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

Optimal. Leaf size=349 \[ \frac {d x \left (a+b x^2\right )^{7/2} \left (15 a^2 d^2-68 a b c d+152 b^2 c^2\right )}{960 b^3}+\frac {x \left (a+b x^2\right )^{5/2} \left (-5 a^3 d^3+36 a^2 b c d^2-120 a b^2 c^2 d+320 b^3 c^3\right )}{1920 b^3}+\frac {a x \left (a+b x^2\right )^{3/2} \left (-5 a^3 d^3+36 a^2 b c d^2-120 a b^2 c^2 d+320 b^3 c^3\right )}{1536 b^3}+\frac {a^2 x \sqrt {a+b x^2} \left (-5 a^3 d^3+36 a^2 b c d^2-120 a b^2 c^2 d+320 b^3 c^3\right )}{1024 b^3}+\frac {a^3 \left (-5 a^3 d^3+36 a^2 b c d^2-120 a b^2 c^2 d+320 b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{1024 b^{7/2}}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right ) (16 b c-5 a d)}{120 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )^2}{12 b} \]

[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

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Rubi [A]  time = 0.25, antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {416, 528, 388, 195, 217, 206} \[ \frac {d x \left (a+b x^2\right )^{7/2} \left (15 a^2 d^2-68 a b c d+152 b^2 c^2\right )}{960 b^3}+\frac {x \left (a+b x^2\right )^{5/2} \left (36 a^2 b c d^2-5 a^3 d^3-120 a b^2 c^2 d+320 b^3 c^3\right )}{1920 b^3}+\frac {a x \left (a+b x^2\right )^{3/2} \left (36 a^2 b c d^2-5 a^3 d^3-120 a b^2 c^2 d+320 b^3 c^3\right )}{1536 b^3}+\frac {a^2 x \sqrt {a+b x^2} \left (36 a^2 b c d^2-5 a^3 d^3-120 a b^2 c^2 d+320 b^3 c^3\right )}{1024 b^3}+\frac {a^3 \left (36 a^2 b c d^2-5 a^3 d^3-120 a b^2 c^2 d+320 b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{1024 b^{7/2}}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right ) (16 b c-5 a d)}{120 b^2}+\frac {d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )^2}{12 b} \]

Antiderivative was successfully verified.

[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 195

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 206

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

Rule 217

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 388

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 416

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 528

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*} \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 (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 ) \operatorname {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*}

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Mathematica [A]  time = 5.18, size = 270, normalized size = 0.77 \[ \frac {\sqrt {b} x \sqrt {a+b x^2} \left (75 a^5 d^3-10 a^4 b d^2 \left (54 c+5 d x^2\right )+40 a^3 b^2 d \left (45 c^2+9 c d x^2+d^2 x^4\right )+48 a^2 b^3 \left (220 c^3+295 c^2 d x^2+186 c d^2 x^4+45 d^3 x^6\right )+64 a b^4 x^2 \left (130 c^3+255 c^2 d x^2+189 c d^2 x^4+50 d^3 x^6\right )+128 b^5 x^4 \left (20 c^3+45 c^2 d x^2+36 c d^2 x^4+10 d^3 x^6\right )\right )-15 a^3 \left (5 a^3 d^3-36 a^2 b c d^2+120 a b^2 c^2 d-320 b^3 c^3\right ) \log \left (\sqrt {b} \sqrt {a+b x^2}+b x\right )}{15360 b^{7/2}} \]

Antiderivative was successfully verified.

[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[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(15360*b^(7
/2))

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fricas [A]  time = 1.58, size = 608, normalized size = 1.74 \[ \left [-\frac {15 \, {\left (320 \, a^{3} b^{3} c^{3} - 120 \, a^{4} b^{2} c^{2} d + 36 \, a^{5} b c d^{2} - 5 \, a^{6} d^{3}\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (1280 \, b^{6} d^{3} x^{11} + 128 \, {\left (36 \, b^{6} c d^{2} + 25 \, a b^{5} d^{3}\right )} x^{9} + 144 \, {\left (40 \, b^{6} c^{2} d + 84 \, a b^{5} c d^{2} + 15 \, a^{2} b^{4} d^{3}\right )} x^{7} + 8 \, {\left (320 \, b^{6} c^{3} + 2040 \, a b^{5} c^{2} d + 1116 \, a^{2} b^{4} c d^{2} + 5 \, a^{3} b^{3} d^{3}\right )} x^{5} + 10 \, {\left (832 \, a b^{5} c^{3} + 1416 \, a^{2} b^{4} c^{2} d + 36 \, a^{3} b^{3} c d^{2} - 5 \, a^{4} b^{2} d^{3}\right )} x^{3} + 15 \, {\left (704 \, a^{2} b^{4} c^{3} + 120 \, a^{3} b^{3} c^{2} d - 36 \, a^{4} b^{2} c d^{2} + 5 \, a^{5} b d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{30720 \, b^{4}}, -\frac {15 \, {\left (320 \, a^{3} b^{3} c^{3} - 120 \, a^{4} b^{2} c^{2} d + 36 \, a^{5} b c d^{2} - 5 \, a^{6} d^{3}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (1280 \, b^{6} d^{3} x^{11} + 128 \, {\left (36 \, b^{6} c d^{2} + 25 \, a b^{5} d^{3}\right )} x^{9} + 144 \, {\left (40 \, b^{6} c^{2} d + 84 \, a b^{5} c d^{2} + 15 \, a^{2} b^{4} d^{3}\right )} x^{7} + 8 \, {\left (320 \, b^{6} c^{3} + 2040 \, a b^{5} c^{2} d + 1116 \, a^{2} b^{4} c d^{2} + 5 \, a^{3} b^{3} d^{3}\right )} x^{5} + 10 \, {\left (832 \, a b^{5} c^{3} + 1416 \, a^{2} b^{4} c^{2} d + 36 \, a^{3} b^{3} c d^{2} - 5 \, a^{4} b^{2} d^{3}\right )} x^{3} + 15 \, {\left (704 \, a^{2} b^{4} c^{3} + 120 \, a^{3} b^{3} c^{2} d - 36 \, a^{4} b^{2} c d^{2} + 5 \, a^{5} b d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{15360 \, b^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[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]

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giac [A]  time = 0.68, size = 321, normalized size = 0.92 \[ \frac {1}{15360} \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, b^{2} d^{3} x^{2} + \frac {36 \, b^{12} c d^{2} + 25 \, a b^{11} d^{3}}{b^{10}}\right )} x^{2} + \frac {9 \, {\left (40 \, b^{12} c^{2} d + 84 \, a b^{11} c d^{2} + 15 \, a^{2} b^{10} d^{3}\right )}}{b^{10}}\right )} x^{2} + \frac {320 \, b^{12} c^{3} + 2040 \, a b^{11} c^{2} d + 1116 \, a^{2} b^{10} c d^{2} + 5 \, a^{3} b^{9} d^{3}}{b^{10}}\right )} x^{2} + \frac {5 \, {\left (832 \, a b^{11} c^{3} + 1416 \, a^{2} b^{10} c^{2} d + 36 \, a^{3} b^{9} c d^{2} - 5 \, a^{4} b^{8} d^{3}\right )}}{b^{10}}\right )} x^{2} + \frac {15 \, {\left (704 \, a^{2} b^{10} c^{3} + 120 \, a^{3} b^{9} c^{2} d - 36 \, a^{4} b^{8} c d^{2} + 5 \, a^{5} b^{7} d^{3}\right )}}{b^{10}}\right )} \sqrt {b x^{2} + a} x - \frac {{\left (320 \, a^{3} b^{3} c^{3} - 120 \, a^{4} b^{2} c^{2} d + 36 \, a^{5} b c d^{2} - 5 \, a^{6} d^{3}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{1024 \, b^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.02, size = 476, normalized size = 1.36 \[ \frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} d^{3} x^{5}}{12 b}-\frac {5 a^{6} d^{3} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{1024 b^{\frac {7}{2}}}+\frac {9 a^{5} c \,d^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{256 b^{\frac {5}{2}}}-\frac {15 a^{4} c^{2} d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{128 b^{\frac {3}{2}}}+\frac {5 a^{3} c^{3} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{16 \sqrt {b}}-\frac {5 \sqrt {b \,x^{2}+a}\, a^{5} d^{3} x}{1024 b^{3}}+\frac {9 \sqrt {b \,x^{2}+a}\, a^{4} c \,d^{2} x}{256 b^{2}}-\frac {15 \sqrt {b \,x^{2}+a}\, a^{3} c^{2} d x}{128 b}+\frac {5 \sqrt {b \,x^{2}+a}\, a^{2} c^{3} x}{16}-\frac {5 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{4} d^{3} x}{1536 b^{3}}+\frac {3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{3} c \,d^{2} x}{128 b^{2}}-\frac {5 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2} c^{2} d x}{64 b}-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} a \,d^{3} x^{3}}{24 b^{2}}+\frac {5 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a \,c^{3} x}{24}+\frac {3 \left (b \,x^{2}+a \right )^{\frac {7}{2}} c \,d^{2} x^{3}}{10 b}-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} a^{3} d^{3} x}{384 b^{3}}+\frac {3 \left (b \,x^{2}+a \right )^{\frac {5}{2}} a^{2} c \,d^{2} x}{160 b^{2}}-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} a \,c^{2} d x}{16 b}+\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} c^{3} x}{6}+\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{2} d^{3} x}{64 b^{3}}-\frac {9 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a c \,d^{2} x}{80 b^{2}}+\frac {3 \left (b \,x^{2}+a \right )^{\frac {7}{2}} c^{2} d x}{8 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.47, size = 447, normalized size = 1.28 \[ \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} d^{3} x^{5}}{12 \, b} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} c d^{2} x^{3}}{10 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} a d^{3} x^{3}}{24 \, b^{2}} + \frac {1}{6} \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} c^{3} x + \frac {5}{24} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a c^{3} x + \frac {5}{16} \, \sqrt {b x^{2} + a} a^{2} c^{3} x + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} c^{2} d x}{8 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} a c^{2} d x}{16 \, b} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} c^{2} d x}{64 \, b} - \frac {15 \, \sqrt {b x^{2} + a} a^{3} c^{2} d x}{128 \, b} - \frac {9 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a c d^{2} x}{80 \, b^{2}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2} c d^{2} x}{160 \, b^{2}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} c d^{2} x}{128 \, b^{2}} + \frac {9 \, \sqrt {b x^{2} + a} a^{4} c d^{2} x}{256 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} d^{3} x}{64 \, b^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3} d^{3} x}{384 \, b^{3}} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4} d^{3} x}{1536 \, b^{3}} - \frac {5 \, \sqrt {b x^{2} + a} a^{5} d^{3} x}{1024 \, b^{3}} + \frac {5 \, a^{3} c^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, \sqrt {b}} - \frac {15 \, a^{4} c^{2} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {3}{2}}} + \frac {9 \, a^{5} c d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{256 \, b^{\frac {5}{2}}} - \frac {5 \, a^{6} d^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{1024 \, b^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (b\,x^2+a\right )}^{5/2}\,{\left (d\,x^2+c\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [B]  time = 102.67, size = 796, normalized size = 2.28 \[ \frac {5 a^{\frac {11}{2}} d^{3} x}{1024 b^{3} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {9 a^{\frac {9}{2}} c d^{2} x}{256 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {5 a^{\frac {9}{2}} d^{3} x^{3}}{3072 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {15 a^{\frac {7}{2}} c^{2} d x}{128 b \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {3 a^{\frac {7}{2}} c d^{2} x^{3}}{256 b \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {a^{\frac {7}{2}} d^{3} x^{5}}{1536 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {a^{\frac {5}{2}} c^{3} x \sqrt {1 + \frac {b x^{2}}{a}}}{2} + \frac {3 a^{\frac {5}{2}} c^{3} x}{16 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {133 a^{\frac {5}{2}} c^{2} d x^{3}}{128 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {387 a^{\frac {5}{2}} c d^{2} x^{5}}{640 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {55 a^{\frac {5}{2}} d^{3} x^{7}}{384 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {35 a^{\frac {3}{2}} b c^{3} x^{3}}{48 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {127 a^{\frac {3}{2}} b c^{2} d x^{5}}{64 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {219 a^{\frac {3}{2}} b c d^{2} x^{7}}{160 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {67 a^{\frac {3}{2}} b d^{3} x^{9}}{192 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {17 \sqrt {a} b^{2} c^{3} x^{5}}{24 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {23 \sqrt {a} b^{2} c^{2} d x^{7}}{16 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {87 \sqrt {a} b^{2} c d^{2} x^{9}}{80 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {7 \sqrt {a} b^{2} d^{3} x^{11}}{24 \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {5 a^{6} d^{3} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{1024 b^{\frac {7}{2}}} + \frac {9 a^{5} c d^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{256 b^{\frac {5}{2}}} - \frac {15 a^{4} c^{2} d \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{128 b^{\frac {3}{2}}} + \frac {5 a^{3} c^{3} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{16 \sqrt {b}} + \frac {b^{3} c^{3} x^{7}}{6 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 b^{3} c^{2} d x^{9}}{8 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 b^{3} c d^{2} x^{11}}{10 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {b^{3} d^{3} x^{13}}{12 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*a**(11/2)*d**3*x/(1024*b**3*sqrt(1 + b*x**2/a)) - 9*a**(9/2)*c*d**2*x/(256*b**2*sqrt(1 + b*x**2/a)) + 5*a**(
9/2)*d**3*x**3/(3072*b**2*sqrt(1 + b*x**2/a)) + 15*a**(7/2)*c**2*d*x/(128*b*sqrt(1 + b*x**2/a)) - 3*a**(7/2)*c
*d**2*x**3/(256*b*sqrt(1 + b*x**2/a)) - a**(7/2)*d**3*x**5/(1536*b*sqrt(1 + b*x**2/a)) + a**(5/2)*c**3*x*sqrt(
1 + b*x**2/a)/2 + 3*a**(5/2)*c**3*x/(16*sqrt(1 + b*x**2/a)) + 133*a**(5/2)*c**2*d*x**3/(128*sqrt(1 + b*x**2/a)
) + 387*a**(5/2)*c*d**2*x**5/(640*sqrt(1 + b*x**2/a)) + 55*a**(5/2)*d**3*x**7/(384*sqrt(1 + b*x**2/a)) + 35*a*
*(3/2)*b*c**3*x**3/(48*sqrt(1 + b*x**2/a)) + 127*a**(3/2)*b*c**2*d*x**5/(64*sqrt(1 + b*x**2/a)) + 219*a**(3/2)
*b*c*d**2*x**7/(160*sqrt(1 + b*x**2/a)) + 67*a**(3/2)*b*d**3*x**9/(192*sqrt(1 + b*x**2/a)) + 17*sqrt(a)*b**2*c
**3*x**5/(24*sqrt(1 + b*x**2/a)) + 23*sqrt(a)*b**2*c**2*d*x**7/(16*sqrt(1 + b*x**2/a)) + 87*sqrt(a)*b**2*c*d**
2*x**9/(80*sqrt(1 + b*x**2/a)) + 7*sqrt(a)*b**2*d**3*x**11/(24*sqrt(1 + b*x**2/a)) - 5*a**6*d**3*asinh(sqrt(b)
*x/sqrt(a))/(1024*b**(7/2)) + 9*a**5*c*d**2*asinh(sqrt(b)*x/sqrt(a))/(256*b**(5/2)) - 15*a**4*c**2*d*asinh(sqr
t(b)*x/sqrt(a))/(128*b**(3/2)) + 5*a**3*c**3*asinh(sqrt(b)*x/sqrt(a))/(16*sqrt(b)) + b**3*c**3*x**7/(6*sqrt(a)
*sqrt(1 + b*x**2/a)) + 3*b**3*c**2*d*x**9/(8*sqrt(a)*sqrt(1 + b*x**2/a)) + 3*b**3*c*d**2*x**11/(10*sqrt(a)*sqr
t(1 + b*x**2/a)) + b**3*d**3*x**13/(12*sqrt(a)*sqrt(1 + b*x**2/a))

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