3.4.44 \(\int x^2 \sqrt {-c+d x} \sqrt {c+d x} (a+b x^2) \, dx\) [344]

Optimal. Leaf size=159 \[ \frac {c^2 \left (b c^2+2 a d^2\right ) x \sqrt {-c+d x} \sqrt {c+d x}}{16 d^4}+\frac {\left (b c^2+2 a d^2\right ) x (-c+d x)^{3/2} (c+d x)^{3/2}}{8 d^4}+\frac {b x^3 (-c+d x)^{3/2} (c+d x)^{3/2}}{6 d^2}-\frac {c^4 \left (b c^2+2 a d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{8 d^5} \]

[Out]

1/8*(2*a*d^2+b*c^2)*x*(d*x-c)^(3/2)*(d*x+c)^(3/2)/d^4+1/6*b*x^3*(d*x-c)^(3/2)*(d*x+c)^(3/2)/d^2-1/8*c^4*(2*a*d
^2+b*c^2)*arctanh((d*x-c)^(1/2)/(d*x+c)^(1/2))/d^5+1/16*c^2*(2*a*d^2+b*c^2)*x*(d*x-c)^(1/2)*(d*x+c)^(1/2)/d^4

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Rubi [A]
time = 0.08, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {471, 92, 12, 38, 65, 223, 212} \begin {gather*} \frac {c^2 x \sqrt {d x-c} \sqrt {c+d x} \left (2 a d^2+b c^2\right )}{16 d^4}+\frac {x (d x-c)^{3/2} (c+d x)^{3/2} \left (2 a d^2+b c^2\right )}{8 d^4}-\frac {c^4 \left (2 a d^2+b c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d x-c}}{\sqrt {c+d x}}\right )}{8 d^5}+\frac {b x^3 (d x-c)^{3/2} (c+d x)^{3/2}}{6 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*Sqrt[-c + d*x]*Sqrt[c + d*x]*(a + b*x^2),x]

[Out]

(c^2*(b*c^2 + 2*a*d^2)*x*Sqrt[-c + d*x]*Sqrt[c + d*x])/(16*d^4) + ((b*c^2 + 2*a*d^2)*x*(-c + d*x)^(3/2)*(c + d
*x)^(3/2))/(8*d^4) + (b*x^3*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(6*d^2) - (c^4*(b*c^2 + 2*a*d^2)*ArcTanh[Sqrt[-c
+ d*x]/Sqrt[c + d*x]])/(8*d^5)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 38

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[x*(a + b*x)^m*((c + d*x)^m/(2*m + 1))
, x] + Dist[2*a*c*(m/(2*m + 1)), Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] &&
EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 92

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a + b*x
)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 471

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_.)*((c_) + (d_.)
*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*((a2 + b2*x^(n/2))^(p + 1)/(b1*b2*e*(
m + n*(p + 1) + 1))), x] - Dist[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(b1*b2*(m + n*(p + 1) + 1)), I
nt[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] &&
EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]

Rubi steps

\begin {align*} \int x^2 \sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right ) \, dx &=\frac {b x^3 (-c+d x)^{3/2} (c+d x)^{3/2}}{6 d^2}+\frac {1}{2} \left (2 a+\frac {b c^2}{d^2}\right ) \int x^2 \sqrt {-c+d x} \sqrt {c+d x} \, dx\\ &=\frac {\left (b c^2+2 a d^2\right ) x (-c+d x)^{3/2} (c+d x)^{3/2}}{8 d^4}+\frac {b x^3 (-c+d x)^{3/2} (c+d x)^{3/2}}{6 d^2}+\frac {\left (b c^2+2 a d^2\right ) \int c^2 \sqrt {-c+d x} \sqrt {c+d x} \, dx}{8 d^4}\\ &=\frac {\left (b c^2+2 a d^2\right ) x (-c+d x)^{3/2} (c+d x)^{3/2}}{8 d^4}+\frac {b x^3 (-c+d x)^{3/2} (c+d x)^{3/2}}{6 d^2}+\frac {\left (c^2 \left (b c^2+2 a d^2\right )\right ) \int \sqrt {-c+d x} \sqrt {c+d x} \, dx}{8 d^4}\\ &=\frac {c^2 \left (b c^2+2 a d^2\right ) x \sqrt {-c+d x} \sqrt {c+d x}}{16 d^4}+\frac {\left (b c^2+2 a d^2\right ) x (-c+d x)^{3/2} (c+d x)^{3/2}}{8 d^4}+\frac {b x^3 (-c+d x)^{3/2} (c+d x)^{3/2}}{6 d^2}-\frac {\left (c^4 \left (b c^2+2 a d^2\right )\right ) \int \frac {1}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx}{16 d^4}\\ &=\frac {c^2 \left (b c^2+2 a d^2\right ) x \sqrt {-c+d x} \sqrt {c+d x}}{16 d^4}+\frac {\left (b c^2+2 a d^2\right ) x (-c+d x)^{3/2} (c+d x)^{3/2}}{8 d^4}+\frac {b x^3 (-c+d x)^{3/2} (c+d x)^{3/2}}{6 d^2}-\frac {\left (c^4 \left (b c^2+2 a d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2 c+x^2}} \, dx,x,\sqrt {-c+d x}\right )}{8 d^5}\\ &=\frac {c^2 \left (b c^2+2 a d^2\right ) x \sqrt {-c+d x} \sqrt {c+d x}}{16 d^4}+\frac {\left (b c^2+2 a d^2\right ) x (-c+d x)^{3/2} (c+d x)^{3/2}}{8 d^4}+\frac {b x^3 (-c+d x)^{3/2} (c+d x)^{3/2}}{6 d^2}-\frac {\left (c^4 \left (b c^2+2 a d^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{8 d^5}\\ &=\frac {c^2 \left (b c^2+2 a d^2\right ) x \sqrt {-c+d x} \sqrt {c+d x}}{16 d^4}+\frac {\left (b c^2+2 a d^2\right ) x (-c+d x)^{3/2} (c+d x)^{3/2}}{8 d^4}+\frac {b x^3 (-c+d x)^{3/2} (c+d x)^{3/2}}{6 d^2}-\frac {c^4 \left (b c^2+2 a d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{8 d^5}\\ \end {align*}

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Mathematica [A]
time = 0.24, size = 116, normalized size = 0.73 \begin {gather*} \frac {d x \sqrt {-c+d x} \sqrt {c+d x} \left (-6 a d^2 \left (c^2-2 d^2 x^2\right )+b \left (-3 c^4-2 c^2 d^2 x^2+8 d^4 x^4\right )\right )-6 c^4 \left (b c^2+2 a d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{48 d^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d*x*Sqrt[-c + d*x]*Sqrt[c + d*x]*(-6*a*d^2*(c^2 - 2*d^2*x^2) + b*(-3*c^4 - 2*c^2*d^2*x^2 + 8*d^4*x^4)) - 6*c^
4*(b*c^2 + 2*a*d^2)*ArcTanh[Sqrt[-c + d*x]/Sqrt[c + d*x]])/(48*d^5)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.29, size = 240, normalized size = 1.51

method	result	size
risch	\(\frac {x \left (-8 b \,d^{4} x^{4}-12 a \,d^{4} x^{2}+2 b \,c^{2} d^{2} x^{2}+6 a \,c^{2} d^{2}+3 b \,c^{4}\right ) \left (-d x +c \right ) \sqrt {d x +c}}{48 d^{4} \sqrt {d x -c}}-\frac {\left (\frac {c^{4} \ln \left (\frac {d^{2} x}{\sqrt {d^{2}}}+\sqrt {d^{2} x^{2}-c^{2}}\right ) a}{8 d^{2} \sqrt {d^{2}}}+\frac {c^{6} \ln \left (\frac {d^{2} x}{\sqrt {d^{2}}}+\sqrt {d^{2} x^{2}-c^{2}}\right ) b}{16 d^{4} \sqrt {d^{2}}}\right ) \sqrt {\left (d x -c \right ) \left (d x +c \right )}}{\sqrt {d x -c}\, \sqrt {d x +c}}\)	\(192\)
default	\(-\frac {\sqrt {d x -c}\, \sqrt {d x +c}\, \left (-8 \,\mathrm {csgn}\left (d \right ) b \,d^{5} x^{5} \sqrt {d^{2} x^{2}-c^{2}}-12 \,\mathrm {csgn}\left (d \right ) a \,d^{5} x^{3} \sqrt {d^{2} x^{2}-c^{2}}+2 \,\mathrm {csgn}\left (d \right ) b \,c^{2} d^{3} x^{3} \sqrt {d^{2} x^{2}-c^{2}}+6 \sqrt {d^{2} x^{2}-c^{2}}\, \mathrm {csgn}\left (d \right ) d^{3} a \,c^{2} x +3 \sqrt {d^{2} x^{2}-c^{2}}\, \mathrm {csgn}\left (d \right ) d b \,c^{4} x +6 \ln \left (\left (\sqrt {d^{2} x^{2}-c^{2}}\, \mathrm {csgn}\left (d \right )+d x \right ) \mathrm {csgn}\left (d \right )\right ) a \,c^{4} d^{2}+3 \ln \left (\left (\sqrt {d^{2} x^{2}-c^{2}}\, \mathrm {csgn}\left (d \right )+d x \right ) \mathrm {csgn}\left (d \right )\right ) b \,c^{6}\right ) \mathrm {csgn}\left (d \right )}{48 \sqrt {d^{2} x^{2}-c^{2}}\, d^{5}}\)	\(240\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*(d*x-c)^(1/2)*(d*x+c)^(1/2)*(-8*csgn(d)*b*d^5*x^5*(d^2*x^2-c^2)^(1/2)-12*csgn(d)*a*d^5*x^3*(d^2*x^2-c^2)
^(1/2)+2*csgn(d)*b*c^2*d^3*x^3*(d^2*x^2-c^2)^(1/2)+6*(d^2*x^2-c^2)^(1/2)*csgn(d)*d^3*a*c^2*x+3*(d^2*x^2-c^2)^(
1/2)*csgn(d)*d*b*c^4*x+6*ln(((d^2*x^2-c^2)^(1/2)*csgn(d)+d*x)*csgn(d))*a*c^4*d^2+3*ln(((d^2*x^2-c^2)^(1/2)*csg
n(d)+d*x)*csgn(d))*b*c^6)*csgn(d)/(d^2*x^2-c^2)^(1/2)/d^5

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Maxima [A]
time = 0.27, size = 192, normalized size = 1.21 \begin {gather*} \frac {{\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} b x^{3}}{6 \, d^{2}} - \frac {b c^{6} \log \left (2 \, d^{2} x + 2 \, \sqrt {d^{2} x^{2} - c^{2}} d\right )}{16 \, d^{5}} - \frac {a c^{4} \log \left (2 \, d^{2} x + 2 \, \sqrt {d^{2} x^{2} - c^{2}} d\right )}{8 \, d^{3}} + \frac {\sqrt {d^{2} x^{2} - c^{2}} b c^{4} x}{16 \, d^{4}} + \frac {\sqrt {d^{2} x^{2} - c^{2}} a c^{2} x}{8 \, d^{2}} + \frac {{\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} b c^{2} x}{8 \, d^{4}} + \frac {{\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} a x}{4 \, d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(d^2*x^2 - c^2)^(3/2)*b*x^3/d^2 - 1/16*b*c^6*log(2*d^2*x + 2*sqrt(d^2*x^2 - c^2)*d)/d^5 - 1/8*a*c^4*log(2*
d^2*x + 2*sqrt(d^2*x^2 - c^2)*d)/d^3 + 1/16*sqrt(d^2*x^2 - c^2)*b*c^4*x/d^4 + 1/8*sqrt(d^2*x^2 - c^2)*a*c^2*x/
d^2 + 1/8*(d^2*x^2 - c^2)^(3/2)*b*c^2*x/d^4 + 1/4*(d^2*x^2 - c^2)^(3/2)*a*x/d^2

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Fricas [A]
time = 3.44, size = 112, normalized size = 0.70 \begin {gather*} \frac {{\left (8 \, b d^{5} x^{5} - 2 \, {\left (b c^{2} d^{3} - 6 \, a d^{5}\right )} x^{3} - 3 \, {\left (b c^{4} d + 2 \, a c^{2} d^{3}\right )} x\right )} \sqrt {d x + c} \sqrt {d x - c} + 3 \, {\left (b c^{6} + 2 \, a c^{4} d^{2}\right )} \log \left (-d x + \sqrt {d x + c} \sqrt {d x - c}\right )}{48 \, d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*((8*b*d^5*x^5 - 2*(b*c^2*d^3 - 6*a*d^5)*x^3 - 3*(b*c^4*d + 2*a*c^2*d^3)*x)*sqrt(d*x + c)*sqrt(d*x - c) +
3*(b*c^6 + 2*a*c^4*d^2)*log(-d*x + sqrt(d*x + c)*sqrt(d*x - c)))/d^5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*(a + b*x**2)*sqrt(-c + d*x)*sqrt(c + d*x), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 432 vs. \(2 (135) = 270\).
time = 0.63, size = 432, normalized size = 2.72 \begin {gather*} \frac {40 \, {\left (\sqrt {d x + c} \sqrt {d x - c} {\left ({\left (d x + c\right )} {\left (\frac {2 \, {\left (d x + c\right )}}{d^{2}} - \frac {7 \, c}{d^{2}}\right )} + \frac {9 \, c^{2}}{d^{2}}\right )} + \frac {6 \, c^{3} \log \left ({\left | -\sqrt {d x + c} + \sqrt {d x - c} \right |}\right )}{d^{2}}\right )} a c + 2 \, {\left ({\left ({\left (2 \, {\left (d x + c\right )} {\left (3 \, {\left (d x + c\right )} {\left (\frac {4 \, {\left (d x + c\right )}}{d^{4}} - \frac {21 \, c}{d^{4}}\right )} + \frac {133 \, c^{2}}{d^{4}}\right )} - \frac {295 \, c^{3}}{d^{4}}\right )} {\left (d x + c\right )} + \frac {195 \, c^{4}}{d^{4}}\right )} \sqrt {d x + c} \sqrt {d x - c} + \frac {90 \, c^{5} \log \left ({\left | -\sqrt {d x + c} + \sqrt {d x - c} \right |}\right )}{d^{4}}\right )} b c + 10 \, {\left ({\left ({\left (d x + c\right )} {\left (2 \, {\left (d x + c\right )} {\left (\frac {3 \, {\left (d x + c\right )}}{d^{3}} - \frac {13 \, c}{d^{3}}\right )} + \frac {43 \, c^{2}}{d^{3}}\right )} - \frac {39 \, c^{3}}{d^{3}}\right )} \sqrt {d x + c} \sqrt {d x - c} - \frac {18 \, c^{4} \log \left ({\left | -\sqrt {d x + c} + \sqrt {d x - c} \right |}\right )}{d^{3}}\right )} a d + {\left ({\left ({\left (2 \, {\left ({\left (d x + c\right )} {\left (4 \, {\left (d x + c\right )} {\left (\frac {5 \, {\left (d x + c\right )}}{d^{5}} - \frac {31 \, c}{d^{5}}\right )} + \frac {321 \, c^{2}}{d^{5}}\right )} - \frac {451 \, c^{3}}{d^{5}}\right )} {\left (d x + c\right )} + \frac {745 \, c^{4}}{d^{5}}\right )} {\left (d x + c\right )} - \frac {405 \, c^{5}}{d^{5}}\right )} \sqrt {d x + c} \sqrt {d x - c} - \frac {150 \, c^{6} \log \left ({\left | -\sqrt {d x + c} + \sqrt {d x - c} \right |}\right )}{d^{5}}\right )} b d}{240 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/240*(40*(sqrt(d*x + c)*sqrt(d*x - c)*((d*x + c)*(2*(d*x + c)/d^2 - 7*c/d^2) + 9*c^2/d^2) + 6*c^3*log(abs(-sq
rt(d*x + c) + sqrt(d*x - c)))/d^2)*a*c + 2*(((2*(d*x + c)*(3*(d*x + c)*(4*(d*x + c)/d^4 - 21*c/d^4) + 133*c^2/
d^4) - 295*c^3/d^4)*(d*x + c) + 195*c^4/d^4)*sqrt(d*x + c)*sqrt(d*x - c) + 90*c^5*log(abs(-sqrt(d*x + c) + sqr
t(d*x - c)))/d^4)*b*c + 10*(((d*x + c)*(2*(d*x + c)*(3*(d*x + c)/d^3 - 13*c/d^3) + 43*c^2/d^3) - 39*c^3/d^3)*s
qrt(d*x + c)*sqrt(d*x - c) - 18*c^4*log(abs(-sqrt(d*x + c) + sqrt(d*x - c)))/d^3)*a*d + (((2*((d*x + c)*(4*(d*
x + c)*(5*(d*x + c)/d^5 - 31*c/d^5) + 321*c^2/d^5) - 451*c^3/d^5)*(d*x + c) + 745*c^4/d^5)*(d*x + c) - 405*c^5
/d^5)*sqrt(d*x + c)*sqrt(d*x - c) - 150*c^6*log(abs(-sqrt(d*x + c) + sqrt(d*x - c)))/d^5)*b*d)/d

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Mupad [B]
time = 42.57, size = 1681, normalized size = 10.57 \begin {gather*} \frac {\frac {35\,b\,c^6\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}{12\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^3}-\frac {b\,c^6\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}{4\,\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}+\frac {757\,b\,c^6\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^5}{4\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^5}+\frac {7339\,b\,c^6\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^7}{4\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^7}+\frac {41929\,b\,c^6\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^9}{6\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^9}+\frac {25661\,b\,c^6\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{11}}{2\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{11}}+\frac {25661\,b\,c^6\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{13}}{2\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{13}}+\frac {41929\,b\,c^6\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{15}}{6\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{15}}+\frac {7339\,b\,c^6\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{17}}{4\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{17}}+\frac {757\,b\,c^6\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{19}}{4\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{19}}+\frac {35\,b\,c^6\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{21}}{12\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{21}}-\frac {b\,c^6\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{23}}{4\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{23}}}{d^5-\frac {12\,d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^2}+\frac {66\,d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^4}-\frac {220\,d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^6}+\frac {495\,d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^8}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^8}-\frac {792\,d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{10}}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{10}}+\frac {924\,d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{12}}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{12}}-\frac {792\,d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{14}}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{14}}+\frac {495\,d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{16}}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{16}}-\frac {220\,d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{18}}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{18}}+\frac {66\,d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{20}}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{20}}-\frac {12\,d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{22}}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{22}}+\frac {d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{24}}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{24}}}-\frac {\frac {a\,c^4\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}{2\,\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}+\frac {35\,a\,c^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}{2\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^3}+\frac {273\,a\,c^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^5}{2\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^5}+\frac {715\,a\,c^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^7}{2\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^7}+\frac {715\,a\,c^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^9}{2\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^9}+\frac {273\,a\,c^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{11}}{2\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{11}}+\frac {35\,a\,c^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{13}}{2\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{13}}+\frac {a\,c^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{15}}{2\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{15}}}{d^3-\frac {8\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^2}+\frac {28\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^4}-\frac {56\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^6}+\frac {70\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^8}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^8}-\frac {56\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{10}}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{10}}+\frac {28\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{12}}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{12}}-\frac {8\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{14}}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{14}}+\frac {d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{16}}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{16}}}+\frac {a\,c^4\,\mathrm {atanh}\left (\frac {\sqrt {c+d\,x}-\sqrt {c}}{\sqrt {-c}-\sqrt {d\,x-c}}\right )}{2\,d^3}+\frac {b\,c^6\,\mathrm {atanh}\left (\frac {\sqrt {c+d\,x}-\sqrt {c}}{\sqrt {-c}-\sqrt {d\,x-c}}\right )}{4\,d^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((35*b*c^6*((c + d*x)^(1/2) - c^(1/2))^3)/(12*((-c)^(1/2) - (d*x - c)^(1/2))^3) - (b*c^6*((c + d*x)^(1/2) - c^
(1/2)))/(4*((-c)^(1/2) - (d*x - c)^(1/2))) + (757*b*c^6*((c + d*x)^(1/2) - c^(1/2))^5)/(4*((-c)^(1/2) - (d*x -
c)^(1/2))^5) + (7339*b*c^6*((c + d*x)^(1/2) - c^(1/2))^7)/(4*((-c)^(1/2) - (d*x - c)^(1/2))^7) + (41929*b*c^6
*((c + d*x)^(1/2) - c^(1/2))^9)/(6*((-c)^(1/2) - (d*x - c)^(1/2))^9) + (25661*b*c^6*((c + d*x)^(1/2) - c^(1/2)
)^11)/(2*((-c)^(1/2) - (d*x - c)^(1/2))^11) + (25661*b*c^6*((c + d*x)^(1/2) - c^(1/2))^13)/(2*((-c)^(1/2) - (d
*x - c)^(1/2))^13) + (41929*b*c^6*((c + d*x)^(1/2) - c^(1/2))^15)/(6*((-c)^(1/2) - (d*x - c)^(1/2))^15) + (733
9*b*c^6*((c + d*x)^(1/2) - c^(1/2))^17)/(4*((-c)^(1/2) - (d*x - c)^(1/2))^17) + (757*b*c^6*((c + d*x)^(1/2) -
c^(1/2))^19)/(4*((-c)^(1/2) - (d*x - c)^(1/2))^19) + (35*b*c^6*((c + d*x)^(1/2) - c^(1/2))^21)/(12*((-c)^(1/2)
- (d*x - c)^(1/2))^21) - (b*c^6*((c + d*x)^(1/2) - c^(1/2))^23)/(4*((-c)^(1/2) - (d*x - c)^(1/2))^23))/(d^5 -
(12*d^5*((c + d*x)^(1/2) - c^(1/2))^2)/((-c)^(1/2) - (d*x - c)^(1/2))^2 + (66*d^5*((c + d*x)^(1/2) - c^(1/2))
^4)/((-c)^(1/2) - (d*x - c)^(1/2))^4 - (220*d^5*((c + d*x)^(1/2) - c^(1/2))^6)/((-c)^(1/2) - (d*x - c)^(1/2))^
6 + (495*d^5*((c + d*x)^(1/2) - c^(1/2))^8)/((-c)^(1/2) - (d*x - c)^(1/2))^8 - (792*d^5*((c + d*x)^(1/2) - c^(
1/2))^10)/((-c)^(1/2) - (d*x - c)^(1/2))^10 + (924*d^5*((c + d*x)^(1/2) - c^(1/2))^12)/((-c)^(1/2) - (d*x - c)
^(1/2))^12 - (792*d^5*((c + d*x)^(1/2) - c^(1/2))^14)/((-c)^(1/2) - (d*x - c)^(1/2))^14 + (495*d^5*((c + d*x)^
(1/2) - c^(1/2))^16)/((-c)^(1/2) - (d*x - c)^(1/2))^16 - (220*d^5*((c + d*x)^(1/2) - c^(1/2))^18)/((-c)^(1/2)
- (d*x - c)^(1/2))^18 + (66*d^5*((c + d*x)^(1/2) - c^(1/2))^20)/((-c)^(1/2) - (d*x - c)^(1/2))^20 - (12*d^5*((
c + d*x)^(1/2) - c^(1/2))^22)/((-c)^(1/2) - (d*x - c)^(1/2))^22 + (d^5*((c + d*x)^(1/2) - c^(1/2))^24)/((-c)^(
1/2) - (d*x - c)^(1/2))^24) - ((a*c^4*((c + d*x)^(1/2) - c^(1/2)))/(2*((-c)^(1/2) - (d*x - c)^(1/2))) + (35*a*
c^4*((c + d*x)^(1/2) - c^(1/2))^3)/(2*((-c)^(1/2) - (d*x - c)^(1/2))^3) + (273*a*c^4*((c + d*x)^(1/2) - c^(1/2
))^5)/(2*((-c)^(1/2) - (d*x - c)^(1/2))^5) + (715*a*c^4*((c + d*x)^(1/2) - c^(1/2))^7)/(2*((-c)^(1/2) - (d*x -
c)^(1/2))^7) + (715*a*c^4*((c + d*x)^(1/2) - c^(1/2))^9)/(2*((-c)^(1/2) - (d*x - c)^(1/2))^9) + (273*a*c^4*((
c + d*x)^(1/2) - c^(1/2))^11)/(2*((-c)^(1/2) - (d*x - c)^(1/2))^11) + (35*a*c^4*((c + d*x)^(1/2) - c^(1/2))^13
)/(2*((-c)^(1/2) - (d*x - c)^(1/2))^13) + (a*c^4*((c + d*x)^(1/2) - c^(1/2))^15)/(2*((-c)^(1/2) - (d*x - c)^(1
/2))^15))/(d^3 - (8*d^3*((c + d*x)^(1/2) - c^(1/2))^2)/((-c)^(1/2) - (d*x - c)^(1/2))^2 + (28*d^3*((c + d*x)^(
1/2) - c^(1/2))^4)/((-c)^(1/2) - (d*x - c)^(1/2))^4 - (56*d^3*((c + d*x)^(1/2) - c^(1/2))^6)/((-c)^(1/2) - (d*
x - c)^(1/2))^6 + (70*d^3*((c + d*x)^(1/2) - c^(1/2))^8)/((-c)^(1/2) - (d*x - c)^(1/2))^8 - (56*d^3*((c + d*x)
^(1/2) - c^(1/2))^10)/((-c)^(1/2) - (d*x - c)^(1/2))^10 + (28*d^3*((c + d*x)^(1/2) - c^(1/2))^12)/((-c)^(1/2)
- (d*x - c)^(1/2))^12 - (8*d^3*((c + d*x)^(1/2) - c^(1/2))^14)/((-c)^(1/2) - (d*x - c)^(1/2))^14 + (d^3*((c +
d*x)^(1/2) - c^(1/2))^16)/((-c)^(1/2) - (d*x - c)^(1/2))^16) + (a*c^4*atanh(((c + d*x)^(1/2) - c^(1/2))/((-c)^
(1/2) - (d*x - c)^(1/2))))/(2*d^3) + (b*c^6*atanh(((c + d*x)^(1/2) - c^(1/2))/((-c)^(1/2) - (d*x - c)^(1/2))))
/(4*d^5)

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