3.1.63 \(\int (a+b x^2)^{5/2} (c+d x^2)^2 \, dx\) [63]

Optimal. Leaf size=241 \[ \frac {a^2 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \sqrt {a+b x^2}}{256 b^2}+\frac {a \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{384 b^2}+\frac {\left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{5/2}}{480 b^2}+\frac {3 d (4 b c-a d) x \left (a+b x^2\right )^{7/2}}{80 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}+\frac {a^3 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{5/2}} \]

[Out]

1/384*a*(3*a^2*d^2-20*a*b*c*d+80*b^2*c^2)*x*(b*x^2+a)^(3/2)/b^2+1/480*(3*a^2*d^2-20*a*b*c*d+80*b^2*c^2)*x*(b*x
^2+a)^(5/2)/b^2+3/80*d*(-a*d+4*b*c)*x*(b*x^2+a)^(7/2)/b^2+1/10*d*x*(b*x^2+a)^(7/2)*(d*x^2+c)/b+1/256*a^3*(3*a^
2*d^2-20*a*b*c*d+80*b^2*c^2)*arctanh(x*b^(1/2)/(b*x^2+a)^(1/2))/b^(5/2)+1/256*a^2*(3*a^2*d^2-20*a*b*c*d+80*b^2
*c^2)*x*(b*x^2+a)^(1/2)/b^2

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Rubi [A]
time = 0.10, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {427, 396, 201, 223, 212} \begin {gather*} \frac {x \left (a+b x^2\right )^{5/2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{480 b^2}+\frac {a x \left (a+b x^2\right )^{3/2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{384 b^2}+\frac {a^2 x \sqrt {a+b x^2} \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right )}{256 b^2}+\frac {a^3 \left (3 a^2 d^2-20 a b c d+80 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{5/2}}+\frac {3 d x \left (a+b x^2\right )^{7/2} (4 b c-a d)}{80 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(5/2)*(c + d*x^2)^2,x]

[Out]

(a^2*(80*b^2*c^2 - 20*a*b*c*d + 3*a^2*d^2)*x*Sqrt[a + b*x^2])/(256*b^2) + (a*(80*b^2*c^2 - 20*a*b*c*d + 3*a^2*
d^2)*x*(a + b*x^2)^(3/2))/(384*b^2) + ((80*b^2*c^2 - 20*a*b*c*d + 3*a^2*d^2)*x*(a + b*x^2)^(5/2))/(480*b^2) +
(3*d*(4*b*c - a*d)*x*(a + b*x^2)^(7/2))/(80*b^2) + (d*x*(a + b*x^2)^(7/2)*(c + d*x^2))/(10*b) + (a^3*(80*b^2*c
^2 - 20*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(256*b^(5/2))

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 396

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

Rule 427

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*x*(a + b*x^n)^(p + 1)*((c
+ d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

Rubi steps

\begin {align*} \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2 \, dx &=\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}+\frac {\int \left (a+b x^2\right )^{5/2} \left (c (10 b c-a d)+3 d (4 b c-a d) x^2\right ) \, dx}{10 b}\\ &=\frac {3 d (4 b c-a d) x \left (a+b x^2\right )^{7/2}}{80 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}-\frac {(3 a d (4 b c-a d)-8 b c (10 b c-a d)) \int \left (a+b x^2\right )^{5/2} \, dx}{80 b^2}\\ &=\frac {\left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{5/2}}{480 b^2}+\frac {3 d (4 b c-a d) x \left (a+b x^2\right )^{7/2}}{80 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}+\frac {\left (a \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right )\right ) \int \left (a+b x^2\right )^{3/2} \, dx}{96 b^2}\\ &=\frac {a \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{384 b^2}+\frac {\left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{5/2}}{480 b^2}+\frac {3 d (4 b c-a d) x \left (a+b x^2\right )^{7/2}}{80 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}+\frac {\left (a^2 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right )\right ) \int \sqrt {a+b x^2} \, dx}{128 b^2}\\ &=\frac {a^2 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \sqrt {a+b x^2}}{256 b^2}+\frac {a \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{384 b^2}+\frac {\left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{5/2}}{480 b^2}+\frac {3 d (4 b c-a d) x \left (a+b x^2\right )^{7/2}}{80 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}+\frac {\left (a^3 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{256 b^2}\\ &=\frac {a^2 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \sqrt {a+b x^2}}{256 b^2}+\frac {a \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{384 b^2}+\frac {\left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{5/2}}{480 b^2}+\frac {3 d (4 b c-a d) x \left (a+b x^2\right )^{7/2}}{80 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}+\frac {\left (a^3 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{256 b^2}\\ &=\frac {a^2 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \sqrt {a+b x^2}}{256 b^2}+\frac {a \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{3/2}}{384 b^2}+\frac {\left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) x \left (a+b x^2\right )^{5/2}}{480 b^2}+\frac {3 d (4 b c-a d) x \left (a+b x^2\right )^{7/2}}{80 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{10 b}+\frac {a^3 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.33, size = 190, normalized size = 0.79 \begin {gather*} \frac {\sqrt {b} x \sqrt {a+b x^2} \left (-45 a^4 d^2+30 a^3 b d \left (10 c+d x^2\right )+64 b^4 x^4 \left (10 c^2+15 c d x^2+6 d^2 x^4\right )+16 a b^3 x^2 \left (130 c^2+170 c d x^2+63 d^2 x^4\right )+8 a^2 b^2 \left (330 c^2+295 c d x^2+93 d^2 x^4\right )\right )-15 a^3 \left (80 b^2 c^2-20 a b c d+3 a^2 d^2\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{3840 b^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(5/2)*(c + d*x^2)^2,x]

[Out]

(Sqrt[b]*x*Sqrt[a + b*x^2]*(-45*a^4*d^2 + 30*a^3*b*d*(10*c + d*x^2) + 64*b^4*x^4*(10*c^2 + 15*c*d*x^2 + 6*d^2*
x^4) + 16*a*b^3*x^2*(130*c^2 + 170*c*d*x^2 + 63*d^2*x^4) + 8*a^2*b^2*(330*c^2 + 295*c*d*x^2 + 93*d^2*x^4)) - 1
5*a^3*(80*b^2*c^2 - 20*a*b*c*d + 3*a^2*d^2)*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(3840*b^(5/2))

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Maple [A]
time = 0.06, size = 283, normalized size = 1.17

method	result	size
risch	\(-\frac {x \left (-384 b^{4} d^{2} x^{8}-1008 a \,b^{3} d^{2} x^{6}-960 b^{4} c d \,x^{6}-744 a^{2} b^{2} d^{2} x^{4}-2720 a \,b^{3} c d \,x^{4}-640 b^{4} c^{2} x^{4}-30 a^{3} b \,d^{2} x^{2}-2360 a^{2} b^{2} c d \,x^{2}-2080 a \,b^{3} c^{2} x^{2}+45 a^{4} d^{2}-300 a^{3} b c d -2640 a^{2} b^{2} c^{2}\right ) \sqrt {b \,x^{2}+a}}{3840 b^{2}}+\frac {3 a^{5} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) d^{2}}{256 b^{\frac {5}{2}}}-\frac {5 a^{4} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) c d}{64 b^{\frac {3}{2}}}+\frac {5 a^{3} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) c^{2}}{16 \sqrt {b}}\)	\(231\)
default	\(d^{2} \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{10 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )}{10 b}\right )+2 c d \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )+c^{2} \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )\)	\(283\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(5/2)*(d*x^2+c)^2,x,method=_RETURNVERBOSE)

[Out]

d^2*(1/10*x^3*(b*x^2+a)^(7/2)/b-3/10*a/b*(1/8*x*(b*x^2+a)^(7/2)/b-1/8*a/b*(1/6*x*(b*x^2+a)^(5/2)+5/6*a*(1/4*x*
(b*x^2+a)^(3/2)+3/4*a*(1/2*x*(b*x^2+a)^(1/2)+1/2*a/b^(1/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2)))))))+2*c*d*(1/8*x*(b*
x^2+a)^(7/2)/b-1/8*a/b*(1/6*x*(b*x^2+a)^(5/2)+5/6*a*(1/4*x*(b*x^2+a)^(3/2)+3/4*a*(1/2*x*(b*x^2+a)^(1/2)+1/2*a/
b^(1/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))))))+c^2*(1/6*x*(b*x^2+a)^(5/2)+5/6*a*(1/4*x*(b*x^2+a)^(3/2)+3/4*a*(1/2*x
*(b*x^2+a)^(1/2)+1/2*a/b^(1/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2)))))

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Maxima [A]
time = 0.31, size = 286, normalized size = 1.19 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} d^{2} x^{3}}{10 \, b} + \frac {1}{6} \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} c^{2} x + \frac {5}{24} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a c^{2} x + \frac {5}{16} \, \sqrt {b x^{2} + a} a^{2} c^{2} x + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} c d x}{4 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} a c d x}{24 \, b} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} c d x}{96 \, b} - \frac {5 \, \sqrt {b x^{2} + a} a^{3} c d x}{64 \, b} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a d^{2} x}{80 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2} d^{2} x}{160 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} d^{2} x}{128 \, b^{2}} + \frac {3 \, \sqrt {b x^{2} + a} a^{4} d^{2} x}{256 \, b^{2}} + \frac {5 \, a^{3} c^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, \sqrt {b}} - \frac {5 \, a^{4} c d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{64 \, b^{\frac {3}{2}}} + \frac {3 \, a^{5} d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{256 \, b^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)*(d*x^2+c)^2,x, algorithm="maxima")

[Out]

1/10*(b*x^2 + a)^(7/2)*d^2*x^3/b + 1/6*(b*x^2 + a)^(5/2)*c^2*x + 5/24*(b*x^2 + a)^(3/2)*a*c^2*x + 5/16*sqrt(b*
x^2 + a)*a^2*c^2*x + 1/4*(b*x^2 + a)^(7/2)*c*d*x/b - 1/24*(b*x^2 + a)^(5/2)*a*c*d*x/b - 5/96*(b*x^2 + a)^(3/2)
*a^2*c*d*x/b - 5/64*sqrt(b*x^2 + a)*a^3*c*d*x/b - 3/80*(b*x^2 + a)^(7/2)*a*d^2*x/b^2 + 1/160*(b*x^2 + a)^(5/2)
*a^2*d^2*x/b^2 + 1/128*(b*x^2 + a)^(3/2)*a^3*d^2*x/b^2 + 3/256*sqrt(b*x^2 + a)*a^4*d^2*x/b^2 + 5/16*a^3*c^2*ar
csinh(b*x/sqrt(a*b))/sqrt(b) - 5/64*a^4*c*d*arcsinh(b*x/sqrt(a*b))/b^(3/2) + 3/256*a^5*d^2*arcsinh(b*x/sqrt(a*
b))/b^(5/2)

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Fricas [A]
time = 0.56, size = 420, normalized size = 1.74 \begin {gather*} \left [\frac {15 \, {\left (80 \, a^{3} b^{2} c^{2} - 20 \, a^{4} b c d + 3 \, a^{5} d^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (384 \, b^{5} d^{2} x^{9} + 48 \, {\left (20 \, b^{5} c d + 21 \, a b^{4} d^{2}\right )} x^{7} + 8 \, {\left (80 \, b^{5} c^{2} + 340 \, a b^{4} c d + 93 \, a^{2} b^{3} d^{2}\right )} x^{5} + 10 \, {\left (208 \, a b^{4} c^{2} + 236 \, a^{2} b^{3} c d + 3 \, a^{3} b^{2} d^{2}\right )} x^{3} + 15 \, {\left (176 \, a^{2} b^{3} c^{2} + 20 \, a^{3} b^{2} c d - 3 \, a^{4} b d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{7680 \, b^{3}}, -\frac {15 \, {\left (80 \, a^{3} b^{2} c^{2} - 20 \, a^{4} b c d + 3 \, a^{5} d^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (384 \, b^{5} d^{2} x^{9} + 48 \, {\left (20 \, b^{5} c d + 21 \, a b^{4} d^{2}\right )} x^{7} + 8 \, {\left (80 \, b^{5} c^{2} + 340 \, a b^{4} c d + 93 \, a^{2} b^{3} d^{2}\right )} x^{5} + 10 \, {\left (208 \, a b^{4} c^{2} + 236 \, a^{2} b^{3} c d + 3 \, a^{3} b^{2} d^{2}\right )} x^{3} + 15 \, {\left (176 \, a^{2} b^{3} c^{2} + 20 \, a^{3} b^{2} c d - 3 \, a^{4} b d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{3840 \, b^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)*(d*x^2+c)^2,x, algorithm="fricas")

[Out]

[1/7680*(15*(80*a^3*b^2*c^2 - 20*a^4*b*c*d + 3*a^5*d^2)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a
) + 2*(384*b^5*d^2*x^9 + 48*(20*b^5*c*d + 21*a*b^4*d^2)*x^7 + 8*(80*b^5*c^2 + 340*a*b^4*c*d + 93*a^2*b^3*d^2)*
x^5 + 10*(208*a*b^4*c^2 + 236*a^2*b^3*c*d + 3*a^3*b^2*d^2)*x^3 + 15*(176*a^2*b^3*c^2 + 20*a^3*b^2*c*d - 3*a^4*
b*d^2)*x)*sqrt(b*x^2 + a))/b^3, -1/3840*(15*(80*a^3*b^2*c^2 - 20*a^4*b*c*d + 3*a^5*d^2)*sqrt(-b)*arctan(sqrt(-
b)*x/sqrt(b*x^2 + a)) - (384*b^5*d^2*x^9 + 48*(20*b^5*c*d + 21*a*b^4*d^2)*x^7 + 8*(80*b^5*c^2 + 340*a*b^4*c*d
+ 93*a^2*b^3*d^2)*x^5 + 10*(208*a*b^4*c^2 + 236*a^2*b^3*c*d + 3*a^3*b^2*d^2)*x^3 + 15*(176*a^2*b^3*c^2 + 20*a^
3*b^2*c*d - 3*a^4*b*d^2)*x)*sqrt(b*x^2 + a))/b^3]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(5/2)*(d*x**2+c)**2,x)

[Out]

Timed out

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Giac [A]
time = 0.60, size = 221, normalized size = 0.92 \begin {gather*} \frac {1}{3840} \, {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, b^{2} d^{2} x^{2} + \frac {20 \, b^{10} c d + 21 \, a b^{9} d^{2}}{b^{8}}\right )} x^{2} + \frac {80 \, b^{10} c^{2} + 340 \, a b^{9} c d + 93 \, a^{2} b^{8} d^{2}}{b^{8}}\right )} x^{2} + \frac {5 \, {\left (208 \, a b^{9} c^{2} + 236 \, a^{2} b^{8} c d + 3 \, a^{3} b^{7} d^{2}\right )}}{b^{8}}\right )} x^{2} + \frac {15 \, {\left (176 \, a^{2} b^{8} c^{2} + 20 \, a^{3} b^{7} c d - 3 \, a^{4} b^{6} d^{2}\right )}}{b^{8}}\right )} \sqrt {b x^{2} + a} x - \frac {{\left (80 \, a^{3} b^{2} c^{2} - 20 \, a^{4} b c d + 3 \, a^{5} d^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{256 \, b^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)*(d*x^2+c)^2,x, algorithm="giac")

[Out]

1/3840*(2*(4*(6*(8*b^2*d^2*x^2 + (20*b^10*c*d + 21*a*b^9*d^2)/b^8)*x^2 + (80*b^10*c^2 + 340*a*b^9*c*d + 93*a^2
*b^8*d^2)/b^8)*x^2 + 5*(208*a*b^9*c^2 + 236*a^2*b^8*c*d + 3*a^3*b^7*d^2)/b^8)*x^2 + 15*(176*a^2*b^8*c^2 + 20*a
^3*b^7*c*d - 3*a^4*b^6*d^2)/b^8)*sqrt(b*x^2 + a)*x - 1/256*(80*a^3*b^2*c^2 - 20*a^4*b*c*d + 3*a^5*d^2)*log(abs
(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(5/2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (b\,x^2+a\right )}^{5/2}\,{\left (d\,x^2+c\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^(5/2)*(c + d*x^2)^2,x)

[Out]

int((a + b*x^2)^(5/2)*(c + d*x^2)^2, x)

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