3.7.39 \(\int \frac {x^3}{(a+b \sqrt {c+d x})^2} \, dx\) [639]

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

[Out]

(3*b^4*c^2-9*a^2*b^2*c+5*a^4)*x/b^6/d^3-4/3*a*(-3*b^2*c+2*a^2)*(d*x+c)^(3/2)/b^5/d^4+3/2*(-b^2*c+a^2)*(d*x+c)^
2/b^4/d^4-4/5*a*(d*x+c)^(5/2)/b^3/d^4+1/3*(d*x+c)^3/b^2/d^4+2*(-b^2*c+a^2)^2*(-b^2*c+7*a^2)*ln(a+b*(d*x+c)^(1/
2))/b^8/d^4-12*a*(-b^2*c+a^2)^2*(d*x+c)^(1/2)/b^7/d^4+2*a*(-b^2*c+a^2)^3/b^8/d^4/(a+b*(d*x+c)^(1/2))

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Rubi [A]
time = 0.20, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {378, 1412, 786} \begin {gather*} \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} \end {gather*}

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)

Rule 378

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coefficient[v, x, 0], d = Coefficient[v,
x, 1]}, Dist[1/d^(m + 1), Subst[Int[SimplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]
] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]

Rule 786

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rule 1412

Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, D
ist[g, Subst[Int[x^(g - 1)*(d + e*x^(g*n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, p
, q}, x] && EqQ[n2, 2*n] && FractionQ[n]

Rubi steps

\begin {align*} \int \frac {x^3}{\left (a+b \sqrt {c+d x}\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {(-c+x)^3}{\left (a+b \sqrt {x}\right )^2} \, dx,x,c+d x\right )}{d^4}\\ &=\frac {2 \text {Subst}\left (\int \frac {x \left (-c+x^2\right )^3}{(a+b x)^2} \, dx,x,\sqrt {c+d x}\right )}{d^4}\\ &=\frac {2 \text {Subst}\left (\int \left (-\frac {6 a \left (a^2-b^2 c\right )^2}{b^7}+\frac {\left (5 a^4-9 a^2 b^2 c+3 b^4 c^2\right ) x}{b^6}-\frac {2 a \left (2 a^2-3 b^2 c\right ) x^2}{b^5}-\frac {3 \left (-a^2+b^2 c\right ) x^3}{b^4}-\frac {2 a x^4}{b^3}+\frac {x^5}{b^2}-\frac {a \left (a^2-b^2 c\right )^3}{b^7 (a+b x)^2}-\frac {\left (-7 a^2+b^2 c\right ) \left (-a^2+b^2 c\right )^2}{b^7 (a+b x)}\right ) \, dx,x,\sqrt {c+d x}\right )}{d^4}\\ &=\frac {\left (5 a^4-9 a^2 b^2 c+3 b^4 c^2\right ) x}{b^6 d^3}-\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 {4 a (c+d x)^{5/2}}{5 b^3 d^4}+\frac {(c+d x)^3}{3 b^2 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 {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}\\ \end {align*}

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Mathematica [A]
time = 0.20, size = 283, normalized size = 1.18 \begin {gather*} \frac {60 a^7-360 a^6 b \sqrt {c+d x}-30 a^5 b^2 (13 c+7 d x)+10 a^4 b^3 \sqrt {c+d x} (79 c+7 d x)-3 a^2 b^5 \sqrt {c+d x} \left (163 c^2+36 c d x-7 d^2 x^2\right )+5 a^3 b^4 \left (119 c^2+76 c d x-7 d^2 x^2\right )+5 b^7 \sqrt {c+d x} \left (11 c^3+6 c^2 d x-3 c d^2 x^2+2 d^3 x^3\right )-a b^6 \left (269 c^3+162 c^2 d x-33 c d^2 x^2+14 d^3 x^3\right )+60 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right ) \log \left (a+b \sqrt {c+d x}\right )}{30 b^8 d^4 \left (a+b \sqrt {c+d x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(60*a^7 - 360*a^6*b*Sqrt[c + d*x] - 30*a^5*b^2*(13*c + 7*d*x) + 10*a^4*b^3*Sqrt[c + d*x]*(79*c + 7*d*x) - 3*a^
2*b^5*Sqrt[c + d*x]*(163*c^2 + 36*c*d*x - 7*d^2*x^2) + 5*a^3*b^4*(119*c^2 + 76*c*d*x - 7*d^2*x^2) + 5*b^7*Sqrt
[c + d*x]*(11*c^3 + 6*c^2*d*x - 3*c*d^2*x^2 + 2*d^3*x^3) - a*b^6*(269*c^3 + 162*c^2*d*x - 33*c*d^2*x^2 + 14*d^
3*x^3) + 60*(a^2 - b^2*c)^2*(7*a^2 - b^2*c)*(a + b*Sqrt[c + d*x])*Log[a + b*Sqrt[c + d*x]])/(30*b^8*d^4*(a + b
*Sqrt[c + d*x]))

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Maple [A]
time = 0.04, size = 277, normalized size = 1.15

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (-\frac {\left (d x +c \right )^{3} b^{5}}{6}+\frac {2 a \left (d x +c \right )^{\frac {5}{2}} b^{4}}{5}+\frac {3 b^{5} c \left (d x +c \right )^{2}}{4}-\frac {3 a^{2} b^{3} \left (d x +c \right )^{2}}{4}-2 a \,b^{4} c \left (d x +c \right )^{\frac {3}{2}}-\frac {3 b^{5} c^{2} \left (d x +c \right )}{2}+\frac {4 a^{3} b^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {9 a^{2} b^{3} c \left (d x +c \right )}{2}+6 a \,c^{2} b^{4} \sqrt {d x +c}-\frac {5 a^{4} b \left (d x +c \right )}{2}-12 a^{3} c \,b^{2} \sqrt {d x +c}+6 a^{5} \sqrt {d x +c}\right )}{b^{7}}+\frac {2 a \left (-b^{6} c^{3}+3 a^{2} b^{4} c^{2}-3 a^{4} b^{2} c +a^{6}\right )}{b^{8} \left (a +b \sqrt {d x +c}\right )}+\frac {2 \left (-b^{6} c^{3}+9 a^{2} b^{4} c^{2}-15 a^{4} b^{2} c +7 a^{6}\right ) \ln \left (a +b \sqrt {d x +c}\right )}{b^{8}}}{d^{4}}\)	\(277\)
default	\(\frac {-\frac {2 \left (-\frac {\left (d x +c \right )^{3} b^{5}}{6}+\frac {2 a \left (d x +c \right )^{\frac {5}{2}} b^{4}}{5}+\frac {3 b^{5} c \left (d x +c \right )^{2}}{4}-\frac {3 a^{2} b^{3} \left (d x +c \right )^{2}}{4}-2 a \,b^{4} c \left (d x +c \right )^{\frac {3}{2}}-\frac {3 b^{5} c^{2} \left (d x +c \right )}{2}+\frac {4 a^{3} b^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {9 a^{2} b^{3} c \left (d x +c \right )}{2}+6 a \,c^{2} b^{4} \sqrt {d x +c}-\frac {5 a^{4} b \left (d x +c \right )}{2}-12 a^{3} c \,b^{2} \sqrt {d x +c}+6 a^{5} \sqrt {d x +c}\right )}{b^{7}}+\frac {2 a \left (-b^{6} c^{3}+3 a^{2} b^{4} c^{2}-3 a^{4} b^{2} c +a^{6}\right )}{b^{8} \left (a +b \sqrt {d x +c}\right )}+\frac {2 \left (-b^{6} c^{3}+9 a^{2} b^{4} c^{2}-15 a^{4} b^{2} c +7 a^{6}\right ) \ln \left (a +b \sqrt {d x +c}\right )}{b^{8}}}{d^{4}}\)	\(277\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a+b*(d*x+c)^(1/2))^2,x,method=_RETURNVERBOSE)

[Out]

2/d^4*(-1/b^7*(-1/6*(d*x+c)^3*b^5+2/5*a*(d*x+c)^(5/2)*b^4+3/4*b^5*c*(d*x+c)^2-3/4*a^2*b^3*(d*x+c)^2-2*a*b^4*c*
(d*x+c)^(3/2)-3/2*b^5*c^2*(d*x+c)+4/3*a^3*b^2*(d*x+c)^(3/2)+9/2*a^2*b^3*c*(d*x+c)+6*a*c^2*b^4*(d*x+c)^(1/2)-5/
2*a^4*b*(d*x+c)-12*a^3*c*b^2*(d*x+c)^(1/2)+6*a^5*(d*x+c)^(1/2))+a*(-b^6*c^3+3*a^2*b^4*c^2-3*a^4*b^2*c+a^6)/b^8
/(a+b*(d*x+c)^(1/2))+1/b^8*(-b^6*c^3+9*a^2*b^4*c^2-15*a^4*b^2*c+7*a^6)*ln(a+b*(d*x+c)^(1/2)))

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Maxima [A]
time = 0.28, size = 251, normalized size = 1.05 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b*(d*x+c)^(1/2))^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 = 0.36, size = 392, normalized size = 1.63 \begin {gather*} \frac {10 \, b^{8} d^{4} x^{4} + 55 \, b^{8} c^{4} - 220 \, a^{2} b^{6} c^{3} + 195 \, a^{4} b^{4} c^{2} + 30 \, a^{6} b^{2} c - 60 \, a^{8} - 5 \, {\left (b^{8} c - 7 \, a^{2} b^{6}\right )} d^{3} x^{3} + 15 \, {\left (b^{8} c^{2} - 8 \, a^{2} b^{6} c + 7 \, a^{4} b^{4}\right )} d^{2} x^{2} + 5 \, {\left (17 \, b^{8} c^{3} - 87 \, a^{2} b^{6} c^{2} + 96 \, a^{4} b^{4} c - 30 \, a^{6} b^{2}\right )} d x - 60 \, {\left (b^{8} c^{4} - 10 \, a^{2} b^{6} c^{3} + 24 \, a^{4} b^{4} c^{2} - 22 \, a^{6} b^{2} c + 7 \, a^{8} + {\left (b^{8} c^{3} - 9 \, a^{2} b^{6} c^{2} + 15 \, a^{4} b^{4} c - 7 \, a^{6} b^{2}\right )} d x\right )} \log \left (\sqrt {d x + c} b + a\right ) - 4 \, {\left (6 \, a b^{7} d^{3} x^{3} + 81 \, a b^{7} c^{3} - 271 \, a^{3} b^{5} c^{2} + 295 \, a^{5} b^{3} c - 105 \, a^{7} b - 2 \, {\left (6 \, a b^{7} c - 7 \, a^{3} b^{5}\right )} d^{2} x^{2} + 2 \, {\left (24 \, a b^{7} c^{2} - 61 \, a^{3} b^{5} c + 35 \, a^{5} b^{3}\right )} d x\right )} \sqrt {d x + c}}{30 \, {\left (b^{10} d^{5} x + {\left (b^{10} c - a^{2} b^{8}\right )} d^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b*(d*x+c)^(1/2))^2,x, algorithm="fricas")

[Out]

1/30*(10*b^8*d^4*x^4 + 55*b^8*c^4 - 220*a^2*b^6*c^3 + 195*a^4*b^4*c^2 + 30*a^6*b^2*c - 60*a^8 - 5*(b^8*c - 7*a
^2*b^6)*d^3*x^3 + 15*(b^8*c^2 - 8*a^2*b^6*c + 7*a^4*b^4)*d^2*x^2 + 5*(17*b^8*c^3 - 87*a^2*b^6*c^2 + 96*a^4*b^4
*c - 30*a^6*b^2)*d*x - 60*(b^8*c^4 - 10*a^2*b^6*c^3 + 24*a^4*b^4*c^2 - 22*a^6*b^2*c + 7*a^8 + (b^8*c^3 - 9*a^2
*b^6*c^2 + 15*a^4*b^4*c - 7*a^6*b^2)*d*x)*log(sqrt(d*x + c)*b + a) - 4*(6*a*b^7*d^3*x^3 + 81*a*b^7*c^3 - 271*a
^3*b^5*c^2 + 295*a^5*b^3*c - 105*a^7*b - 2*(6*a*b^7*c - 7*a^3*b^5)*d^2*x^2 + 2*(24*a*b^7*c^2 - 61*a^3*b^5*c +
35*a^5*b^3)*d*x)*sqrt(d*x + c))/(b^10*d^5*x + (b^10*c - a^2*b^8)*d^4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\left (a + b \sqrt {c + d x}\right )^{2}}\, dx \end {gather*}

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 [A]
time = 3.73, size = 324, normalized size = 1.35 \begin {gather*} -\frac {2 \, {\left (b^{6} c^{3} - 9 \, a^{2} b^{4} c^{2} + 15 \, a^{4} b^{2} c - 7 \, a^{6}\right )} \log \left ({\left | \sqrt {d x + c} b + a \right |}\right )}{b^{8} d^{4}} - \frac {2 \, {\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )}}{{\left (\sqrt {d x + c} b + a\right )} b^{8} d^{4}} + \frac {10 \, {\left (d x + c\right )}^{3} b^{10} d^{20} - 45 \, {\left (d x + c\right )}^{2} b^{10} c d^{20} + 90 \, {\left (d x + c\right )} b^{10} c^{2} d^{20} - 24 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{9} d^{20} + 120 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{9} c d^{20} - 360 \, \sqrt {d x + c} a b^{9} c^{2} d^{20} + 45 \, {\left (d x + c\right )}^{2} a^{2} b^{8} d^{20} - 270 \, {\left (d x + c\right )} a^{2} b^{8} c d^{20} - 80 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{3} b^{7} d^{20} + 720 \, \sqrt {d x + c} a^{3} b^{7} c d^{20} + 150 \, {\left (d x + c\right )} a^{4} b^{6} d^{20} - 360 \, \sqrt {d x + c} a^{5} b^{5} d^{20}}{30 \, b^{12} d^{24}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b*(d*x+c)^(1/2))^2,x, algorithm="giac")

[Out]

-2*(b^6*c^3 - 9*a^2*b^4*c^2 + 15*a^4*b^2*c - 7*a^6)*log(abs(sqrt(d*x + c)*b + a))/(b^8*d^4) - 2*(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 + a)*b^8*d^4) + 1/30*(10*(d*x + c)^3*b^10*d^20 - 45*(d*x +
c)^2*b^10*c*d^20 + 90*(d*x + c)*b^10*c^2*d^20 - 24*(d*x + c)^(5/2)*a*b^9*d^20 + 120*(d*x + c)^(3/2)*a*b^9*c*d
^20 - 360*sqrt(d*x + c)*a*b^9*c^2*d^20 + 45*(d*x + c)^2*a^2*b^8*d^20 - 270*(d*x + c)*a^2*b^8*c*d^20 - 80*(d*x
+ c)^(3/2)*a^3*b^7*d^20 + 720*sqrt(d*x + c)*a^3*b^7*c*d^20 + 150*(d*x + c)*a^4*b^6*d^20 - 360*sqrt(d*x + c)*a^
5*b^5*d^20)/(b^12*d^24)

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Mupad [B]
time = 0.09, size = 461, normalized size = 1.92 \begin {gather*} \left (\frac {4\,a^3}{3\,b^5\,d^4}+\frac {2\,a\,\left (\frac {6\,c}{b^2\,d^4}-\frac {6\,a^2}{b^4\,d^4}\right )}{3\,b}\right )\,{\left (c+d\,x\right )}^{3/2}-\left (\frac {3\,c}{2\,b^2\,d^4}-\frac {3\,a^2}{2\,b^4\,d^4}\right )\,{\left (c+d\,x\right )}^2-\left (\frac {2\,a\,\left (\frac {a^2\,\left (\frac {6\,c}{b^2\,d^4}-\frac {6\,a^2}{b^4\,d^4}\right )}{b^2}-\frac {2\,a\,\left (\frac {4\,a^3}{b^5\,d^4}+\frac {2\,a\,\left (\frac {6\,c}{b^2\,d^4}-\frac {6\,a^2}{b^4\,d^4}\right )}{b}\right )}{b}+\frac {6\,c^2}{b^2\,d^4}\right )}{b}+\frac {a^2\,\left (\frac {4\,a^3}{b^5\,d^4}+\frac {2\,a\,\left (\frac {6\,c}{b^2\,d^4}-\frac {6\,a^2}{b^4\,d^4}\right )}{b}\right )}{b^2}\right )\,\sqrt {c+d\,x}+\frac {{\left (c+d\,x\right )}^3}{3\,b^2\,d^4}+\frac {2\,\left (a^7-3\,a^5\,b^2\,c+3\,a^3\,b^4\,c^2-a\,b^6\,c^3\right )}{b\,\left (b^8\,d^4\,\sqrt {c+d\,x}+a\,b^7\,d^4\right )}+d\,x\,\left (\frac {a^2\,\left (\frac {6\,c}{b^2\,d^4}-\frac {6\,a^2}{b^4\,d^4}\right )}{2\,b^2}-\frac {a\,\left (\frac {4\,a^3}{b^5\,d^4}+\frac {2\,a\,\left (\frac {6\,c}{b^2\,d^4}-\frac {6\,a^2}{b^4\,d^4}\right )}{b}\right )}{b}+\frac {3\,c^2}{b^2\,d^4}\right )+\frac {\ln \left (a+b\,\sqrt {c+d\,x}\right )\,\left (14\,a^6-30\,a^4\,b^2\,c+18\,a^2\,b^4\,c^2-2\,b^6\,c^3\right )}{b^8\,d^4}-\frac {4\,a\,{\left (c+d\,x\right )}^{5/2}}{5\,b^3\,d^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + b*(c + d*x)^(1/2))^2,x)

[Out]

((4*a^3)/(3*b^5*d^4) + (2*a*((6*c)/(b^2*d^4) - (6*a^2)/(b^4*d^4)))/(3*b))*(c + d*x)^(3/2) - ((3*c)/(2*b^2*d^4)
- (3*a^2)/(2*b^4*d^4))*(c + d*x)^2 - ((2*a*((a^2*((6*c)/(b^2*d^4) - (6*a^2)/(b^4*d^4)))/b^2 - (2*a*((4*a^3)/(
b^5*d^4) + (2*a*((6*c)/(b^2*d^4) - (6*a^2)/(b^4*d^4)))/b))/b + (6*c^2)/(b^2*d^4)))/b + (a^2*((4*a^3)/(b^5*d^4)
+ (2*a*((6*c)/(b^2*d^4) - (6*a^2)/(b^4*d^4)))/b))/b^2)*(c + d*x)^(1/2) + (c + d*x)^3/(3*b^2*d^4) + (2*(a^7 -
3*a^5*b^2*c - a*b^6*c^3 + 3*a^3*b^4*c^2))/(b*(b^8*d^4*(c + d*x)^(1/2) + a*b^7*d^4)) + d*x*((a^2*((6*c)/(b^2*d^
4) - (6*a^2)/(b^4*d^4)))/(2*b^2) - (a*((4*a^3)/(b^5*d^4) + (2*a*((6*c)/(b^2*d^4) - (6*a^2)/(b^4*d^4)))/b))/b +
(3*c^2)/(b^2*d^4)) + (log(a + b*(c + d*x)^(1/2))*(14*a^6 - 2*b^6*c^3 - 30*a^4*b^2*c + 18*a^2*b^4*c^2))/(b^8*d
^4) - (4*a*(c + d*x)^(5/2))/(5*b^3*d^4)

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