x-72 c d^2 x^2-48 d^3 x^3\right )\right )}{384 b^{9/2} d^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]*(105*a^3*d^3 - 5*a^2*b*d^2*(29*c
+ 14*d*x) + a*b^2*d*(15*c^2 + 92*c*d*x + 56*d^2*x^2) + b^3*(9*c^3 - 6*c^2*d*x -
72*c*d^2*x^2 - 48*d^3*x^3)) + 3*(b*c - a*d)^2*(3*b^2*c^2 + 10*a*b*c*d + 35*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^(9/2)*d^(5/2))

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Maple [B]  time = 0.03, size = 574, normalized size = 2.3 \[{\frac{1}{384\,{b}^{4}{d}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( 96\,{x}^{3}{b}^{3}{d}^{3}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}-112\,{x}^{2}a{b}^{2}{d}^{3}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+144\,{x}^{2}{b}^{3}c{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+105\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{4}{d}^{4}-180\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}bc{d}^{3}+54\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}{b}^{2}{c}^{2}{d}^{2}+12\,{c}^{3}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) a{b}^{3}d+9\,{c}^{4}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{4}+140\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }x{a}^{2}b{d}^{3}-184\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }xa{b}^{2}c{d}^{2}+12\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }x{b}^{3}{c}^{2}d-210\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{a}^{3}{d}^{3}+290\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{a}^{2}bc{d}^{2}-30\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }a{b}^{2}{c}^{2}d-18\,{c}^{3}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{b}^{3}\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/384*(d*x+c)^(1/2)*(b*x+a)^(1/2)*(96*x^3*b^3*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^
(1/2)-112*x^2*a*b^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+144*x^2*b^3*c*d^2*((
b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+105*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^4-180*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+12*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+
9*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+140*(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+12*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x*b^3*c^2
*d-210*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*d^3+290*(b*d)^(1/2)*((b*x+a)*(d*x
+c))^(1/2)*a^2*b*c*d^2-30*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b^2*c^2*d-18*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^4/d^2/(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((d*x + c)^(3/2)*x^2/sqrt(b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.300996, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (48 \, b^{3} d^{3} x^{3} - 9 \, b^{3} c^{3} - 15 \, a b^{2} c^{2} d + 145 \, a^{2} b c d^{2} - 105 \, a^{3} d^{3} + 8 \,{\left (9 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} + 2 \,{\left (3 \, b^{3} c^{2} d - 46 \, a b^{2} c d^{2} + 35 \, a^{2} b d^{3}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (3 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 60 \, a^{3} b c d^{3} + 35 \, a^{4} d^{4}\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{768 \, \sqrt{b d} b^{4} d^{2}}, \frac{2 \,{\left (48 \, b^{3} d^{3} x^{3} - 9 \, b^{3} c^{3} - 15 \, a b^{2} c^{2} d + 145 \, a^{2} b c d^{2} - 105 \, a^{3} d^{3} + 8 \,{\left (9 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} + 2 \,{\left (3 \, b^{3} c^{2} d - 46 \, a b^{2} c d^{2} + 35 \, a^{2} b d^{3}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (3 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 60 \, a^{3} b c d^{3} + 35 \, a^{4} d^{4}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{384 \, \sqrt{-b d} b^{4} d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.276038, size = 676, normalized size = 2.66 \[ \frac{\frac{8 \,{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b x + a\right )}}{b^{2}} + \frac{b^{6} c d^{3} - 13 \, a b^{5} d^{4}}{b^{7} d^{4}}\right )} - \frac{3 \,{\left (b^{7} c^{2} d^{2} + 2 \, a b^{6} c d^{3} - 11 \, a^{2} b^{5} d^{4}\right )}}{b^{7} d^{4}}\right )} - \frac{3 \,{\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b d^{2}}\right )} c{\left | b \right |}}{b^{2}} + \frac{{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{6 \,{\left (b x + a\right )}}{b^{3}} + \frac{b^{12} c d^{5} - 25 \, a b^{11} d^{6}}{b^{14} d^{6}}\right )} - \frac{5 \, b^{13} c^{2} d^{4} + 14 \, a b^{12} c d^{5} - 163 \, a^{2} b^{11} d^{6}}{b^{14} d^{6}}\right )} + \frac{3 \,{\left (5 \, b^{14} c^{3} d^{3} + 9 \, a b^{13} c^{2} d^{4} + 15 \, a^{2} b^{12} c d^{5} - 93 \, a^{3} b^{11} d^{6}\right )}}{b^{14} d^{6}}\right )} \sqrt{b x + a} + \frac{3 \,{\left (5 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 35 \, a^{4} d^{4}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b^{2} d^{3}}\right )} d{\left | b \right |}}{b^{2}}}{192 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/192*(8*(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^2 + (b^6*c*d^3 - 13*a*b^5*d^4)/(b^7*d^4)) - 3*(b^7*c^2*d^2 + 2*a*b^6*c*d
^3 - 11*a^2*b^5*d^4)/(b^7*d^4)) - 3*(b^3*c^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*d^2))*c*abs(b)/b^2 + (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 + (b^12*c*d^5 - 25*a*b^11*d^6)/(b^14*d^6)) - (
5*b^13*c^2*d^4 + 14*a*b^12*c*d^5 - 163*a^2*b^11*d^6)/(b^14*d^6)) + 3*(5*b^14*c^3
*d^3 + 9*a*b^13*c^2*d^4 + 15*a^2*b^12*c*d^5 - 93*a^3*b^11*d^6)/(b^14*d^6))*sqrt(
b*x + a) + 3*(5*b^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 3
5*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^3))*d*abs(b)/b^2)/b