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

Optimal. Leaf size=230 \[ -\frac{2 a \left (a^2-b^2 c\right )^3 \log \left (a+b \sqrt{c+d x}\right )}{b^8 d^4}+\frac{2 \left (a^2-b^2 c\right )^3 \sqrt{c+d x}}{b^7 d^4}-\frac{a \left (a^2-3 b^2 c\right ) (c+d x)^2}{2 b^4 d^4}+\frac{2 \left (a^2-3 b^2 c\right ) (c+d x)^{5/2}}{5 b^3 d^4}-\frac{a x \left (a^4-3 a^2 b^2 c+3 b^4 c^2\right )}{b^6 d^3}+\frac{2 \left (a^4-3 a^2 b^2 c+3 b^4 c^2\right ) (c+d x)^{3/2}}{3 b^5 d^4}-\frac{a (c+d x)^3}{3 b^2 d^4}+\frac{2 (c+d x)^{7/2}}{7 b d^4} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.52168, antiderivative size = 230, 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 \[ -\frac{2 a \left (a^2-b^2 c\right )^3 \log \left (a+b \sqrt{c+d x}\right )}{b^8 d^4}+\frac{2 \left (a^2-b^2 c\right )^3 \sqrt{c+d x}}{b^7 d^4}-\frac{a \left (a^2-3 b^2 c\right ) (c+d x)^2}{2 b^4 d^4}+\frac{2 \left (a^2-3 b^2 c\right ) (c+d x)^{5/2}}{5 b^3 d^4}-\frac{a x \left (a^4-3 a^2 b^2 c+3 b^4 c^2\right )}{b^6 d^3}+\frac{2 \left (a^4-3 a^2 b^2 c+3 b^4 c^2\right ) (c+d x)^{3/2}}{3 b^5 d^4}-\frac{a (c+d x)^3}{3 b^2 d^4}+\frac{2 (c+d x)^{7/2}}{7 b d^4} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a \left (c + d x\right )^{3}}{3 b^{2} d^{4}} - \frac{a \left (a^{2} - 3 b^{2} c\right ) \left (c + d x\right )^{2}}{2 b^{4} d^{4}} - \frac{2 a \left (a^{4} - 3 a^{2} b^{2} c + 3 b^{4} c^{2}\right ) \int ^{\sqrt{c + d x}} x\, dx}{b^{6} d^{4}} - \frac{2 a \left (a^{2} - b^{2} c\right )^{3} \log{\left (a + b \sqrt{c + d x} \right )}}{b^{8} d^{4}} + \frac{2 \left (a^{2} - b^{2} c\right )^{3} \int ^{\sqrt{c + d x}} \frac{1}{b^{7}}\, dx}{d^{4}} + \frac{2 \left (c + d x\right )^{\frac{7}{2}}}{7 b d^{4}} + \frac{2 \left (a^{2} - 3 b^{2} c\right ) \left (c + d x\right )^{\frac{5}{2}}}{5 b^{3} d^{4}} + \frac{2 \left (c + d x\right )^{\frac{3}{2}} \left (a^{4} - 3 a^{2} b^{2} c + 3 b^{4} c^{2}\right )}{3 b^{5} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.69608, size = 244, normalized size = 1.06 \[ \frac{-210 a \left (a^2-b^2 c\right )^3 \log \left (a^2-b^2 (c+d x)\right )-420 a \left (a^2-b^2 c\right )^3 \tanh ^{-1}\left (\frac{b \sqrt{c+d x}}{a}\right )+b \left (420 a^6 \sqrt{c+d x}-210 a^5 b d x-140 a^4 b^2 (8 c-d x) \sqrt{c+d x}-105 a^3 b^3 d x (d x-4 c)+84 a^2 b^4 \sqrt{c+d x} \left (11 c^2-3 c d x+d^2 x^2\right )-35 a b^5 d x \left (6 c^2-3 c d x+2 d^2 x^2\right )+12 b^6 \sqrt{c+d x} \left (-16 c^3+8 c^2 d x-6 c d^2 x^2+5 d^3 x^3\right )\right )}{210 b^8 d^4} \]

Antiderivative was successfully verified.

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

[Out]

(b*(-210*a^5*b*d*x - 105*a^3*b^3*d*x*(-4*c + d*x) + 420*a^6*Sqrt[c + d*x] - 140*
a^4*b^2*(8*c - d*x)*Sqrt[c + d*x] + 84*a^2*b^4*Sqrt[c + d*x]*(11*c^2 - 3*c*d*x +
d^2*x^2) - 35*a*b^5*d*x*(6*c^2 - 3*c*d*x + 2*d^2*x^2) + 12*b^6*Sqrt[c + d*x]*(-
16*c^3 + 8*c^2*d*x - 6*c*d^2*x^2 + 5*d^3*x^3)) - 420*a*(a^2 - b^2*c)^3*ArcTanh[(
b*Sqrt[c + d*x])/a] - 210*a*(a^2 - b^2*c)^3*Log[a^2 - b^2*(c + d*x)])/(210*b^8*d
^4)

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 394, normalized size = 1.7 \[ -{\frac{a{x}^{3}}{3\,{b}^{2}d}}+{\frac{2}{7\,b{d}^{4}} \left ( dx+c \right ) ^{{\frac{7}{2}}}}+2\,{\frac{{a}^{6}\sqrt{dx+c}}{{d}^{4}{b}^{7}}}+{\frac{5\,{c}^{2}{a}^{3}}{2\,{d}^{4}{b}^{4}}}-2\,{\frac{ \left ( dx+c \right ) ^{3/2}{a}^{2}c}{{d}^{4}{b}^{3}}}+6\,{\frac{{a}^{2}{c}^{2}\sqrt{dx+c}}{{d}^{4}{b}^{3}}}-6\,{\frac{{a}^{4}c\sqrt{dx+c}}{{d}^{4}{b}^{5}}}+2\,{\frac{{a}^{3}xc}{{b}^{4}{d}^{3}}}-{\frac{11\,a{c}^{3}}{6\,{d}^{4}{b}^{2}}}-{\frac{{a}^{5}c}{{d}^{4}{b}^{6}}}-{\frac{ax{c}^{2}}{{b}^{2}{d}^{3}}}-{\frac{6\,c}{5\,b{d}^{4}} \left ( dx+c \right ) ^{{\frac{5}{2}}}}+{\frac{2\,{a}^{2}}{5\,{d}^{4}{b}^{3}} \left ( dx+c \right ) ^{{\frac{5}{2}}}}+2\,{\frac{{c}^{2} \left ( dx+c \right ) ^{3/2}}{b{d}^{4}}}-2\,{\frac{{c}^{3}\sqrt{dx+c}}{b{d}^{4}}}+{\frac{2\,{a}^{4}}{3\,{d}^{4}{b}^{5}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{x{a}^{5}}{{d}^{3}{b}^{6}}}+{\frac{a{x}^{2}c}{2\,{b}^{2}{d}^{2}}}-{\frac{{x}^{2}{a}^{3}}{2\,{b}^{4}{d}^{2}}}+2\,{\frac{a\ln \left ( a+b\sqrt{dx+c} \right ){c}^{3}}{{d}^{4}{b}^{2}}}-6\,{\frac{{a}^{3}\ln \left ( a+b\sqrt{dx+c} \right ){c}^{2}}{{d}^{4}{b}^{4}}}+6\,{\frac{{a}^{5}\ln \left ( a+b\sqrt{dx+c} \right ) c}{{d}^{4}{b}^{6}}}-2\,{\frac{{a}^{7}\ln \left ( a+b\sqrt{dx+c} \right ) }{{d}^{4}{b}^{8}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(a+b*(d*x+c)^(1/2)),x)

[Out]

-1/3/d/b^2*x^3*a+2/7*(d*x+c)^(7/2)/b/d^4+2/d^4/b^7*a^6*(d*x+c)^(1/2)+5/2/d^4/b^4
*a^3*c^2-2/d^4/b^3*(d*x+c)^(3/2)*a^2*c+6/d^4/b^3*a^2*c^2*(d*x+c)^(1/2)-6/d^4/b^5
*a^4*c*(d*x+c)^(1/2)+2/d^3/b^4*x*a^3*c-11/6/d^4/b^2*a*c^3-1/d^4/b^6*c*a^5-1/d^3/
b^2*x*a*c^2-6/5/d^4/b*(d*x+c)^(5/2)*c+2/5/d^4/b^3*a^2*(d*x+c)^(5/2)+2/d^4/b*(d*x
+c)^(3/2)*c^2-2/d^4/b*c^3*(d*x+c)^(1/2)+2/3/d^4/b^5*a^4*(d*x+c)^(3/2)-1/d^3/b^6*
x*a^5+1/2/d^2/b^2*x^2*a*c-1/2/d^2/b^4*x^2*a^3+2/d^4*a/b^2*ln(a+b*(d*x+c)^(1/2))*
c^3-6/d^4*a^3/b^4*ln(a+b*(d*x+c)^(1/2))*c^2+6/d^4*a^5/b^6*ln(a+b*(d*x+c)^(1/2))*
c-2/d^4*a^7/b^8*ln(a+b*(d*x+c)^(1/2))

_______________________________________________________________________________________

Maxima [A]  time = 0.706678, size = 328, normalized size = 1.43 \[ \frac{\frac{60 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{6} - 70 \,{\left (d x + c\right )}^{3} a b^{5} - 84 \,{\left (3 \, b^{6} c - a^{2} b^{4}\right )}{\left (d x + c\right )}^{\frac{5}{2}} + 105 \,{\left (3 \, a b^{5} c - a^{3} b^{3}\right )}{\left (d x + c\right )}^{2} + 140 \,{\left (3 \, b^{6} c^{2} - 3 \, a^{2} b^{4} c + a^{4} b^{2}\right )}{\left (d x + c\right )}^{\frac{3}{2}} - 210 \,{\left (3 \, a b^{5} c^{2} - 3 \, a^{3} b^{3} c + a^{5} b\right )}{\left (d x + c\right )} - 420 \,{\left (b^{6} c^{3} - 3 \, a^{2} b^{4} c^{2} + 3 \, a^{4} b^{2} c - a^{6}\right )} \sqrt{d x + c}}{b^{7}} + \frac{420 \,{\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )} \log \left (\sqrt{d x + c} b + a\right )}{b^{8}}}{210 \, d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/210*((60*(d*x + c)^(7/2)*b^6 - 70*(d*x + c)^3*a*b^5 - 84*(3*b^6*c - a^2*b^4)*(
d*x + c)^(5/2) + 105*(3*a*b^5*c - a^3*b^3)*(d*x + c)^2 + 140*(3*b^6*c^2 - 3*a^2*
b^4*c + a^4*b^2)*(d*x + c)^(3/2) - 210*(3*a*b^5*c^2 - 3*a^3*b^3*c + a^5*b)*(d*x
+ c) - 420*(b^6*c^3 - 3*a^2*b^4*c^2 + 3*a^4*b^2*c - a^6)*sqrt(d*x + c))/b^7 + 42
0*(a*b^6*c^3 - 3*a^3*b^4*c^2 + 3*a^5*b^2*c - a^7)*log(sqrt(d*x + c)*b + a)/b^8)/
d^4

_______________________________________________________________________________________

Fricas [A]  time = 0.2837, size = 308, normalized size = 1.34 \[ -\frac{70 \, a b^{6} d^{3} x^{3} - 105 \,{\left (a b^{6} c - a^{3} b^{4}\right )} d^{2} x^{2} + 210 \,{\left (a b^{6} c^{2} - 2 \, a^{3} b^{4} c + a^{5} b^{2}\right )} d x - 420 \,{\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )} \log \left (\sqrt{d x + c} b + a\right ) - 4 \,{\left (15 \, b^{7} d^{3} x^{3} - 48 \, b^{7} c^{3} + 231 \, a^{2} b^{5} c^{2} - 280 \, a^{4} b^{3} c + 105 \, a^{6} b - 3 \,{\left (6 \, b^{7} c - 7 \, a^{2} b^{5}\right )} d^{2} x^{2} +{\left (24 \, b^{7} c^{2} - 63 \, a^{2} b^{5} c + 35 \, a^{4} b^{3}\right )} d x\right )} \sqrt{d x + c}}{210 \, b^{8} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/210*(70*a*b^6*d^3*x^3 - 105*(a*b^6*c - a^3*b^4)*d^2*x^2 + 210*(a*b^6*c^2 - 2*
a^3*b^4*c + a^5*b^2)*d*x - 420*(a*b^6*c^3 - 3*a^3*b^4*c^2 + 3*a^5*b^2*c - a^7)*l
og(sqrt(d*x + c)*b + a) - 4*(15*b^7*d^3*x^3 - 48*b^7*c^3 + 231*a^2*b^5*c^2 - 280
*a^4*b^3*c + 105*a^6*b - 3*(6*b^7*c - 7*a^2*b^5)*d^2*x^2 + (24*b^7*c^2 - 63*a^2*
b^5*c + 35*a^4*b^3)*d*x)*sqrt(d*x + c))/(b^8*d^4)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.306543, size = 533, normalized size = 2.32 \[ \frac{2 \,{\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )}{\rm ln}\left ({\left | \sqrt{d x + c} b + a \right |}\right )}{b^{8} d^{4}} - \frac{2 \,{\left (a b^{6} c^{3}{\rm ln}\left ({\left | a \right |}\right ) - 3 \, a^{3} b^{4} c^{2}{\rm ln}\left ({\left | a \right |}\right ) + 3 \, a^{5} b^{2} c{\rm ln}\left ({\left | a \right |}\right ) - a^{7}{\rm ln}\left ({\left | a \right |}\right )\right )}}{b^{8} d^{4}} + \frac{60 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{6} d^{24} - 252 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{6} c d^{24} + 420 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{6} c^{2} d^{24} - 420 \, \sqrt{d x + c} b^{6} c^{3} d^{24} - 70 \,{\left (d x + c\right )}^{3} a b^{5} d^{24} + 315 \,{\left (d x + c\right )}^{2} a b^{5} c d^{24} - 630 \,{\left (d x + c\right )} a b^{5} c^{2} d^{24} + 84 \,{\left (d x + c\right )}^{\frac{5}{2}} a^{2} b^{4} d^{24} - 420 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} b^{4} c d^{24} + 1260 \, \sqrt{d x + c} a^{2} b^{4} c^{2} d^{24} - 105 \,{\left (d x + c\right )}^{2} a^{3} b^{3} d^{24} + 630 \,{\left (d x + c\right )} a^{3} b^{3} c d^{24} + 140 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{4} b^{2} d^{24} - 1260 \, \sqrt{d x + c} a^{4} b^{2} c d^{24} - 210 \,{\left (d x + c\right )} a^{5} b d^{24} + 420 \, \sqrt{d x + c} a^{6} d^{24}}{210 \, b^{7} d^{28}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(a*b^6*c^3 - 3*a^3*b^4*c^2 + 3*a^5*b^2*c - a^7)*ln(abs(sqrt(d*x + c)*b + a))/(
b^8*d^4) - 2*(a*b^6*c^3*ln(abs(a)) - 3*a^3*b^4*c^2*ln(abs(a)) + 3*a^5*b^2*c*ln(a
bs(a)) - a^7*ln(abs(a)))/(b^8*d^4) + 1/210*(60*(d*x + c)^(7/2)*b^6*d^24 - 252*(d
*x + c)^(5/2)*b^6*c*d^24 + 420*(d*x + c)^(3/2)*b^6*c^2*d^24 - 420*sqrt(d*x + c)*
b^6*c^3*d^24 - 70*(d*x + c)^3*a*b^5*d^24 + 315*(d*x + c)^2*a*b^5*c*d^24 - 630*(d
*x + c)*a*b^5*c^2*d^24 + 84*(d*x + c)^(5/2)*a^2*b^4*d^24 - 420*(d*x + c)^(3/2)*a
^2*b^4*c*d^24 + 1260*sqrt(d*x + c)*a^2*b^4*c^2*d^24 - 105*(d*x + c)^2*a^3*b^3*d^
24 + 630*(d*x + c)*a^3*b^3*c*d^24 + 140*(d*x + c)^(3/2)*a^4*b^2*d^24 - 1260*sqrt
(d*x + c)*a^4*b^2*c*d^24 - 210*(d*x + c)*a^5*b*d^24 + 420*sqrt(d*x + c)*a^6*d^24
)/(b^7*d^28)