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

Optimal. Leaf size=326 \[ \frac {12 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{13/2}}{13 b^8 d^4}-\frac {20 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{11/2}}{11 b^8 d^4}-\frac {12 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{7/2}}{7 b^8 d^4}+\frac {4 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^8 d^4}-\frac {4 a \left (a^2-b^2 c\right )^3 \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^8 d^4}+\frac {4 \left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^8 d^4}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{17/2}}{17 b^8 d^4}-\frac {28 a \left (a+b \sqrt {c+d x}\right )^{15/2}}{15 b^8 d^4} \]

[Out]

-4/3*a*(-b^2*c+a^2)^3*(a+b*(d*x+c)^(1/2))^(3/2)/b^8/d^4+4/5*(-b^2*c+a^2)^2*(-b^2*c+7*a^2)*(a+b*(d*x+c)^(1/2))^
(5/2)/b^8/d^4-12/7*a*(-3*b^2*c+7*a^2)*(-b^2*c+a^2)*(a+b*(d*x+c)^(1/2))^(7/2)/b^8/d^4+4/9*(3*b^4*c^2-30*a^2*b^2
*c+35*a^4)*(a+b*(d*x+c)^(1/2))^(9/2)/b^8/d^4-20/11*a*(-3*b^2*c+7*a^2)*(a+b*(d*x+c)^(1/2))^(11/2)/b^8/d^4+12/13
*(-b^2*c+7*a^2)*(a+b*(d*x+c)^(1/2))^(13/2)/b^8/d^4-28/15*a*(a+b*(d*x+c)^(1/2))^(15/2)/b^8/d^4+4/17*(a+b*(d*x+c
)^(1/2))^(17/2)/b^8/d^4

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Rubi [A]  time = 0.24, antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {371, 1398, 772} \[ \frac {4 \left (-30 a^2 b^2 c+35 a^4+3 b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^8 d^4}+\frac {12 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{13/2}}{13 b^8 d^4}-\frac {20 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{11/2}}{11 b^8 d^4}-\frac {12 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{7/2}}{7 b^8 d^4}+\frac {4 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^8 d^4}-\frac {4 a \left (a^2-b^2 c\right )^3 \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^8 d^4}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{17/2}}{17 b^8 d^4}-\frac {28 a \left (a+b \sqrt {c+d x}\right )^{15/2}}{15 b^8 d^4} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Sqrt[a + b*Sqrt[c + d*x]],x]

[Out]

(-4*a*(a^2 - b^2*c)^3*(a + b*Sqrt[c + d*x])^(3/2))/(3*b^8*d^4) + (4*(a^2 - b^2*c)^2*(7*a^2 - b^2*c)*(a + b*Sqr
t[c + d*x])^(5/2))/(5*b^8*d^4) - (12*a*(7*a^2 - 3*b^2*c)*(a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(7/2))/(7*b^8*d^4
) + (4*(35*a^4 - 30*a^2*b^2*c + 3*b^4*c^2)*(a + b*Sqrt[c + d*x])^(9/2))/(9*b^8*d^4) - (20*a*(7*a^2 - 3*b^2*c)*
(a + b*Sqrt[c + d*x])^(11/2))/(11*b^8*d^4) + (12*(7*a^2 - b^2*c)*(a + b*Sqrt[c + d*x])^(13/2))/(13*b^8*d^4) -
(28*a*(a + b*Sqrt[c + d*x])^(15/2))/(15*b^8*d^4) + (4*(a + b*Sqrt[c + d*x])^(17/2))/(17*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 x^3 \sqrt {a+b \sqrt {c+d x}} \, dx &=\frac {\operatorname {Subst}\left (\int \sqrt {a+b \sqrt {x}} (-c+x)^3 \, dx,x,c+d x\right )}{d^4}\\ &=\frac {2 \operatorname {Subst}\left (\int x \sqrt {a+b x} \left (-c+x^2\right )^3 \, dx,x,\sqrt {c+d x}\right )}{d^4}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (-\frac {a \left (a^2-b^2 c\right )^3 \sqrt {a+b x}}{b^7}-\frac {\left (-7 a^2+b^2 c\right ) \left (-a^2+b^2 c\right )^2 (a+b x)^{3/2}}{b^7}-\frac {3 \left (7 a^5-10 a^3 b^2 c+3 a b^4 c^2\right ) (a+b x)^{5/2}}{b^7}+\frac {\left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) (a+b x)^{7/2}}{b^7}-\frac {5 a \left (7 a^2-3 b^2 c\right ) (a+b x)^{9/2}}{b^7}-\frac {3 \left (-7 a^2+b^2 c\right ) (a+b x)^{11/2}}{b^7}-\frac {7 a (a+b x)^{13/2}}{b^7}+\frac {(a+b x)^{15/2}}{b^7}\right ) \, dx,x,\sqrt {c+d x}\right )}{d^4}\\ &=-\frac {4 a \left (a^2-b^2 c\right )^3 \left (a+b \sqrt {c+d x}\right )^{3/2}}{3 b^8 d^4}+\frac {4 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{5/2}}{5 b^8 d^4}-\frac {12 a \left (7 a^2-3 b^2 c\right ) \left (a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{7/2}}{7 b^8 d^4}+\frac {4 \left (35 a^4-30 a^2 b^2 c+3 b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right )^{9/2}}{9 b^8 d^4}-\frac {20 a \left (7 a^2-3 b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{11/2}}{11 b^8 d^4}+\frac {12 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{13/2}}{13 b^8 d^4}-\frac {28 a \left (a+b \sqrt {c+d x}\right )^{15/2}}{15 b^8 d^4}+\frac {4 \left (a+b \sqrt {c+d x}\right )^{17/2}}{17 b^8 d^4}\\ \end {align*}

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Mathematica [A]  time = 0.42, size = 232, normalized size = 0.71 \[ \frac {4 \left (a+b \sqrt {c+d x}\right )^{3/2} \left (-14336 a^7+21504 a^6 b \sqrt {c+d x}+3840 a^5 b^2 (10 c-7 d x)-640 a^4 b^3 (104 c-49 d x) \sqrt {c+d x}-48 a^3 b^4 \left (616 c^2-1080 c d x+735 d^2 x^2\right )+24 a^2 b^5 \sqrt {c+d x} \left (2960 c^2-2716 c d x+1617 d^2 x^2\right )+6 a b^6 \left (320 c^3-3936 c^2 d x+5754 c d^2 x^2-7007 d^3 x^3\right )-231 b^7 \sqrt {c+d x} \left (128 c^3-160 c^2 d x+180 c d^2 x^2-195 d^3 x^3\right )\right )}{765765 b^8 d^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Sqrt[a + b*Sqrt[c + d*x]],x]

[Out]

(4*(a + b*Sqrt[c + d*x])^(3/2)*(-14336*a^7 + 3840*a^5*b^2*(10*c - 7*d*x) + 21504*a^6*b*Sqrt[c + d*x] - 640*a^4
*b^3*(104*c - 49*d*x)*Sqrt[c + d*x] - 48*a^3*b^4*(616*c^2 - 1080*c*d*x + 735*d^2*x^2) + 24*a^2*b^5*Sqrt[c + d*
x]*(2960*c^2 - 2716*c*d*x + 1617*d^2*x^2) + 6*a*b^6*(320*c^3 - 3936*c^2*d*x + 5754*c*d^2*x^2 - 7007*d^3*x^3) -
231*b^7*Sqrt[c + d*x]*(128*c^3 - 160*c^2*d*x + 180*c*d^2*x^2 - 195*d^3*x^3)))/(765765*b^8*d^4)

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fricas [A]  time = 0.54, size = 286, normalized size = 0.88 \[ \frac {4 \, {\left (45045 \, b^{8} d^{4} x^{4} - 29568 \, b^{8} c^{4} + 72960 \, a^{2} b^{6} c^{3} - 96128 \, a^{4} b^{4} c^{2} + 59904 \, a^{6} b^{2} c - 14336 \, a^{8} + 231 \, {\left (15 \, b^{8} c - 14 \, a^{2} b^{6}\right )} d^{3} x^{3} - 28 \, {\left (165 \, b^{8} c^{2} - 291 \, a^{2} b^{6} c + 140 \, a^{4} b^{4}\right )} d^{2} x^{2} + 32 \, {\left (231 \, b^{8} c^{3} - 555 \, a^{2} b^{6} c^{2} + 520 \, a^{4} b^{4} c - 168 \, a^{6} b^{2}\right )} d x + {\left (3003 \, a b^{7} d^{3} x^{3} - 27648 \, a b^{7} c^{3} + 41472 \, a^{3} b^{5} c^{2} - 28160 \, a^{5} b^{3} c + 7168 \, a^{7} b - 3528 \, {\left (2 \, a b^{7} c - a^{3} b^{5}\right )} d^{2} x^{2} + 32 \, {\left (417 \, a b^{7} c^{2} - 417 \, a^{3} b^{5} c + 140 \, a^{5} b^{3}\right )} d x\right )} \sqrt {d x + c}\right )} \sqrt {\sqrt {d x + c} b + a}}{765765 \, b^{8} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/765765*(45045*b^8*d^4*x^4 - 29568*b^8*c^4 + 72960*a^2*b^6*c^3 - 96128*a^4*b^4*c^2 + 59904*a^6*b^2*c - 14336*
a^8 + 231*(15*b^8*c - 14*a^2*b^6)*d^3*x^3 - 28*(165*b^8*c^2 - 291*a^2*b^6*c + 140*a^4*b^4)*d^2*x^2 + 32*(231*b
^8*c^3 - 555*a^2*b^6*c^2 + 520*a^4*b^4*c - 168*a^6*b^2)*d*x + (3003*a*b^7*d^3*x^3 - 27648*a*b^7*c^3 + 41472*a^
3*b^5*c^2 - 28160*a^5*b^3*c + 7168*a^7*b - 3528*(2*a*b^7*c - a^3*b^5)*d^2*x^2 + 32*(417*a*b^7*c^2 - 417*a^3*b^
5*c + 140*a^5*b^3)*d*x)*sqrt(d*x + c))*sqrt(sqrt(d*x + c)*b + a)/(b^8*d^4)

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giac [B]  time = 0.63, size = 915, normalized size = 2.81 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/765765*(17*(15015*(sqrt(d*x + c)*b + a)^(3/2)*b^6*c^3 - 45045*sqrt(sqrt(d*x + c)*b + a)*a*b^6*c^3 - 19305*(
sqrt(d*x + c)*b + a)^(7/2)*b^4*c^2 + 81081*(sqrt(d*x + c)*b + a)^(5/2)*a*b^4*c^2 - 135135*(sqrt(d*x + c)*b + a
)^(3/2)*a^2*b^4*c^2 + 135135*sqrt(sqrt(d*x + c)*b + a)*a^3*b^4*c^2 + 12285*(sqrt(d*x + c)*b + a)^(11/2)*b^2*c
- 75075*(sqrt(d*x + c)*b + a)^(9/2)*a*b^2*c + 193050*(sqrt(d*x + c)*b + a)^(7/2)*a^2*b^2*c - 270270*(sqrt(d*x
+ c)*b + a)^(5/2)*a^3*b^2*c + 225225*(sqrt(d*x + c)*b + a)^(3/2)*a^4*b^2*c - 135135*sqrt(sqrt(d*x + c)*b + a)*
a^5*b^2*c - 3003*(sqrt(d*x + c)*b + a)^(15/2) + 24255*(sqrt(d*x + c)*b + a)^(13/2)*a - 85995*(sqrt(d*x + c)*b
+ a)^(11/2)*a^2 + 175175*(sqrt(d*x + c)*b + a)^(9/2)*a^3 - 225225*(sqrt(d*x + c)*b + a)^(7/2)*a^4 + 189189*(sq
rt(d*x + c)*b + a)^(5/2)*a^5 - 105105*(sqrt(d*x + c)*b + a)^(3/2)*a^6 + 45045*sqrt(sqrt(d*x + c)*b + a)*a^7)*a
/(b^7*d^3) + (153153*(sqrt(d*x + c)*b + a)^(5/2)*b^6*c^3 - 510510*(sqrt(d*x + c)*b + a)^(3/2)*a*b^6*c^3 + 7657
65*sqrt(sqrt(d*x + c)*b + a)*a^2*b^6*c^3 - 255255*(sqrt(d*x + c)*b + a)^(9/2)*b^4*c^2 + 1312740*(sqrt(d*x + c)
*b + a)^(7/2)*a*b^4*c^2 - 2756754*(sqrt(d*x + c)*b + a)^(5/2)*a^2*b^4*c^2 + 3063060*(sqrt(d*x + c)*b + a)^(3/2
)*a^3*b^4*c^2 - 2297295*sqrt(sqrt(d*x + c)*b + a)*a^4*b^4*c^2 + 176715*(sqrt(d*x + c)*b + a)^(13/2)*b^2*c - 12
53070*(sqrt(d*x + c)*b + a)^(11/2)*a*b^2*c + 3828825*(sqrt(d*x + c)*b + a)^(9/2)*a^2*b^2*c - 6563700*(sqrt(d*x
+ c)*b + a)^(7/2)*a^3*b^2*c + 6891885*(sqrt(d*x + c)*b + a)^(5/2)*a^4*b^2*c - 4594590*(sqrt(d*x + c)*b + a)^(
3/2)*a^5*b^2*c + 2297295*sqrt(sqrt(d*x + c)*b + a)*a^6*b^2*c - 45045*(sqrt(d*x + c)*b + a)^(17/2) + 408408*(sq
rt(d*x + c)*b + a)^(15/2)*a - 1649340*(sqrt(d*x + c)*b + a)^(13/2)*a^2 + 3898440*(sqrt(d*x + c)*b + a)^(11/2)*
a^3 - 5955950*(sqrt(d*x + c)*b + a)^(9/2)*a^4 + 6126120*(sqrt(d*x + c)*b + a)^(7/2)*a^5 - 4288284*(sqrt(d*x +
c)*b + a)^(5/2)*a^6 + 2042040*(sqrt(d*x + c)*b + a)^(3/2)*a^7 - 765765*sqrt(sqrt(d*x + c)*b + a)*a^8)/(b^7*d^3
))/(b*d)

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maple [A]  time = 0.01, size = 383, normalized size = 1.17 \[ \frac {-\frac {28 \left (a +\sqrt {d x +c}\, b \right )^{\frac {15}{2}} a}{15}-\frac {4 \left (-b^{2} c +a^{2}\right )^{3} \left (a +\sqrt {d x +c}\, b \right )^{\frac {3}{2}} a}{3}+\frac {4 \left (a +\sqrt {d x +c}\, b \right )^{\frac {17}{2}}}{17}+\frac {4 \left (-3 b^{2} c +21 a^{2}\right ) \left (a +\sqrt {d x +c}\, b \right )^{\frac {13}{2}}}{13}+\frac {4 \left (-8 \left (-b^{2} c +a^{2}\right ) a -2 \left (-2 b^{2} c +6 a^{2}\right ) a -\left (-3 b^{2} c +15 a^{2}\right ) a \right ) \left (a +\sqrt {d x +c}\, b \right )^{\frac {11}{2}}}{11}+\frac {4 \left (8 \left (-b^{2} c +a^{2}\right ) a^{2}-\left (-8 \left (-b^{2} c +a^{2}\right ) a -2 \left (-2 b^{2} c +6 a^{2}\right ) a \right ) a +\left (-b^{2} c +a^{2}\right ) \left (-2 b^{2} c +6 a^{2}\right )+\left (-b^{2} c +a^{2}\right )^{2}\right ) \left (a +\sqrt {d x +c}\, b \right )^{\frac {9}{2}}}{9}+\frac {4 \left (-6 \left (-b^{2} c +a^{2}\right )^{2} a -\left (8 \left (-b^{2} c +a^{2}\right ) a^{2}+\left (-b^{2} c +a^{2}\right ) \left (-2 b^{2} c +6 a^{2}\right )+\left (-b^{2} c +a^{2}\right )^{2}\right ) a \right ) \left (a +\sqrt {d x +c}\, b \right )^{\frac {7}{2}}}{7}+\frac {4 \left (6 \left (-b^{2} c +a^{2}\right )^{2} a^{2}+\left (-b^{2} c +a^{2}\right )^{3}\right ) \left (a +\sqrt {d x +c}\, b \right )^{\frac {5}{2}}}{5}}{b^{8} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/d^4/b^8*(1/17*(a+b*(d*x+c)^(1/2))^(17/2)-7/15*a*(a+b*(d*x+c)^(1/2))^(15/2)+1/13*(-3*b^2*c+21*a^2)*(a+b*(d*x+
c)^(1/2))^(13/2)+1/11*(-8*(-b^2*c+a^2)*a-2*a*(-2*b^2*c+6*a^2)-(-3*b^2*c+15*a^2)*a)*(a+b*(d*x+c)^(1/2))^(11/2)+
1/9*((-b^2*c+a^2)*(-2*b^2*c+6*a^2)+8*a^2*(-b^2*c+a^2)+(-b^2*c+a^2)^2-(-8*(-b^2*c+a^2)*a-2*a*(-2*b^2*c+6*a^2))*
a)*(a+b*(d*x+c)^(1/2))^(9/2)+1/7*(-6*(-b^2*c+a^2)^2*a-((-b^2*c+a^2)*(-2*b^2*c+6*a^2)+8*a^2*(-b^2*c+a^2)+(-b^2*
c+a^2)^2)*a)*(a+b*(d*x+c)^(1/2))^(7/2)+1/5*((-b^2*c+a^2)^3+6*(-b^2*c+a^2)^2*a^2)*(a+b*(d*x+c)^(1/2))^(5/2)-1/3
*(-b^2*c+a^2)^3*a*(a+b*(d*x+c)^(1/2))^(3/2))

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maxima [A]  time = 0.96, size = 268, normalized size = 0.82 \[ \frac {4 \, {\left (45045 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {17}{2}} - 357357 \, {\left (\sqrt {d x + c} b + a\right )}^{\frac {15}{2}} a - 176715 \, {\left (b^{2} c - 7 \, a^{2}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {13}{2}} + 348075 \, {\left (3 \, a b^{2} c - 7 \, a^{3}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {11}{2}} + 85085 \, {\left (3 \, b^{4} c^{2} - 30 \, a^{2} b^{2} c + 35 \, a^{4}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {9}{2}} - 328185 \, {\left (3 \, a b^{4} c^{2} - 10 \, a^{3} b^{2} c + 7 \, a^{5}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {7}{2}} - 153153 \, {\left (b^{6} c^{3} - 9 \, a^{2} b^{4} c^{2} + 15 \, a^{4} b^{2} c - 7 \, a^{6}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {5}{2}} + 255255 \, {\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )} {\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}}\right )}}{765765 \, b^{8} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/765765*(45045*(sqrt(d*x + c)*b + a)^(17/2) - 357357*(sqrt(d*x + c)*b + a)^(15/2)*a - 176715*(b^2*c - 7*a^2)*
(sqrt(d*x + c)*b + a)^(13/2) + 348075*(3*a*b^2*c - 7*a^3)*(sqrt(d*x + c)*b + a)^(11/2) + 85085*(3*b^4*c^2 - 30
*a^2*b^2*c + 35*a^4)*(sqrt(d*x + c)*b + a)^(9/2) - 328185*(3*a*b^4*c^2 - 10*a^3*b^2*c + 7*a^5)*(sqrt(d*x + c)*
b + a)^(7/2) - 153153*(b^6*c^3 - 9*a^2*b^4*c^2 + 15*a^4*b^2*c - 7*a^6)*(sqrt(d*x + c)*b + a)^(5/2) + 255255*(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)^(3/2))/(b^8*d^4)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,\sqrt {a+b\,\sqrt {c+d\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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