3.478 \(\int \frac{x^3}{\left (a+b \sqrt{c+d x}\right )^2} \, dx\)

Optimal. Leaf size=240 \[ \frac{2 a \left (a^2-b^2 c\right )^3}{b^8 d^4 \left (a+b \sqrt{c+d x}\right )}+\frac{2 \left (7 a^2-b^2 c\right ) \left (a^2-b^2 c\right )^2 \log \left (a+b \sqrt{c+d x}\right )}{b^8 d^4}-\frac{12 a \left (a^2-b^2 c\right )^2 \sqrt{c+d x}}{b^7 d^4}-\frac{4 a \left (2 a^2-3 b^2 c\right ) (c+d x)^{3/2}}{3 b^5 d^4}+\frac{3 \left (a^2-b^2 c\right ) (c+d x)^2}{2 b^4 d^4}+\frac{x \left (5 a^4-9 a^2 b^2 c+3 b^4 c^2\right )}{b^6 d^3}-\frac{4 a (c+d x)^{5/2}}{5 b^3 d^4}+\frac{(c+d x)^3}{3 b^2 d^4} \]

[Out]

((5*a^4 - 9*a^2*b^2*c + 3*b^4*c^2)*x)/(b^6*d^3) - (12*a*(a^2 - b^2*c)^2*Sqrt[c +
d*x])/(b^7*d^4) - (4*a*(2*a^2 - 3*b^2*c)*(c + d*x)^(3/2))/(3*b^5*d^4) + (3*(a^2
- b^2*c)*(c + d*x)^2)/(2*b^4*d^4) - (4*a*(c + d*x)^(5/2))/(5*b^3*d^4) + (c + d*
x)^3/(3*b^2*d^4) + (2*a*(a^2 - b^2*c)^3)/(b^8*d^4*(a + b*Sqrt[c + d*x])) + (2*(a
^2 - b^2*c)^2*(7*a^2 - b^2*c)*Log[a + b*Sqrt[c + d*x]])/(b^8*d^4)

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Rubi [A]  time = 0.575377, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{2 a \left (a^2-b^2 c\right )^3}{b^8 d^4 \left (a+b \sqrt{c+d x}\right )}+\frac{2 \left (7 a^2-b^2 c\right ) \left (a^2-b^2 c\right )^2 \log \left (a+b \sqrt{c+d x}\right )}{b^8 d^4}-\frac{12 a \left (a^2-b^2 c\right )^2 \sqrt{c+d x}}{b^7 d^4}-\frac{4 a \left (2 a^2-3 b^2 c\right ) (c+d x)^{3/2}}{3 b^5 d^4}+\frac{3 \left (a^2-b^2 c\right ) (c+d x)^2}{2 b^4 d^4}+\frac{x \left (5 a^4-9 a^2 b^2 c+3 b^4 c^2\right )}{b^6 d^3}-\frac{4 a (c+d x)^{5/2}}{5 b^3 d^4}+\frac{(c+d x)^3}{3 b^2 d^4} \]

Antiderivative was successfully verified.

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

[Out]

((5*a^4 - 9*a^2*b^2*c + 3*b^4*c^2)*x)/(b^6*d^3) - (12*a*(a^2 - b^2*c)^2*Sqrt[c +
d*x])/(b^7*d^4) - (4*a*(2*a^2 - 3*b^2*c)*(c + d*x)^(3/2))/(3*b^5*d^4) + (3*(a^2
- b^2*c)*(c + d*x)^2)/(2*b^4*d^4) - (4*a*(c + d*x)^(5/2))/(5*b^3*d^4) + (c + d*
x)^3/(3*b^2*d^4) + (2*a*(a^2 - b^2*c)^3)/(b^8*d^4*(a + b*Sqrt[c + d*x])) + (2*(a
^2 - b^2*c)^2*(7*a^2 - b^2*c)*Log[a + b*Sqrt[c + d*x]])/(b^8*d^4)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{4 a \left (c + d x\right )^{\frac{5}{2}}}{5 b^{3} d^{4}} - \frac{4 a \left (2 a^{2} - 3 b^{2} c\right ) \left (c + d x\right )^{\frac{3}{2}}}{3 b^{5} d^{4}} - \frac{12 a \left (a^{2} - b^{2} c\right )^{2} \sqrt{c + d x}}{b^{7} d^{4}} + \frac{2 a \left (a^{2} - b^{2} c\right )^{3}}{b^{8} d^{4} \left (a + b \sqrt{c + d x}\right )} + \frac{\left (c + d x\right )^{3}}{3 b^{2} d^{4}} + \frac{3 \left (a^{2} - b^{2} c\right ) \left (c + d x\right )^{2}}{2 b^{4} d^{4}} + \frac{2 \left (5 a^{4} - 9 a^{2} b^{2} c + 3 b^{4} c^{2}\right ) \int ^{\sqrt{c + d x}} x\, dx}{b^{6} d^{4}} + \frac{2 \left (a^{2} - b^{2} c\right )^{2} \left (7 a^{2} - b^{2} c\right ) \log{\left (a + b \sqrt{c + d x} \right )}}{b^{8} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*a*(c + d*x)**(5/2)/(5*b**3*d**4) - 4*a*(2*a**2 - 3*b**2*c)*(c + d*x)**(3/2)/(
3*b**5*d**4) - 12*a*(a**2 - b**2*c)**2*sqrt(c + d*x)/(b**7*d**4) + 2*a*(a**2 - b
**2*c)**3/(b**8*d**4*(a + b*sqrt(c + d*x))) + (c + d*x)**3/(3*b**2*d**4) + 3*(a*
*2 - b**2*c)*(c + d*x)**2/(2*b**4*d**4) + 2*(5*a**4 - 9*a**2*b**2*c + 3*b**4*c**
2)*Integral(x, (x, sqrt(c + d*x)))/(b**6*d**4) + 2*(a**2 - b**2*c)**2*(7*a**2 -
b**2*c)*log(a + b*sqrt(c + d*x))/(b**8*d**4)

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Mathematica [A]  time = 0.451785, size = 301, normalized size = 1.25 \[ \frac{\frac{60 a^2 \left (a^2-b^2 c\right )^3}{a^2-b^2 (c+d x)}+30 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \log \left (a^2-b^2 (c+d x)\right )+60 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \tanh ^{-1}\left (\frac{b \sqrt{c+d x}}{a}\right )-15 b^4 d^2 x^2 \left (b^2 c-3 a^2\right )+30 b^2 d x \left (5 a^4-6 a^2 b^2 c+b^4 c^2\right )-\frac{4 a b \sqrt{c+d x} \left (-105 a^6+5 a^4 b^2 (59 c+14 d x)+a^2 b^4 \left (-271 c^2-122 c d x+14 d^2 x^2\right )+3 b^6 \left (27 c^3+16 c^2 d x-4 c d^2 x^2+2 d^3 x^3\right )\right )}{b^2 (c+d x)-a^2}+10 b^6 d^3 x^3}{30 b^8 d^4} \]

Antiderivative was successfully verified.

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

[Out]

(30*b^2*(5*a^4 - 6*a^2*b^2*c + b^4*c^2)*d*x - 15*b^4*(-3*a^2 + b^2*c)*d^2*x^2 +
10*b^6*d^3*x^3 + (60*a^2*(a^2 - b^2*c)^3)/(a^2 - b^2*(c + d*x)) - (4*a*b*Sqrt[c
+ d*x]*(-105*a^6 + 5*a^4*b^2*(59*c + 14*d*x) + a^2*b^4*(-271*c^2 - 122*c*d*x + 1
4*d^2*x^2) + 3*b^6*(27*c^3 + 16*c^2*d*x - 4*c*d^2*x^2 + 2*d^3*x^3)))/(-a^2 + b^2
*(c + d*x)) + 60*(a^2 - b^2*c)^2*(7*a^2 - b^2*c)*ArcTanh[(b*Sqrt[c + d*x])/a] +
30*(a^2 - b^2*c)^2*(7*a^2 - b^2*c)*Log[a^2 - b^2*(c + d*x)])/(30*b^8*d^4)

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Maple [A]  time = 0.014, size = 416, normalized size = 1.7 \[{\frac{{x}^{3}}{3\,{b}^{2}d}}-{\frac{c{x}^{2}}{2\,{b}^{2}{d}^{2}}}+{\frac{{c}^{2}x}{{b}^{2}{d}^{3}}}+{\frac{11\,{c}^{3}}{6\,{d}^{4}{b}^{2}}}-{\frac{4\,a}{5\,{b}^{3}{d}^{4}} \left ( dx+c \right ) ^{{\frac{5}{2}}}}+{\frac{3\,{a}^{2}{x}^{2}}{2\,{b}^{4}{d}^{2}}}-6\,{\frac{{a}^{2}xc}{{b}^{4}{d}^{3}}}-{\frac{15\,{a}^{2}{c}^{2}}{2\,{d}^{4}{b}^{4}}}+4\,{\frac{ \left ( dx+c \right ) ^{3/2}ac}{{b}^{3}{d}^{4}}}-{\frac{8\,{a}^{3}}{3\,{d}^{4}{b}^{5}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-12\,{\frac{a{c}^{2}\sqrt{dx+c}}{{b}^{3}{d}^{4}}}+5\,{\frac{x{a}^{4}}{{d}^{3}{b}^{6}}}+5\,{\frac{{a}^{4}c}{{d}^{4}{b}^{6}}}+24\,{\frac{{a}^{3}c\sqrt{dx+c}}{{d}^{4}{b}^{5}}}-12\,{\frac{{a}^{5}\sqrt{dx+c}}{{d}^{4}{b}^{7}}}-2\,{\frac{\ln \left ( a+b\sqrt{dx+c} \right ){c}^{3}}{{d}^{4}{b}^{2}}}+18\,{\frac{\ln \left ( a+b\sqrt{dx+c} \right ){a}^{2}{c}^{2}}{{d}^{4}{b}^{4}}}-30\,{\frac{\ln \left ( a+b\sqrt{dx+c} \right ){a}^{4}c}{{d}^{4}{b}^{6}}}+14\,{\frac{\ln \left ( a+b\sqrt{dx+c} \right ){a}^{6}}{{d}^{4}{b}^{8}}}-2\,{\frac{a{c}^{3}}{{d}^{4}{b}^{2} \left ( a+b\sqrt{dx+c} \right ) }}+6\,{\frac{{c}^{2}{a}^{3}}{{d}^{4}{b}^{4} \left ( a+b\sqrt{dx+c} \right ) }}-6\,{\frac{{a}^{5}c}{{d}^{4}{b}^{6} \left ( a+b\sqrt{dx+c} \right ) }}+2\,{\frac{{a}^{7}}{{d}^{4}{b}^{8} \left ( a+b\sqrt{dx+c} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3/d/b^2*x^3-1/2/d^2/b^2*x^2*c+1/d^3/b^2*x*c^2+11/6/d^4*c^3/b^2-4/5*a*(d*x+c)^(
5/2)/b^3/d^4+3/2/d^2/b^4*x^2*a^2-6/d^3/b^4*x*a^2*c-15/2/d^4/b^4*a^2*c^2+4/d^4/b^
3*(d*x+c)^(3/2)*a*c-8/3/d^4/b^5*a^3*(d*x+c)^(3/2)-12/d^4/b^3*a*c^2*(d*x+c)^(1/2)
+5/d^3/b^6*x*a^4+5/d^4/b^6*a^4*c+24/d^4/b^5*a^3*c*(d*x+c)^(1/2)-12/d^4/b^7*a^5*(
d*x+c)^(1/2)-2/d^4/b^2*ln(a+b*(d*x+c)^(1/2))*c^3+18/d^4/b^4*ln(a+b*(d*x+c)^(1/2)
)*a^2*c^2-30/d^4/b^6*ln(a+b*(d*x+c)^(1/2))*a^4*c+14/d^4/b^8*ln(a+b*(d*x+c)^(1/2)
)*a^6-2/d^4*a/b^2/(a+b*(d*x+c)^(1/2))*c^3+6/d^4*a^3/b^4/(a+b*(d*x+c)^(1/2))*c^2-
6/d^4*a^5/b^6/(a+b*(d*x+c)^(1/2))*c+2/d^4*a^7/b^8/(a+b*(d*x+c)^(1/2))

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Maxima [A]  time = 0.698218, size = 339, normalized size = 1.41 \[ -\frac{\frac{60 \,{\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )}}{\sqrt{d x + c} b^{9} + a b^{8}} - \frac{10 \,{\left (d x + c\right )}^{3} b^{5} - 24 \,{\left (d x + c\right )}^{\frac{5}{2}} a b^{4} - 45 \,{\left (b^{5} c - a^{2} b^{3}\right )}{\left (d x + c\right )}^{2} + 40 \,{\left (3 \, a b^{4} c - 2 \, a^{3} b^{2}\right )}{\left (d x + c\right )}^{\frac{3}{2}} + 30 \,{\left (3 \, b^{5} c^{2} - 9 \, a^{2} b^{3} c + 5 \, a^{4} b\right )}{\left (d x + c\right )} - 360 \,{\left (a b^{4} c^{2} - 2 \, a^{3} b^{2} c + a^{5}\right )} \sqrt{d x + c}}{b^{7}} + \frac{60 \,{\left (b^{6} c^{3} - 9 \, a^{2} b^{4} c^{2} + 15 \, a^{4} b^{2} c - 7 \, a^{6}\right )} \log \left (\sqrt{d x + c} b + a\right )}{b^{8}}}{30 \, d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30*(60*(a*b^6*c^3 - 3*a^3*b^4*c^2 + 3*a^5*b^2*c - a^7)/(sqrt(d*x + c)*b^9 + a
*b^8) - (10*(d*x + c)^3*b^5 - 24*(d*x + c)^(5/2)*a*b^4 - 45*(b^5*c - a^2*b^3)*(d
*x + c)^2 + 40*(3*a*b^4*c - 2*a^3*b^2)*(d*x + c)^(3/2) + 30*(3*b^5*c^2 - 9*a^2*b
^3*c + 5*a^4*b)*(d*x + c) - 360*(a*b^4*c^2 - 2*a^3*b^2*c + a^5)*sqrt(d*x + c))/b
^7 + 60*(b^6*c^3 - 9*a^2*b^4*c^2 + 15*a^4*b^2*c - 7*a^6)*log(sqrt(d*x + c)*b + a
)/b^8)/d^4

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Fricas [A]  time = 0.278942, size = 441, normalized size = 1.84 \[ -\frac{14 \, a b^{6} d^{3} x^{3} + 269 \, a b^{6} c^{3} - 595 \, a^{3} b^{4} c^{2} + 390 \, a^{5} b^{2} c - 60 \, a^{7} -{\left (33 \, a b^{6} c - 35 \, a^{3} b^{4}\right )} d^{2} x^{2} + 2 \,{\left (81 \, a b^{6} c^{2} - 190 \, a^{3} b^{4} c + 105 \, a^{5} b^{2}\right )} d x + 60 \,{\left (a b^{6} c^{3} - 9 \, a^{3} b^{4} c^{2} + 15 \, a^{5} b^{2} c - 7 \, a^{7} +{\left (b^{7} c^{3} - 9 \, a^{2} b^{5} c^{2} + 15 \, a^{4} b^{3} c - 7 \, a^{6} b\right )} \sqrt{d x + c}\right )} \log \left (\sqrt{d x + c} b + a\right ) -{\left (10 \, b^{7} d^{3} x^{3} + 55 \, b^{7} c^{3} - 489 \, a^{2} b^{5} c^{2} + 790 \, a^{4} b^{3} c - 360 \, a^{6} b - 3 \,{\left (5 \, b^{7} c - 7 \, a^{2} b^{5}\right )} d^{2} x^{2} + 2 \,{\left (15 \, b^{7} c^{2} - 54 \, a^{2} b^{5} c + 35 \, a^{4} b^{3}\right )} d x\right )} \sqrt{d x + c}}{30 \,{\left (\sqrt{d x + c} b^{9} d^{4} + a b^{8} d^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30*(14*a*b^6*d^3*x^3 + 269*a*b^6*c^3 - 595*a^3*b^4*c^2 + 390*a^5*b^2*c - 60*a
^7 - (33*a*b^6*c - 35*a^3*b^4)*d^2*x^2 + 2*(81*a*b^6*c^2 - 190*a^3*b^4*c + 105*a
^5*b^2)*d*x + 60*(a*b^6*c^3 - 9*a^3*b^4*c^2 + 15*a^5*b^2*c - 7*a^7 + (b^7*c^3 -
9*a^2*b^5*c^2 + 15*a^4*b^3*c - 7*a^6*b)*sqrt(d*x + c))*log(sqrt(d*x + c)*b + a)
- (10*b^7*d^3*x^3 + 55*b^7*c^3 - 489*a^2*b^5*c^2 + 790*a^4*b^3*c - 360*a^6*b - 3
*(5*b^7*c - 7*a^2*b^5)*d^2*x^2 + 2*(15*b^7*c^2 - 54*a^2*b^5*c + 35*a^4*b^3)*d*x)
*sqrt(d*x + c))/(sqrt(d*x + c)*b^9*d^4 + a*b^8*d^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\left (a + b \sqrt{c + d x}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

undef