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

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

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Rubi [A]  time = 0.26, antiderivative size = 230, 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 = {371, 1398, 772} \[ \frac {2 \left (-3 a^2 b^2 c+a^4+3 b^4 c^2\right ) (c+d x)^{3/2}}{3 b^5 d^4}-\frac {a x \left (-3 a^2 b^2 c+a^4+3 b^4 c^2\right )}{b^6 d^3}+\frac {2 \left (a^2-3 b^2 c\right ) (c+d x)^{5/2}}{5 b^3 d^4}-\frac {a \left (a^2-3 b^2 c\right ) (c+d x)^2}{2 b^4 d^4}+\frac {2 \left (a^2-b^2 c\right )^3 \sqrt {c+d x}}{b^7 d^4}-\frac {2 a \left (a^2-b^2 c\right )^3 \log \left (a+b \sqrt {c+d x}\right )}{b^8 d^4}-\frac {a (c+d x)^3}{3 b^2 d^4}+\frac {2 (c+d x)^{7/2}}{7 b d^4} \]

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)

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 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]

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]

Rubi steps

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

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Mathematica [A]  time = 0.20, size = 213, normalized size = 0.93 \[ \frac {b \left (420 a^6 \sqrt {c+d x}-210 a^5 b d x-140 a^4 b^2 (8 c-d x) \sqrt {c+d x}-105 a^3 b^3 d x (d x-4 c)+84 a^2 b^4 \sqrt {c+d x} \left (11 c^2-3 c d x+d^2 x^2\right )-35 a b^5 d x \left (6 c^2-3 c d x+2 d^2 x^2\right )+12 b^6 \sqrt {c+d x} \left (-16 c^3+8 c^2 d x-6 c d^2 x^2+5 d^3 x^3\right )\right )-420 a \left (a^2-b^2 c\right )^3 \log \left (a+b \sqrt {c+d x}\right )}{210 b^8 d^4} \]

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*Log[a + b*Sqrt[c +
d*x]])/(210*b^8*d^4)

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fricas [A]  time = 0.45, size = 228, normalized size = 0.99 \[ -\frac {70 \, a b^{6} d^{3} x^{3} - 105 \, {\left (a b^{6} c - a^{3} b^{4}\right )} d^{2} x^{2} + 210 \, {\left (a b^{6} c^{2} - 2 \, a^{3} b^{4} c + a^{5} b^{2}\right )} d x - 420 \, {\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )} \log \left (\sqrt {d x + c} b + a\right ) - 4 \, {\left (15 \, b^{7} d^{3} x^{3} - 48 \, b^{7} c^{3} + 231 \, a^{2} b^{5} c^{2} - 280 \, a^{4} b^{3} c + 105 \, a^{6} b - 3 \, {\left (6 \, b^{7} c - 7 \, a^{2} b^{5}\right )} d^{2} x^{2} + {\left (24 \, b^{7} c^{2} - 63 \, a^{2} b^{5} c + 35 \, a^{4} b^{3}\right )} d x\right )} \sqrt {d x + c}}{210 \, b^{8} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b*(d*x+c)^(1/2)),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)*log(sqrt(d*x + c)*b + a) - 4*(15*b^7*d^3*x^3 - 48*b^7*c^3 + 2
31*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)

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giac [A]  time = 0.37, size = 341, normalized size = 1.48 \[ \frac {2 \, {\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )} \log \left ({\left | \sqrt {d x + c} b + a \right |}\right )}{b^{8} d^{4}} + \frac {60 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{6} d^{24} - 252 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{6} c d^{24} + 420 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{6} c^{2} d^{24} - 420 \, \sqrt {d x + c} b^{6} c^{3} d^{24} - 70 \, {\left (d x + c\right )}^{3} a b^{5} d^{24} + 315 \, {\left (d x + c\right )}^{2} a b^{5} c d^{24} - 630 \, {\left (d x + c\right )} a b^{5} c^{2} d^{24} + 84 \, {\left (d x + c\right )}^{\frac {5}{2}} a^{2} b^{4} d^{24} - 420 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b^{4} c d^{24} + 1260 \, \sqrt {d x + c} a^{2} b^{4} c^{2} d^{24} - 105 \, {\left (d x + c\right )}^{2} a^{3} b^{3} d^{24} + 630 \, {\left (d x + c\right )} a^{3} b^{3} c d^{24} + 140 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{4} b^{2} d^{24} - 1260 \, \sqrt {d x + c} a^{4} b^{2} c d^{24} - 210 \, {\left (d x + c\right )} a^{5} b d^{24} + 420 \, \sqrt {d x + c} a^{6} d^{24}}{210 \, b^{7} d^{28}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b*(d*x+c)^(1/2)),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)*log(abs(sqrt(d*x + c)*b + 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)

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maple [A]  time = 0.01, size = 394, normalized size = 1.71 \[ -\frac {a \,x^{3}}{3 b^{2} d}+\frac {a c \,x^{2}}{2 b^{2} d^{2}}-\frac {a^{3} x^{2}}{2 b^{4} d^{2}}+\frac {2 a \,c^{3} \ln \left (a +\sqrt {d x +c}\, b \right )}{b^{2} d^{4}}-\frac {a \,c^{2} x}{b^{2} d^{3}}-\frac {6 a^{3} c^{2} \ln \left (a +\sqrt {d x +c}\, b \right )}{b^{4} d^{4}}+\frac {2 a^{3} c x}{b^{4} d^{3}}-\frac {11 a \,c^{3}}{6 b^{2} d^{4}}-\frac {2 \sqrt {d x +c}\, c^{3}}{b \,d^{4}}+\frac {6 a^{5} c \ln \left (a +\sqrt {d x +c}\, b \right )}{b^{6} d^{4}}-\frac {a^{5} x}{b^{6} d^{3}}+\frac {5 a^{3} c^{2}}{2 b^{4} d^{4}}+\frac {6 \sqrt {d x +c}\, a^{2} c^{2}}{b^{3} d^{4}}+\frac {2 \left (d x +c \right )^{\frac {3}{2}} c^{2}}{b \,d^{4}}-\frac {2 a^{7} \ln \left (a +\sqrt {d x +c}\, b \right )}{b^{8} d^{4}}-\frac {a^{5} c}{b^{6} d^{4}}-\frac {6 \sqrt {d x +c}\, a^{4} c}{b^{5} d^{4}}-\frac {2 \left (d x +c \right )^{\frac {3}{2}} a^{2} c}{b^{3} d^{4}}-\frac {6 \left (d x +c \right )^{\frac {5}{2}} c}{5 b \,d^{4}}+\frac {2 \sqrt {d x +c}\, a^{6}}{b^{7} d^{4}}+\frac {2 \left (d x +c \right )^{\frac {3}{2}} a^{4}}{3 b^{5} d^{4}}+\frac {2 \left (d x +c \right )^{\frac {5}{2}} a^{2}}{5 b^{3} d^{4}}+\frac {2 \left (d x +c \right )^{\frac {7}{2}}}{7 b \,d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.90, size = 243, normalized size = 1.06 \[ \frac {\frac {60 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{6} - 70 \, {\left (d x + c\right )}^{3} a b^{5} - 84 \, {\left (3 \, b^{6} c - a^{2} b^{4}\right )} {\left (d x + c\right )}^{\frac {5}{2}} + 105 \, {\left (3 \, a b^{5} c - a^{3} b^{3}\right )} {\left (d x + c\right )}^{2} + 140 \, {\left (3 \, b^{6} c^{2} - 3 \, a^{2} b^{4} c + a^{4} b^{2}\right )} {\left (d x + c\right )}^{\frac {3}{2}} - 210 \, {\left (3 \, a b^{5} c^{2} - 3 \, a^{3} b^{3} c + a^{5} b\right )} {\left (d x + c\right )} - 420 \, {\left (b^{6} c^{3} - 3 \, a^{2} b^{4} c^{2} + 3 \, a^{4} b^{2} c - a^{6}\right )} \sqrt {d x + c}}{b^{7}} + \frac {420 \, {\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )} \log \left (\sqrt {d x + c} b + a\right )}{b^{8}}}{210 \, d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b*(d*x+c)^(1/2)),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 + 420*(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

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mupad [B]  time = 0.07, size = 317, normalized size = 1.38 \[ \frac {2\,{\left (c+d\,x\right )}^{7/2}}{7\,b\,d^4}-\left (\frac {a^2\,\left (\frac {a^2\,\left (\frac {6\,c}{b\,d^4}-\frac {2\,a^2}{b^3\,d^4}\right )}{b^2}-\frac {6\,c^2}{b\,d^4}\right )}{b^2}+\frac {2\,c^3}{b\,d^4}\right )\,\sqrt {c+d\,x}-\left (\frac {a^2\,\left (\frac {6\,c}{b\,d^4}-\frac {2\,a^2}{b^3\,d^4}\right )}{3\,b^2}-\frac {2\,c^2}{b\,d^4}\right )\,{\left (c+d\,x\right )}^{3/2}-\left (\frac {6\,c}{5\,b\,d^4}-\frac {2\,a^2}{5\,b^3\,d^4}\right )\,{\left (c+d\,x\right )}^{5/2}+\frac {a\,\left (\frac {6\,c}{b\,d^4}-\frac {2\,a^2}{b^3\,d^4}\right )\,{\left (c+d\,x\right )}^2}{4\,b}-\frac {a\,{\left (c+d\,x\right )}^3}{3\,b^2\,d^4}-\frac {\ln \left (a+b\,\sqrt {c+d\,x}\right )\,\left (2\,a^7-6\,a^5\,b^2\,c+6\,a^3\,b^4\,c^2-2\,a\,b^6\,c^3\right )}{b^8\,d^4}+\frac {a\,d\,x\,\left (\frac {a^2\,\left (\frac {6\,c}{b\,d^4}-\frac {2\,a^2}{b^3\,d^4}\right )}{b^2}-\frac {6\,c^2}{b\,d^4}\right )}{2\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + b*(c + d*x)^(1/2)),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{a + b \sqrt {c + d x}}\, dx \]

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)

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