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

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

[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))

________________________________________________________________________________________

Rubi [A]  time = 0.247141, antiderivative size = 349, normalized size of antiderivative = 1., 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 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]

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 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 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 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])

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*}

Mathematica [A]  time = 5.17283, size = 270, normalized size = 0.77 \[ \frac{\sqrt{b} x \sqrt{a+b x^2} \left (40 a^3 b^2 d \left (45 c^2+9 c d x^2+d^2 x^4\right )+48 a^2 b^3 \left (295 c^2 d x^2+220 c^3+186 c d^2 x^4+45 d^3 x^6\right )-10 a^4 b d^2 \left (54 c+5 d x^2\right )+75 a^5 d^3+64 a b^4 x^2 \left (255 c^2 d x^2+130 c^3+189 c d^2 x^4+50 d^3 x^6\right )+128 b^5 x^4 \left (45 c^2 d x^2+20 c^3+36 c d^2 x^4+10 d^3 x^6\right )\right )-15 a^3 \left (-36 a^2 b c d^2+5 a^3 d^3+120 a b^2 c^2 d-320 b^3 c^3\right ) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{15360 b^{7/2}} \]

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))

________________________________________________________________________________________

Maple [A]  time = 0.015, size = 476, normalized size = 1.4 \begin{align*}{\frac{{d}^{3}{x}^{5}}{12\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{a{d}^{3}{x}^{3}}{24\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{{d}^{3}{a}^{2}x}{64\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{{a}^{3}{d}^{3}x}{384\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{d}^{3}{a}^{4}x}{1536\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{d}^{3}{a}^{5}x}{1024\,{b}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{5\,{d}^{3}{a}^{6}}{1024}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}}+{\frac{3\,c{d}^{2}{x}^{3}}{10\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{9\,c{d}^{2}ax}{80\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{3\,{a}^{2}c{d}^{2}x}{160\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{3\,c{d}^{2}{a}^{3}x}{128\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{9\,c{d}^{2}{a}^{4}x}{256\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{9\,c{d}^{2}{a}^{5}}{256}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}}+{\frac{3\,{c}^{2}dx}{8\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{a{c}^{2}dx}{16\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{c}^{2}d{a}^{2}x}{64\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{15\,{c}^{2}d{a}^{3}x}{128\,b}\sqrt{b{x}^{2}+a}}-{\frac{15\,{c}^{2}d{a}^{4}}{128}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{{c}^{3}x}{6} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,a{c}^{3}x}{24} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}{c}^{3}x}{16}\sqrt{b{x}^{2}+a}}+{\frac{5\,{c}^{3}{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}} \end{align*}

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 4.51186, size = 1385, normalized size = 3.97 \begin{align*} \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 ] \end{align*}

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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Sympy [B]  time = 87.8307, size = 796, normalized size = 2.28 \begin{align*} \text{result too large to display} \end{align*}

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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Giac [A]  time = 1.18821, size = 433, normalized size = 1.24 \begin{align*} \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}}} \end{align*}

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)