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

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

[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 = 0.279523, 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, Rules used = {371, 1398, 772} \[ \frac{x \left (-9 a^2 b^2 c+5 a^4+3 b^4 c^2\right )}{b^6 d^3}+\frac{2 a \left (a^2-b^2 c\right )^3}{b^8 d^4 \left (a+b \sqrt{c+d x}\right )}-\frac{12 a \left (a^2-b^2 c\right )^2 \sqrt{c+d x}}{b^7 d^4}+\frac{3 \left (a^2-b^2 c\right ) (c+d x)^2}{2 b^4 d^4}-\frac{4 a \left (2 a^2-3 b^2 c\right ) (c+d x)^{3/2}}{3 b^5 d^4}+\frac{2 \left (7 a^2-b^2 c\right ) \left (a^2-b^2 c\right )^2 \log \left (a+b \sqrt{c+d x}\right )}{b^8 d^4}-\frac{4 a (c+d x)^{5/2}}{5 b^3 d^4}+\frac{(c+d x)^3}{3 b^2 d^4} \]

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)

Rule 371

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 1398

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]

Rule 772

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]

Rubi steps

\begin{align*} \int \frac{x^3}{\left (a+b \sqrt{c+d x}\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(-c+x)^3}{\left (a+b \sqrt{x}\right )^2} \, dx,x,c+d x\right )}{d^4}\\ &=\frac{2 \operatorname{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 \operatorname{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*}

Mathematica [A]  time = 0.281365, size = 273, normalized size = 1.14 \[ \frac{a^3 b^4 \left (856 c^2+380 c d x-35 d^2 x^2\right )-3 a^2 b^5 \sqrt{c+d x} \left (76 c^2+36 c d x-7 d^2 x^2\right )-6 a^5 b^2 (102 c+35 d x)+2 a^4 b^3 \sqrt{c+d x} (284 c+35 d x)+60 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right ) \log \left (a+b \sqrt{c+d x}\right )-324 a^6 b \sqrt{c+d x}+96 a^7-a b^6 \left (162 c^2 d x+324 c^3-33 c d^2 x^2+14 d^3 x^3\right )+5 b^7 d x \sqrt{c+d x} \left (6 c^2-3 c d x+2 d^2 x^2\right )}{30 b^8 d^4 \left (a+b \sqrt{c+d x}\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(96*a^7 - 324*a^6*b*Sqrt[c + d*x] - 6*a^5*b^2*(102*c + 35*d*x) + 2*a^4*b^3*Sqrt[c + d*x]*(284*c + 35*d*x) + a^
3*b^4*(856*c^2 + 380*c*d*x - 35*d^2*x^2) - 3*a^2*b^5*Sqrt[c + d*x]*(76*c^2 + 36*c*d*x - 7*d^2*x^2) + 5*b^7*d*x
*Sqrt[c + d*x]*(6*c^2 - 3*c*d*x + 2*d^2*x^2) - a*b^6*(324*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 = 0.008, size = 416, normalized size = 1.7 \begin{align*}{\frac{{x}^{3}}{3\,{b}^{2}d}}-{\frac{c{x}^{2}}{2\,{b}^{2}{d}^{2}}}+{\frac{{c}^{2}x}{{d}^{3}{b}^{2}}}+{\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{x{a}^{2}c}{{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{{c}^{2}a\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{a{c}^{3}}{{d}^{4}{b}^{2} \left ( a+b\sqrt{dx+c} \right ) }}+6\,{\frac{{a}^{3}{c}^{2}}{{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 ) }}-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}}} \end{align*}

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/b^2*c^3-4/5*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*(d*x+c)^(3/2)*a^3-12/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*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))-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

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Maxima [A]  time = 1.06348, size = 339, normalized size = 1.41 \begin{align*} -\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{align*}

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 = 1.61897, size = 838, normalized size = 3.49 \begin{align*} \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{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\left (a + b \sqrt{c + d x}\right )^{2}}\, dx \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}

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]

undef