\(\int (a+b x^2)^{5/2} (c+d x^2)^3 \, dx\) [62]

Optimal result

Integrand size = 21, antiderivative size = 349 \[ \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^3 \, dx=\frac {a^2 \left (320 b^3 c^3-120 a b^2 c^2 d+36 a^2 b c d^2-5 a^3 d^3\right ) x \sqrt {a+b x^2}}{1024 b^3}+\frac {a \left (320 b^3 c^3-120 a b^2 c^2 d+36 a^2 b c d^2-5 a^3 d^3\right ) x \left (a+b x^2\right )^{3/2}}{1536 b^3}+\frac {\left (320 b^3 c^3-120 a b^2 c^2 d+36 a^2 b c d^2-5 a^3 d^3\right ) x \left (a+b x^2\right )^{5/2}}{1920 b^3}+\frac {d \left (152 b^2 c^2-68 a b c d+15 a^2 d^2\right ) x \left (a+b x^2\right )^{7/2}}{960 b^3}+\frac {d (16 b c-5 a d) x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{120 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )^2}{12 b}+\frac {a^3 \left (320 b^3 c^3-120 a b^2 c^2 d+36 a^2 b c d^2-5 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{1024 b^{7/2}} \]

[Out]

1/1536*a*(-5*a^3*d^3+36*a^2*b*c*d^2-120*a*b^2*c^2*d+320*b^3*c^3)*x*(b*x^2+a)^(3/2)/b^3+1/1920*(-5*a^3*d^3+36*a
^2*b*c*d^2-120*a*b^2*c^2*d+320*b^3*c^3)*x*(b*x^2+a)^(5/2)/b^3+1/960*d*(15*a^2*d^2-68*a*b*c*d+152*b^2*c^2)*x*(b
*x^2+a)^(7/2)/b^3+1/120*d*(-5*a*d+16*b*c)*x*(b*x^2+a)^(7/2)*(d*x^2+c)/b^2+1/12*d*x*(b*x^2+a)^(7/2)*(d*x^2+c)^2
/b+1/1024*a^3*(-5*a^3*d^3+36*a^2*b*c*d^2-120*a*b^2*c^2*d+320*b^3*c^3)*arctanh(x*b^(1/2)/(b*x^2+a)^(1/2))/b^(7/
2)+1/1024*a^2*(-5*a^3*d^3+36*a^2*b*c*d^2-120*a*b^2*c^2*d+320*b^3*c^3)*x*(b*x^2+a)^(1/2)/b^3

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {427, 542, 396, 201, 223, 212} \[ \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^3 \, dx=\frac {d x \left (a+b x^2\right )^{7/2} \left (15 a^2 d^2-68 a b c d+152 b^2 c^2\right )}{960 b^3}+\frac {a^3 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (-5 a^3 d^3+36 a^2 b c d^2-120 a b^2 c^2 d+320 b^3 c^3\right )}{1024 b^{7/2}}+\frac {x \left (a+b x^2\right )^{5/2} \left (-5 a^3 d^3+36 a^2 b c d^2-120 a b^2 c^2 d+320 b^3 c^3\right )}{1920 b^3}+\frac {a x \left (a+b x^2\right )^{3/2} \left (-5 a^3 d^3+36 a^2 b c d^2-120 a b^2 c^2 d+320 b^3 c^3\right )}{1536 b^3}+\frac {a^2 x \sqrt {a+b x^2} \left (-5 a^3 d^3+36 a^2 b c d^2-120 a b^2 c^2 d+320 b^3 c^3\right )}{1024 b^3}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right ) (16 b c-5 a d)}{120 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )^2}{12 b} \]

[In]

Int[(a + b*x^2)^(5/2)*(c + d*x^2)^3,x]

[Out]

(a^2*(320*b^3*c^3 - 120*a*b^2*c^2*d + 36*a^2*b*c*d^2 - 5*a^3*d^3)*x*Sqrt[a + b*x^2])/(1024*b^3) + (a*(320*b^3*
c^3 - 120*a*b^2*c^2*d + 36*a^2*b*c*d^2 - 5*a^3*d^3)*x*(a + b*x^2)^(3/2))/(1536*b^3) + ((320*b^3*c^3 - 120*a*b^
2*c^2*d + 36*a^2*b*c*d^2 - 5*a^3*d^3)*x*(a + b*x^2)^(5/2))/(1920*b^3) + (d*(152*b^2*c^2 - 68*a*b*c*d + 15*a^2*
d^2)*x*(a + b*x^2)^(7/2))/(960*b^3) + (d*(16*b*c - 5*a*d)*x*(a + b*x^2)^(7/2)*(c + d*x^2))/(120*b^2) + (d*x*(a
+ b*x^2)^(7/2)*(c + d*x^2)^2)/(12*b) + (a^3*(320*b^3*c^3 - 120*a*b^2*c^2*d + 36*a^2*b*c*d^2 - 5*a^3*d^3)*ArcT
anh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(1024*b^(7/2))

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]

Rule 542

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

Rubi steps \begin{align*} \text {integral}& = \frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )^2}{12 b}+\frac {\int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right ) \left (c (12 b c-a d)+d (16 b c-5 a d) x^2\right ) \, dx}{12 b} \\ & = \frac {d (16 b c-5 a d) x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{120 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )^2}{12 b}+\frac {\int \left (a+b x^2\right )^{5/2} \left (c \left (120 b^2 c^2-26 a b c d+5 a^2 d^2\right )+d \left (152 b^2 c^2-68 a b c d+15 a^2 d^2\right ) x^2\right ) \, dx}{120 b^2} \\ & = \frac {d \left (152 b^2 c^2-68 a b c d+15 a^2 d^2\right ) x \left (a+b x^2\right )^{7/2}}{960 b^3}+\frac {d (16 b c-5 a d) x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{120 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )^2}{12 b}+\frac {\left (320 b^3 c^3-120 a b^2 c^2 d+36 a^2 b c d^2-5 a^3 d^3\right ) \int \left (a+b x^2\right )^{5/2} \, dx}{320 b^3} \\ & = \frac {\left (320 b^3 c^3-120 a b^2 c^2 d+36 a^2 b c d^2-5 a^3 d^3\right ) x \left (a+b x^2\right )^{5/2}}{1920 b^3}+\frac {d \left (152 b^2 c^2-68 a b c d+15 a^2 d^2\right ) x \left (a+b x^2\right )^{7/2}}{960 b^3}+\frac {d (16 b c-5 a d) x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{120 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )^2}{12 b}+\frac {\left (a \left (320 b^3 c^3-120 a b^2 c^2 d+36 a^2 b c d^2-5 a^3 d^3\right )\right ) \int \left (a+b x^2\right )^{3/2} \, dx}{384 b^3} \\ & = \frac {a \left (320 b^3 c^3-120 a b^2 c^2 d+36 a^2 b c d^2-5 a^3 d^3\right ) x \left (a+b x^2\right )^{3/2}}{1536 b^3}+\frac {\left (320 b^3 c^3-120 a b^2 c^2 d+36 a^2 b c d^2-5 a^3 d^3\right ) x \left (a+b x^2\right )^{5/2}}{1920 b^3}+\frac {d \left (152 b^2 c^2-68 a b c d+15 a^2 d^2\right ) x \left (a+b x^2\right )^{7/2}}{960 b^3}+\frac {d (16 b c-5 a d) x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{120 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )^2}{12 b}+\frac {\left (a^2 \left (320 b^3 c^3-120 a b^2 c^2 d+36 a^2 b c d^2-5 a^3 d^3\right )\right ) \int \sqrt {a+b x^2} \, dx}{512 b^3} \\ & = \frac {a^2 \left (320 b^3 c^3-120 a b^2 c^2 d+36 a^2 b c d^2-5 a^3 d^3\right ) x \sqrt {a+b x^2}}{1024 b^3}+\frac {a \left (320 b^3 c^3-120 a b^2 c^2 d+36 a^2 b c d^2-5 a^3 d^3\right ) x \left (a+b x^2\right )^{3/2}}{1536 b^3}+\frac {\left (320 b^3 c^3-120 a b^2 c^2 d+36 a^2 b c d^2-5 a^3 d^3\right ) x \left (a+b x^2\right )^{5/2}}{1920 b^3}+\frac {d \left (152 b^2 c^2-68 a b c d+15 a^2 d^2\right ) x \left (a+b x^2\right )^{7/2}}{960 b^3}+\frac {d (16 b c-5 a d) x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{120 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )^2}{12 b}+\frac {\left (a^3 \left (320 b^3 c^3-120 a b^2 c^2 d+36 a^2 b c d^2-5 a^3 d^3\right )\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{1024 b^3} \\ & = \frac {a^2 \left (320 b^3 c^3-120 a b^2 c^2 d+36 a^2 b c d^2-5 a^3 d^3\right ) x \sqrt {a+b x^2}}{1024 b^3}+\frac {a \left (320 b^3 c^3-120 a b^2 c^2 d+36 a^2 b c d^2-5 a^3 d^3\right ) x \left (a+b x^2\right )^{3/2}}{1536 b^3}+\frac {\left (320 b^3 c^3-120 a b^2 c^2 d+36 a^2 b c d^2-5 a^3 d^3\right ) x \left (a+b x^2\right )^{5/2}}{1920 b^3}+\frac {d \left (152 b^2 c^2-68 a b c d+15 a^2 d^2\right ) x \left (a+b x^2\right )^{7/2}}{960 b^3}+\frac {d (16 b c-5 a d) x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{120 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )^2}{12 b}+\frac {\left (a^3 \left (320 b^3 c^3-120 a b^2 c^2 d+36 a^2 b c d^2-5 a^3 d^3\right )\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{1024 b^3} \\ & = \frac {a^2 \left (320 b^3 c^3-120 a b^2 c^2 d+36 a^2 b c d^2-5 a^3 d^3\right ) x \sqrt {a+b x^2}}{1024 b^3}+\frac {a \left (320 b^3 c^3-120 a b^2 c^2 d+36 a^2 b c d^2-5 a^3 d^3\right ) x \left (a+b x^2\right )^{3/2}}{1536 b^3}+\frac {\left (320 b^3 c^3-120 a b^2 c^2 d+36 a^2 b c d^2-5 a^3 d^3\right ) x \left (a+b x^2\right )^{5/2}}{1920 b^3}+\frac {d \left (152 b^2 c^2-68 a b c d+15 a^2 d^2\right ) x \left (a+b x^2\right )^{7/2}}{960 b^3}+\frac {d (16 b c-5 a d) x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )}{120 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )^2}{12 b}+\frac {a^3 \left (320 b^3 c^3-120 a b^2 c^2 d+36 a^2 b c d^2-5 a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{1024 b^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.77 \[ \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^3 \, dx=\frac {\sqrt {b} x \sqrt {a+b x^2} \left (75 a^5 d^3-10 a^4 b d^2 \left (54 c+5 d x^2\right )+40 a^3 b^2 d \left (45 c^2+9 c d x^2+d^2 x^4\right )+128 b^5 x^4 \left (20 c^3+45 c^2 d x^2+36 c d^2 x^4+10 d^3 x^6\right )+48 a^2 b^3 \left (220 c^3+295 c^2 d x^2+186 c d^2 x^4+45 d^3 x^6\right )+64 a b^4 x^2 \left (130 c^3+255 c^2 d x^2+189 c d^2 x^4+50 d^3 x^6\right )\right )+15 a^3 \left (-320 b^3 c^3+120 a b^2 c^2 d-36 a^2 b c d^2+5 a^3 d^3\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{15360 b^{7/2}} \]

[In]

Integrate[(a + b*x^2)^(5/2)*(c + d*x^2)^3,x]

[Out]

(Sqrt[b]*x*Sqrt[a + b*x^2]*(75*a^5*d^3 - 10*a^4*b*d^2*(54*c + 5*d*x^2) + 40*a^3*b^2*d*(45*c^2 + 9*c*d*x^2 + d^
2*x^4) + 128*b^5*x^4*(20*c^3 + 45*c^2*d*x^2 + 36*c*d^2*x^4 + 10*d^3*x^6) + 48*a^2*b^3*(220*c^3 + 295*c^2*d*x^2
+ 186*c*d^2*x^4 + 45*d^3*x^6) + 64*a*b^4*x^2*(130*c^3 + 255*c^2*d*x^2 + 189*c*d^2*x^4 + 50*d^3*x^6)) + 15*a^3
*(-320*b^3*c^3 + 120*a*b^2*c^2*d - 36*a^2*b*c*d^2 + 5*a^3*d^3)*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(15360*b^(
7/2))

Maple [A] (verified)

Time = 2.54 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.71

method	result	size
pseudoelliptic	\(-\frac {5 \left (a^{3} \left (a^{3} d^{3}-\frac {36}{5} a^{2} b c \,d^{2}+24 a \,b^{2} c^{2} d -64 b^{3} c^{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )-x \left (\frac {512 x^{4} \left (\frac {1}{2} d^{3} x^{6}+\frac {9}{5} c \,d^{2} x^{4}+\frac {9}{4} c^{2} d \,x^{2}+c^{3}\right ) b^{\frac {11}{2}}}{15}+\left (\frac {704 a \left (\frac {9}{44} d^{3} x^{6}+\frac {93}{110} c \,d^{2} x^{4}+\frac {59}{44} c^{2} d \,x^{2}+c^{3}\right ) b^{\frac {7}{2}}}{5}+\frac {1664 x^{2} \left (\frac {5}{13} d^{3} x^{6}+\frac {189}{130} c \,d^{2} x^{4}+\frac {51}{26} c^{2} d \,x^{2}+c^{3}\right ) b^{\frac {9}{2}}}{15}+\left (\left (\frac {8}{15} d^{2} x^{4}+\frac {24}{5} c d \,x^{2}+24 c^{2}\right ) b^{\frac {5}{2}}+\left (\left (-\frac {2 d \,x^{2}}{3}-\frac {36 c}{5}\right ) b^{\frac {3}{2}}+a d \sqrt {b}\right ) d a \right ) d \,a^{2}\right ) a \right ) \sqrt {b \,x^{2}+a}\right )}{1024 b^{\frac {7}{2}}}\)	\(247\)
risch	\(\frac {x \left (1280 b^{5} d^{3} x^{10}+3200 a \,b^{4} d^{3} x^{8}+4608 b^{5} d^{2} c \,x^{8}+2160 a^{2} b^{3} d^{3} x^{6}+12096 a \,b^{4} c \,d^{2} x^{6}+5760 b^{5} c^{2} d \,x^{6}+40 a^{3} b^{2} d^{3} x^{4}+8928 a^{2} b^{3} c \,d^{2} x^{4}+16320 a \,b^{4} c^{2} d \,x^{4}+2560 b^{5} c^{3} x^{4}-50 x^{2} a^{4} b \,d^{3}+360 x^{2} a^{3} b^{2} c \,d^{2}+14160 x^{2} a^{2} b^{3} c^{2} d +8320 x^{2} a \,b^{4} c^{3}+75 a^{5} d^{3}-540 a^{4} b c \,d^{2}+1800 a^{3} b^{2} c^{2} d +10560 a^{2} b^{3} c^{3}\right ) \sqrt {b \,x^{2}+a}}{15360 b^{3}}-\frac {a^{3} \left (5 a^{3} d^{3}-36 a^{2} b c \,d^{2}+120 a \,b^{2} c^{2} d -320 b^{3} c^{3}\right ) \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{1024 b^{\frac {7}{2}}}\)	\(301\)
default	\(c^{3} \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )+d^{3} \left (\frac {x^{5} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{12 b}-\frac {5 a \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{10 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )}{10 b}\right )}{12 b}\right )+3 c \,d^{2} \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{10 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )}{10 b}\right )+3 c^{2} d \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )\)	\(428\)

[In]

int((b*x^2+a)^(5/2)*(d*x^2+c)^3,x,method=_RETURNVERBOSE)

[Out]

-5/1024*(a^3*(a^3*d^3-36/5*a^2*b*c*d^2+24*a*b^2*c^2*d-64*b^3*c^3)*arctanh((b*x^2+a)^(1/2)/x/b^(1/2))-x*(512/15
*x^4*(1/2*d^3*x^6+9/5*c*d^2*x^4+9/4*c^2*d*x^2+c^3)*b^(11/2)+(704/5*a*(9/44*d^3*x^6+93/110*c*d^2*x^4+59/44*c^2*
d*x^2+c^3)*b^(7/2)+1664/15*x^2*(5/13*d^3*x^6+189/130*c*d^2*x^4+51/26*c^2*d*x^2+c^3)*b^(9/2)+((8/15*d^2*x^4+24/
5*c*d*x^2+24*c^2)*b^(5/2)+((-2/3*d*x^2-36/5*c)*b^(3/2)+a*d*b^(1/2))*d*a)*d*a^2)*a)*(b*x^2+a)^(1/2))/b^(7/2)

Fricas [A] (verification not implemented)

none

Time = 0.52 (sec) , antiderivative size = 608, normalized size of antiderivative = 1.74 \[ \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^3 \, dx=\left [-\frac {15 \, {\left (320 \, a^{3} b^{3} c^{3} - 120 \, a^{4} b^{2} c^{2} d + 36 \, a^{5} b c d^{2} - 5 \, a^{6} d^{3}\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (1280 \, b^{6} d^{3} x^{11} + 128 \, {\left (36 \, b^{6} c d^{2} + 25 \, a b^{5} d^{3}\right )} x^{9} + 144 \, {\left (40 \, b^{6} c^{2} d + 84 \, a b^{5} c d^{2} + 15 \, a^{2} b^{4} d^{3}\right )} x^{7} + 8 \, {\left (320 \, b^{6} c^{3} + 2040 \, a b^{5} c^{2} d + 1116 \, a^{2} b^{4} c d^{2} + 5 \, a^{3} b^{3} d^{3}\right )} x^{5} + 10 \, {\left (832 \, a b^{5} c^{3} + 1416 \, a^{2} b^{4} c^{2} d + 36 \, a^{3} b^{3} c d^{2} - 5 \, a^{4} b^{2} d^{3}\right )} x^{3} + 15 \, {\left (704 \, a^{2} b^{4} c^{3} + 120 \, a^{3} b^{3} c^{2} d - 36 \, a^{4} b^{2} c d^{2} + 5 \, a^{5} b d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{30720 \, b^{4}}, -\frac {15 \, {\left (320 \, a^{3} b^{3} c^{3} - 120 \, a^{4} b^{2} c^{2} d + 36 \, a^{5} b c d^{2} - 5 \, a^{6} d^{3}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (1280 \, b^{6} d^{3} x^{11} + 128 \, {\left (36 \, b^{6} c d^{2} + 25 \, a b^{5} d^{3}\right )} x^{9} + 144 \, {\left (40 \, b^{6} c^{2} d + 84 \, a b^{5} c d^{2} + 15 \, a^{2} b^{4} d^{3}\right )} x^{7} + 8 \, {\left (320 \, b^{6} c^{3} + 2040 \, a b^{5} c^{2} d + 1116 \, a^{2} b^{4} c d^{2} + 5 \, a^{3} b^{3} d^{3}\right )} x^{5} + 10 \, {\left (832 \, a b^{5} c^{3} + 1416 \, a^{2} b^{4} c^{2} d + 36 \, a^{3} b^{3} c d^{2} - 5 \, a^{4} b^{2} d^{3}\right )} x^{3} + 15 \, {\left (704 \, a^{2} b^{4} c^{3} + 120 \, a^{3} b^{3} c^{2} d - 36 \, a^{4} b^{2} c d^{2} + 5 \, a^{5} b d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{15360 \, b^{4}}\right ] \]

[In]

integrate((b*x^2+a)^(5/2)*(d*x^2+c)^3,x, algorithm="fricas")

[Out]

[-1/30720*(15*(320*a^3*b^3*c^3 - 120*a^4*b^2*c^2*d + 36*a^5*b*c*d^2 - 5*a^6*d^3)*sqrt(b)*log(-2*b*x^2 + 2*sqrt
(b*x^2 + a)*sqrt(b)*x - a) - 2*(1280*b^6*d^3*x^11 + 128*(36*b^6*c*d^2 + 25*a*b^5*d^3)*x^9 + 144*(40*b^6*c^2*d
+ 84*a*b^5*c*d^2 + 15*a^2*b^4*d^3)*x^7 + 8*(320*b^6*c^3 + 2040*a*b^5*c^2*d + 1116*a^2*b^4*c*d^2 + 5*a^3*b^3*d^
3)*x^5 + 10*(832*a*b^5*c^3 + 1416*a^2*b^4*c^2*d + 36*a^3*b^3*c*d^2 - 5*a^4*b^2*d^3)*x^3 + 15*(704*a^2*b^4*c^3
+ 120*a^3*b^3*c^2*d - 36*a^4*b^2*c*d^2 + 5*a^5*b*d^3)*x)*sqrt(b*x^2 + a))/b^4, -1/15360*(15*(320*a^3*b^3*c^3 -
120*a^4*b^2*c^2*d + 36*a^5*b*c*d^2 - 5*a^6*d^3)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (1280*b^6*d^3*x
^11 + 128*(36*b^6*c*d^2 + 25*a*b^5*d^3)*x^9 + 144*(40*b^6*c^2*d + 84*a*b^5*c*d^2 + 15*a^2*b^4*d^3)*x^7 + 8*(32
0*b^6*c^3 + 2040*a*b^5*c^2*d + 1116*a^2*b^4*c*d^2 + 5*a^3*b^3*d^3)*x^5 + 10*(832*a*b^5*c^3 + 1416*a^2*b^4*c^2*
d + 36*a^3*b^3*c*d^2 - 5*a^4*b^2*d^3)*x^3 + 15*(704*a^2*b^4*c^3 + 120*a^3*b^3*c^2*d - 36*a^4*b^2*c*d^2 + 5*a^5
*b*d^3)*x)*sqrt(b*x^2 + a))/b^4]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 823 vs. \(2 (352) = 704\).

Time = 0.66 (sec) , antiderivative size = 823, normalized size of antiderivative = 2.36 \[ \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^3 \, dx=\begin {cases} \sqrt {a + b x^{2}} \left (\frac {b^{2} d^{3} x^{11}}{12} + \frac {x^{9} \cdot \left (\frac {25 a b^{2} d^{3}}{12} + 3 b^{3} c d^{2}\right )}{10 b} + \frac {x^{7} \cdot \left (3 a^{2} b d^{3} + 9 a b^{2} c d^{2} - \frac {9 a \left (\frac {25 a b^{2} d^{3}}{12} + 3 b^{3} c d^{2}\right )}{10 b} + 3 b^{3} c^{2} d\right )}{8 b} + \frac {x^{5} \left (a^{3} d^{3} + 9 a^{2} b c d^{2} + 9 a b^{2} c^{2} d - \frac {7 a \left (3 a^{2} b d^{3} + 9 a b^{2} c d^{2} - \frac {9 a \left (\frac {25 a b^{2} d^{3}}{12} + 3 b^{3} c d^{2}\right )}{10 b} + 3 b^{3} c^{2} d\right )}{8 b} + b^{3} c^{3}\right )}{6 b} + \frac {x^{3} \cdot \left (3 a^{3} c d^{2} + 9 a^{2} b c^{2} d + 3 a b^{2} c^{3} - \frac {5 a \left (a^{3} d^{3} + 9 a^{2} b c d^{2} + 9 a b^{2} c^{2} d - \frac {7 a \left (3 a^{2} b d^{3} + 9 a b^{2} c d^{2} - \frac {9 a \left (\frac {25 a b^{2} d^{3}}{12} + 3 b^{3} c d^{2}\right )}{10 b} + 3 b^{3} c^{2} d\right )}{8 b} + b^{3} c^{3}\right )}{6 b}\right )}{4 b} + \frac {x \left (3 a^{3} c^{2} d + 3 a^{2} b c^{3} - \frac {3 a \left (3 a^{3} c d^{2} + 9 a^{2} b c^{2} d + 3 a b^{2} c^{3} - \frac {5 a \left (a^{3} d^{3} + 9 a^{2} b c d^{2} + 9 a b^{2} c^{2} d - \frac {7 a \left (3 a^{2} b d^{3} + 9 a b^{2} c d^{2} - \frac {9 a \left (\frac {25 a b^{2} d^{3}}{12} + 3 b^{3} c d^{2}\right )}{10 b} + 3 b^{3} c^{2} d\right )}{8 b} + b^{3} c^{3}\right )}{6 b}\right )}{4 b}\right )}{2 b}\right ) + \left (a^{3} c^{3} - \frac {a \left (3 a^{3} c^{2} d + 3 a^{2} b c^{3} - \frac {3 a \left (3 a^{3} c d^{2} + 9 a^{2} b c^{2} d + 3 a b^{2} c^{3} - \frac {5 a \left (a^{3} d^{3} + 9 a^{2} b c d^{2} + 9 a b^{2} c^{2} d - \frac {7 a \left (3 a^{2} b d^{3} + 9 a b^{2} c d^{2} - \frac {9 a \left (\frac {25 a b^{2} d^{3}}{12} + 3 b^{3} c d^{2}\right )}{10 b} + 3 b^{3} c^{2} d\right )}{8 b} + b^{3} c^{3}\right )}{6 b}\right )}{4 b}\right )}{2 b}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: b \neq 0 \\a^{\frac {5}{2}} \left (c^{3} x + c^{2} d x^{3} + \frac {3 c d^{2} x^{5}}{5} + \frac {d^{3} x^{7}}{7}\right ) & \text {otherwise} \end {cases} \]

[In]

integrate((b*x**2+a)**(5/2)*(d*x**2+c)**3,x)

[Out]

Piecewise((sqrt(a + b*x**2)*(b**2*d**3*x**11/12 + x**9*(25*a*b**2*d**3/12 + 3*b**3*c*d**2)/(10*b) + x**7*(3*a*
*2*b*d**3 + 9*a*b**2*c*d**2 - 9*a*(25*a*b**2*d**3/12 + 3*b**3*c*d**2)/(10*b) + 3*b**3*c**2*d)/(8*b) + x**5*(a*
*3*d**3 + 9*a**2*b*c*d**2 + 9*a*b**2*c**2*d - 7*a*(3*a**2*b*d**3 + 9*a*b**2*c*d**2 - 9*a*(25*a*b**2*d**3/12 +
3*b**3*c*d**2)/(10*b) + 3*b**3*c**2*d)/(8*b) + b**3*c**3)/(6*b) + x**3*(3*a**3*c*d**2 + 9*a**2*b*c**2*d + 3*a*
b**2*c**3 - 5*a*(a**3*d**3 + 9*a**2*b*c*d**2 + 9*a*b**2*c**2*d - 7*a*(3*a**2*b*d**3 + 9*a*b**2*c*d**2 - 9*a*(2
5*a*b**2*d**3/12 + 3*b**3*c*d**2)/(10*b) + 3*b**3*c**2*d)/(8*b) + b**3*c**3)/(6*b))/(4*b) + x*(3*a**3*c**2*d +
3*a**2*b*c**3 - 3*a*(3*a**3*c*d**2 + 9*a**2*b*c**2*d + 3*a*b**2*c**3 - 5*a*(a**3*d**3 + 9*a**2*b*c*d**2 + 9*a
*b**2*c**2*d - 7*a*(3*a**2*b*d**3 + 9*a*b**2*c*d**2 - 9*a*(25*a*b**2*d**3/12 + 3*b**3*c*d**2)/(10*b) + 3*b**3*
c**2*d)/(8*b) + b**3*c**3)/(6*b))/(4*b))/(2*b)) + (a**3*c**3 - a*(3*a**3*c**2*d + 3*a**2*b*c**3 - 3*a*(3*a**3*
c*d**2 + 9*a**2*b*c**2*d + 3*a*b**2*c**3 - 5*a*(a**3*d**3 + 9*a**2*b*c*d**2 + 9*a*b**2*c**2*d - 7*a*(3*a**2*b*
d**3 + 9*a*b**2*c*d**2 - 9*a*(25*a*b**2*d**3/12 + 3*b**3*c*d**2)/(10*b) + 3*b**3*c**2*d)/(8*b) + b**3*c**3)/(6
*b))/(4*b))/(2*b))*Piecewise((log(2*sqrt(b)*sqrt(a + b*x**2) + 2*b*x)/sqrt(b), Ne(a, 0)), (x*log(x)/sqrt(b*x**
2), True)), Ne(b, 0)), (a**(5/2)*(c**3*x + c**2*d*x**3 + 3*c*d**2*x**5/5 + d**3*x**7/7), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.28 \[ \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^3 \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} d^{3} x^{5}}{12 \, b} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} c d^{2} x^{3}}{10 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} a d^{3} x^{3}}{24 \, b^{2}} + \frac {1}{6} \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} c^{3} x + \frac {5}{24} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a c^{3} x + \frac {5}{16} \, \sqrt {b x^{2} + a} a^{2} c^{3} x + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} c^{2} d x}{8 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} a c^{2} d x}{16 \, b} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} c^{2} d x}{64 \, b} - \frac {15 \, \sqrt {b x^{2} + a} a^{3} c^{2} d x}{128 \, b} - \frac {9 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a c d^{2} x}{80 \, b^{2}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2} c d^{2} x}{160 \, b^{2}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} c d^{2} x}{128 \, b^{2}} + \frac {9 \, \sqrt {b x^{2} + a} a^{4} c d^{2} x}{256 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} d^{3} x}{64 \, b^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3} d^{3} x}{384 \, b^{3}} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4} d^{3} x}{1536 \, b^{3}} - \frac {5 \, \sqrt {b x^{2} + a} a^{5} d^{3} x}{1024 \, b^{3}} + \frac {5 \, a^{3} c^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, \sqrt {b}} - \frac {15 \, a^{4} c^{2} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {3}{2}}} + \frac {9 \, a^{5} c d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{256 \, b^{\frac {5}{2}}} - \frac {5 \, a^{6} d^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{1024 \, b^{\frac {7}{2}}} \]

[In]

integrate((b*x^2+a)^(5/2)*(d*x^2+c)^3,x, algorithm="maxima")

[Out]

1/12*(b*x^2 + a)^(7/2)*d^3*x^5/b + 3/10*(b*x^2 + a)^(7/2)*c*d^2*x^3/b - 1/24*(b*x^2 + a)^(7/2)*a*d^3*x^3/b^2 +
1/6*(b*x^2 + a)^(5/2)*c^3*x + 5/24*(b*x^2 + a)^(3/2)*a*c^3*x + 5/16*sqrt(b*x^2 + a)*a^2*c^3*x + 3/8*(b*x^2 +
a)^(7/2)*c^2*d*x/b - 1/16*(b*x^2 + a)^(5/2)*a*c^2*d*x/b - 5/64*(b*x^2 + a)^(3/2)*a^2*c^2*d*x/b - 15/128*sqrt(b
*x^2 + a)*a^3*c^2*d*x/b - 9/80*(b*x^2 + a)^(7/2)*a*c*d^2*x/b^2 + 3/160*(b*x^2 + a)^(5/2)*a^2*c*d^2*x/b^2 + 3/1
28*(b*x^2 + a)^(3/2)*a^3*c*d^2*x/b^2 + 9/256*sqrt(b*x^2 + a)*a^4*c*d^2*x/b^2 + 1/64*(b*x^2 + a)^(7/2)*a^2*d^3*
x/b^3 - 1/384*(b*x^2 + a)^(5/2)*a^3*d^3*x/b^3 - 5/1536*(b*x^2 + a)^(3/2)*a^4*d^3*x/b^3 - 5/1024*sqrt(b*x^2 + a
)*a^5*d^3*x/b^3 + 5/16*a^3*c^3*arcsinh(b*x/sqrt(a*b))/sqrt(b) - 15/128*a^4*c^2*d*arcsinh(b*x/sqrt(a*b))/b^(3/2
) + 9/256*a^5*c*d^2*arcsinh(b*x/sqrt(a*b))/b^(5/2) - 5/1024*a^6*d^3*arcsinh(b*x/sqrt(a*b))/b^(7/2)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.92 \[ \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^3 \, dx=\frac {1}{15360} \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, b^{2} d^{3} x^{2} + \frac {36 \, b^{12} c d^{2} + 25 \, a b^{11} d^{3}}{b^{10}}\right )} x^{2} + \frac {9 \, {\left (40 \, b^{12} c^{2} d + 84 \, a b^{11} c d^{2} + 15 \, a^{2} b^{10} d^{3}\right )}}{b^{10}}\right )} x^{2} + \frac {320 \, b^{12} c^{3} + 2040 \, a b^{11} c^{2} d + 1116 \, a^{2} b^{10} c d^{2} + 5 \, a^{3} b^{9} d^{3}}{b^{10}}\right )} x^{2} + \frac {5 \, {\left (832 \, a b^{11} c^{3} + 1416 \, a^{2} b^{10} c^{2} d + 36 \, a^{3} b^{9} c d^{2} - 5 \, a^{4} b^{8} d^{3}\right )}}{b^{10}}\right )} x^{2} + \frac {15 \, {\left (704 \, a^{2} b^{10} c^{3} + 120 \, a^{3} b^{9} c^{2} d - 36 \, a^{4} b^{8} c d^{2} + 5 \, a^{5} b^{7} d^{3}\right )}}{b^{10}}\right )} \sqrt {b x^{2} + a} x - \frac {{\left (320 \, a^{3} b^{3} c^{3} - 120 \, a^{4} b^{2} c^{2} d + 36 \, a^{5} b c d^{2} - 5 \, a^{6} d^{3}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{1024 \, b^{\frac {7}{2}}} \]

[In]

integrate((b*x^2+a)^(5/2)*(d*x^2+c)^3,x, algorithm="giac")

[Out]

1/15360*(2*(4*(2*(8*(10*b^2*d^3*x^2 + (36*b^12*c*d^2 + 25*a*b^11*d^3)/b^10)*x^2 + 9*(40*b^12*c^2*d + 84*a*b^11
*c*d^2 + 15*a^2*b^10*d^3)/b^10)*x^2 + (320*b^12*c^3 + 2040*a*b^11*c^2*d + 1116*a^2*b^10*c*d^2 + 5*a^3*b^9*d^3)
/b^10)*x^2 + 5*(832*a*b^11*c^3 + 1416*a^2*b^10*c^2*d + 36*a^3*b^9*c*d^2 - 5*a^4*b^8*d^3)/b^10)*x^2 + 15*(704*a
^2*b^10*c^3 + 120*a^3*b^9*c^2*d - 36*a^4*b^8*c*d^2 + 5*a^5*b^7*d^3)/b^10)*sqrt(b*x^2 + a)*x - 1/1024*(320*a^3*
b^3*c^3 - 120*a^4*b^2*c^2*d + 36*a^5*b*c*d^2 - 5*a^6*d^3)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(7/2)

Mupad [F(-1)]

Timed out. \[ \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^3 \, dx=\int {\left (b\,x^2+a\right )}^{5/2}\,{\left (d\,x^2+c\right )}^3 \,d x \]

[In]

int((a + b*x^2)^(5/2)*(c + d*x^2)^3,x)

[Out]

int((a + b*x^2)^(5/2)*(c + d*x^2)^3, x)