\(\int (a+b x^2)^{7/4} (c+d x^2)^3 \, dx\) [420]

Optimal result

Integrand size = 21, antiderivative size = 361 \[ \int \left (a+b x^2\right )^{7/4} \left (c+d x^2\right )^3 \, dx=\frac {2 a^2 \left (1547 b^3 c^3-2 a d \left (357 b^2 c^2-126 a b c d+20 a^2 d^2\right )\right ) x}{3315 b^3 \sqrt [4]{a+b x^2}}+\frac {2 a \left (1547 b^3 c^3-2 a d \left (357 b^2 c^2-126 a b c d+20 a^2 d^2\right )\right ) x \left (a+b x^2\right )^{3/4}}{9945 b^3}+\frac {2 \left (1547 c^3-\frac {2 a d \left (357 b^2 c^2-126 a b c d+20 a^2 d^2\right )}{b^3}\right ) x \left (a+b x^2\right )^{7/4}}{13923}+\frac {2 d \left (357 b^2 c^2-126 a b c d+20 a^2 d^2\right ) x \left (a+b x^2\right )^{11/4}}{1547 b^3}+\frac {2 d^2 (63 b c-10 a d) x^3 \left (a+b x^2\right )^{11/4}}{357 b^2}+\frac {2 d^3 x^5 \left (a+b x^2\right )^{11/4}}{21 b}-\frac {2 a^{5/2} \left (1547 b^3 c^3-2 a d \left (357 b^2 c^2-126 a b c d+20 a^2 d^2\right )\right ) \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{3315 b^{7/2} \sqrt [4]{a+b x^2}} \] Output:

2/3315*a^2*(1547*b^3*c^3-2*a*d*(20*a^2*d^2-126*a*b*c*d+357*b^2*c^2))*x/b^3 
/(b*x^2+a)^(1/4)+2/9945*a*(1547*b^3*c^3-2*a*d*(20*a^2*d^2-126*a*b*c*d+357* 
b^2*c^2))*x*(b*x^2+a)^(3/4)/b^3+2/13923*(1547*c^3-2*a*d*(20*a^2*d^2-126*a* 
b*c*d+357*b^2*c^2)/b^3)*x*(b*x^2+a)^(7/4)+2/1547*d*(20*a^2*d^2-126*a*b*c*d 
+357*b^2*c^2)*x*(b*x^2+a)^(11/4)/b^3+2/357*d^2*(-10*a*d+63*b*c)*x^3*(b*x^2 
+a)^(11/4)/b^2+2/21*d^3*x^5*(b*x^2+a)^(11/4)/b-2/3315*a^(5/2)*(1547*b^3*c^ 
3-2*a*d*(20*a^2*d^2-126*a*b*c*d+357*b^2*c^2))*(1+b*x^2/a)^(1/4)*EllipticE( 
sin(1/2*arctan(b^(1/2)*x/a^(1/2))),2^(1/2))/b^(7/2)/(b*x^2+a)^(1/4)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 14.61 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.66 \[ \int \left (a+b x^2\right )^{7/4} \left (c+d x^2\right )^3 \, dx=\frac {x \left (2 \left (a+b x^2\right ) \left (420 a^4 d^3-14 a^3 b d^2 \left (189 c+25 d x^2\right )+63 a^2 b^2 d \left (119 c^2+35 c d x^2+5 d^2 x^4\right )+12 a b^3 \left (1547 c^3+2380 c^2 d x^2+1575 c d^2 x^4+390 d^3 x^6\right )+5 b^4 x^2 \left (1547 c^3+3213 c^2 d x^2+2457 c d^2 x^4+663 d^3 x^6\right )\right )+21 a^2 \left (1547 b^3 c^3-714 a b^2 c^2 d+252 a^2 b c d^2-40 a^3 d^3\right ) \sqrt [4]{1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {3}{2},-\frac {b x^2}{a}\right )\right )}{69615 b^3 \sqrt [4]{a+b x^2}} \] Input:

Integrate[(a + b*x^2)^(7/4)*(c + d*x^2)^3,x]
 

Output:

(x*(2*(a + b*x^2)*(420*a^4*d^3 - 14*a^3*b*d^2*(189*c + 25*d*x^2) + 63*a^2* 
b^2*d*(119*c^2 + 35*c*d*x^2 + 5*d^2*x^4) + 12*a*b^3*(1547*c^3 + 2380*c^2*d 
*x^2 + 1575*c*d^2*x^4 + 390*d^3*x^6) + 5*b^4*x^2*(1547*c^3 + 3213*c^2*d*x^ 
2 + 2457*c*d^2*x^4 + 663*d^3*x^6)) + 21*a^2*(1547*b^3*c^3 - 714*a*b^2*c^2* 
d + 252*a^2*b*c*d^2 - 40*a^3*d^3)*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[ 
1/4, 1/2, 3/2, -((b*x^2)/a)]))/(69615*b^3*(a + b*x^2)^(1/4))
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.80, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {318, 27, 403, 27, 299, 211, 211, 227, 225, 212}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \left (a+b x^2\right )^{7/4} \left (c+d x^2\right )^3 \, dx\)

\(\Big \downarrow \) 318

\(\displaystyle \frac {2 \int \frac {1}{2} \left (b x^2+a\right )^{7/4} \left (d x^2+c\right ) \left (d (29 b c-10 a d) x^2+c (21 b c-2 a d)\right )dx}{21 b}+\frac {2 d x \left (a+b x^2\right )^{11/4} \left (c+d x^2\right )^2}{21 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \left (b x^2+a\right )^{7/4} \left (d x^2+c\right ) \left (d (29 b c-10 a d) x^2+c (21 b c-2 a d)\right )dx}{21 b}+\frac {2 d x \left (a+b x^2\right )^{11/4} \left (c+d x^2\right )^2}{21 b}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {\frac {2 \int \frac {1}{2} \left (b x^2+a\right )^{7/4} \left (d \left (473 b^2 c^2-248 a b d c+60 a^2 d^2\right ) x^2+c \left (357 b^2 c^2-92 a b d c+20 a^2 d^2\right )\right )dx}{17 b}+\frac {2 d x \left (a+b x^2\right )^{11/4} \left (c+d x^2\right ) (29 b c-10 a d)}{17 b}}{21 b}+\frac {2 d x \left (a+b x^2\right )^{11/4} \left (c+d x^2\right )^2}{21 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \left (b x^2+a\right )^{7/4} \left (d \left (473 b^2 c^2-248 a b d c+60 a^2 d^2\right ) x^2+c \left (357 b^2 c^2-92 a b d c+20 a^2 d^2\right )\right )dx}{17 b}+\frac {2 d x \left (a+b x^2\right )^{11/4} \left (c+d x^2\right ) (29 b c-10 a d)}{17 b}}{21 b}+\frac {2 d x \left (a+b x^2\right )^{11/4} \left (c+d x^2\right )^2}{21 b}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {\frac {\frac {3 \left (-40 a^3 d^3+252 a^2 b c d^2-714 a b^2 c^2 d+1547 b^3 c^3\right ) \int \left (b x^2+a\right )^{7/4}dx}{13 b}+\frac {2 d x \left (a+b x^2\right )^{11/4} \left (60 a^2 d^2-248 a b c d+473 b^2 c^2\right )}{13 b}}{17 b}+\frac {2 d x \left (a+b x^2\right )^{11/4} \left (c+d x^2\right ) (29 b c-10 a d)}{17 b}}{21 b}+\frac {2 d x \left (a+b x^2\right )^{11/4} \left (c+d x^2\right )^2}{21 b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {\frac {3 \left (-40 a^3 d^3+252 a^2 b c d^2-714 a b^2 c^2 d+1547 b^3 c^3\right ) \left (\frac {7}{9} a \int \left (b x^2+a\right )^{3/4}dx+\frac {2}{9} x \left (a+b x^2\right )^{7/4}\right )}{13 b}+\frac {2 d x \left (a+b x^2\right )^{11/4} \left (60 a^2 d^2-248 a b c d+473 b^2 c^2\right )}{13 b}}{17 b}+\frac {2 d x \left (a+b x^2\right )^{11/4} \left (c+d x^2\right ) (29 b c-10 a d)}{17 b}}{21 b}+\frac {2 d x \left (a+b x^2\right )^{11/4} \left (c+d x^2\right )^2}{21 b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {\frac {3 \left (-40 a^3 d^3+252 a^2 b c d^2-714 a b^2 c^2 d+1547 b^3 c^3\right ) \left (\frac {7}{9} a \left (\frac {3}{5} a \int \frac {1}{\sqrt [4]{b x^2+a}}dx+\frac {2}{5} x \left (a+b x^2\right )^{3/4}\right )+\frac {2}{9} x \left (a+b x^2\right )^{7/4}\right )}{13 b}+\frac {2 d x \left (a+b x^2\right )^{11/4} \left (60 a^2 d^2-248 a b c d+473 b^2 c^2\right )}{13 b}}{17 b}+\frac {2 d x \left (a+b x^2\right )^{11/4} \left (c+d x^2\right ) (29 b c-10 a d)}{17 b}}{21 b}+\frac {2 d x \left (a+b x^2\right )^{11/4} \left (c+d x^2\right )^2}{21 b}\)

\(\Big \downarrow \) 227

\(\displaystyle \frac {\frac {\frac {3 \left (-40 a^3 d^3+252 a^2 b c d^2-714 a b^2 c^2 d+1547 b^3 c^3\right ) \left (\frac {7}{9} a \left (\frac {3 a \sqrt [4]{\frac {b x^2}{a}+1} \int \frac {1}{\sqrt [4]{\frac {b x^2}{a}+1}}dx}{5 \sqrt [4]{a+b x^2}}+\frac {2}{5} x \left (a+b x^2\right )^{3/4}\right )+\frac {2}{9} x \left (a+b x^2\right )^{7/4}\right )}{13 b}+\frac {2 d x \left (a+b x^2\right )^{11/4} \left (60 a^2 d^2-248 a b c d+473 b^2 c^2\right )}{13 b}}{17 b}+\frac {2 d x \left (a+b x^2\right )^{11/4} \left (c+d x^2\right ) (29 b c-10 a d)}{17 b}}{21 b}+\frac {2 d x \left (a+b x^2\right )^{11/4} \left (c+d x^2\right )^2}{21 b}\)

\(\Big \downarrow \) 225

\(\displaystyle \frac {\frac {\frac {3 \left (-40 a^3 d^3+252 a^2 b c d^2-714 a b^2 c^2 d+1547 b^3 c^3\right ) \left (\frac {7}{9} a \left (\frac {3 a \sqrt [4]{\frac {b x^2}{a}+1} \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\int \frac {1}{\left (\frac {b x^2}{a}+1\right )^{5/4}}dx\right )}{5 \sqrt [4]{a+b x^2}}+\frac {2}{5} x \left (a+b x^2\right )^{3/4}\right )+\frac {2}{9} x \left (a+b x^2\right )^{7/4}\right )}{13 b}+\frac {2 d x \left (a+b x^2\right )^{11/4} \left (60 a^2 d^2-248 a b c d+473 b^2 c^2\right )}{13 b}}{17 b}+\frac {2 d x \left (a+b x^2\right )^{11/4} \left (c+d x^2\right ) (29 b c-10 a d)}{17 b}}{21 b}+\frac {2 d x \left (a+b x^2\right )^{11/4} \left (c+d x^2\right )^2}{21 b}\)

\(\Big \downarrow \) 212

\(\displaystyle \frac {\frac {\frac {2 d x \left (a+b x^2\right )^{11/4} \left (60 a^2 d^2-248 a b c d+473 b^2 c^2\right )}{13 b}+\frac {3 \left (-40 a^3 d^3+252 a^2 b c d^2-714 a b^2 c^2 d+1547 b^3 c^3\right ) \left (\frac {7}{9} a \left (\frac {3 a \sqrt [4]{\frac {b x^2}{a}+1} \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{5 \sqrt [4]{a+b x^2}}+\frac {2}{5} x \left (a+b x^2\right )^{3/4}\right )+\frac {2}{9} x \left (a+b x^2\right )^{7/4}\right )}{13 b}}{17 b}+\frac {2 d x \left (a+b x^2\right )^{11/4} \left (c+d x^2\right ) (29 b c-10 a d)}{17 b}}{21 b}+\frac {2 d x \left (a+b x^2\right )^{11/4} \left (c+d x^2\right )^2}{21 b}\)

Input:

Int[(a + b*x^2)^(7/4)*(c + d*x^2)^3,x]
 

Output:

(2*d*x*(a + b*x^2)^(11/4)*(c + d*x^2)^2)/(21*b) + ((2*d*(29*b*c - 10*a*d)* 
x*(a + b*x^2)^(11/4)*(c + d*x^2))/(17*b) + ((2*d*(473*b^2*c^2 - 248*a*b*c* 
d + 60*a^2*d^2)*x*(a + b*x^2)^(11/4))/(13*b) + (3*(1547*b^3*c^3 - 714*a*b^ 
2*c^2*d + 252*a^2*b*c*d^2 - 40*a^3*d^3)*((2*x*(a + b*x^2)^(7/4))/9 + (7*a* 
((2*x*(a + b*x^2)^(3/4))/5 + (3*a*(1 + (b*x^2)/a)^(1/4)*((2*x)/(1 + (b*x^2 
)/a)^(1/4) - (2*Sqrt[a]*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/Sqrt[ 
b]))/(5*(a + b*x^2)^(1/4))))/9))/(13*b))/(17*b))/(21*b)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 212

Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2/(a^(5/4)*Rt[b/a, 2]) 
)*EllipticE[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a 
, 0] && PosQ[b/a]
 

rule 225

Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[2*(x/(a + b*x^2)^(1/4)) 
, x] - Simp[a   Int[1/(a + b*x^2)^(5/4), x], x] /; FreeQ[{a, b}, x] && GtQ[ 
a, 0] && PosQ[b/a]
 

rule 227

Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(1 + b*(x^2/a))^(1/4)/( 
a + b*x^2)^(1/4)   Int[1/(1 + b*(x^2/a))^(1/4), x], x] /; FreeQ[{a, b}, x] 
&& PosQ[a]
 

rule 299

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

rule 318

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S 
imp[1/(b*(2*(p + q) + 1))   Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b 
*c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 
 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G 
tQ[q, 1] && NeQ[2*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntBinomialQ[a, b, c, 
d, 2, p, q, x]
 

rule 403

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

Maple [F]

\[\int \left (b \,x^{2}+a \right )^{\frac {7}{4}} \left (x^{2} d +c \right )^{3}d x\]

Input:

int((b*x^2+a)^(7/4)*(d*x^2+c)^3,x)
 

Output:

int((b*x^2+a)^(7/4)*(d*x^2+c)^3,x)
 

Fricas [F]

\[ \int \left (a+b x^2\right )^{7/4} \left (c+d x^2\right )^3 \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {7}{4}} {\left (d x^{2} + c\right )}^{3} \,d x } \] Input:

integrate((b*x^2+a)^(7/4)*(d*x^2+c)^3,x, algorithm="fricas")
 

Output:

integral((b*d^3*x^8 + (3*b*c*d^2 + a*d^3)*x^6 + 3*(b*c^2*d + a*c*d^2)*x^4 
+ a*c^3 + (b*c^3 + 3*a*c^2*d)*x^2)*(b*x^2 + a)^(3/4), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.63 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.79 \[ \int \left (a+b x^2\right )^{7/4} \left (c+d x^2\right )^3 \, dx=a^{\frac {7}{4}} c^{3} x {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )} + a^{\frac {7}{4}} c^{2} d x^{3} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )} + \frac {3 a^{\frac {7}{4}} c d^{2} x^{5} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{5} + \frac {a^{\frac {7}{4}} d^{3} x^{7} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{7} + \frac {a^{\frac {3}{4}} b c^{3} x^{3} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{3} + \frac {3 a^{\frac {3}{4}} b c^{2} d x^{5} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{5} + \frac {3 a^{\frac {3}{4}} b c d^{2} x^{7} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{7} + \frac {a^{\frac {3}{4}} b d^{3} x^{9} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {9}{2} \\ \frac {11}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{9} \] Input:

integrate((b*x**2+a)**(7/4)*(d*x**2+c)**3,x)
 

Output:

a**(7/4)*c**3*x*hyper((-3/4, 1/2), (3/2,), b*x**2*exp_polar(I*pi)/a) + a** 
(7/4)*c**2*d*x**3*hyper((-3/4, 3/2), (5/2,), b*x**2*exp_polar(I*pi)/a) + 3 
*a**(7/4)*c*d**2*x**5*hyper((-3/4, 5/2), (7/2,), b*x**2*exp_polar(I*pi)/a) 
/5 + a**(7/4)*d**3*x**7*hyper((-3/4, 7/2), (9/2,), b*x**2*exp_polar(I*pi)/ 
a)/7 + a**(3/4)*b*c**3*x**3*hyper((-3/4, 3/2), (5/2,), b*x**2*exp_polar(I* 
pi)/a)/3 + 3*a**(3/4)*b*c**2*d*x**5*hyper((-3/4, 5/2), (7/2,), b*x**2*exp_ 
polar(I*pi)/a)/5 + 3*a**(3/4)*b*c*d**2*x**7*hyper((-3/4, 7/2), (9/2,), b*x 
**2*exp_polar(I*pi)/a)/7 + a**(3/4)*b*d**3*x**9*hyper((-3/4, 9/2), (11/2,) 
, b*x**2*exp_polar(I*pi)/a)/9
 

Maxima [F]

\[ \int \left (a+b x^2\right )^{7/4} \left (c+d x^2\right )^3 \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {7}{4}} {\left (d x^{2} + c\right )}^{3} \,d x } \] Input:

integrate((b*x^2+a)^(7/4)*(d*x^2+c)^3,x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^(7/4)*(d*x^2 + c)^3, x)
 

Giac [F]

\[ \int \left (a+b x^2\right )^{7/4} \left (c+d x^2\right )^3 \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {7}{4}} {\left (d x^{2} + c\right )}^{3} \,d x } \] Input:

integrate((b*x^2+a)^(7/4)*(d*x^2+c)^3,x, algorithm="giac")
 

Output:

integrate((b*x^2 + a)^(7/4)*(d*x^2 + c)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (a+b x^2\right )^{7/4} \left (c+d x^2\right )^3 \, dx=\int {\left (b\,x^2+a\right )}^{7/4}\,{\left (d\,x^2+c\right )}^3 \,d x \] Input:

int((a + b*x^2)^(7/4)*(c + d*x^2)^3,x)
 

Output:

int((a + b*x^2)^(7/4)*(c + d*x^2)^3, x)
 

Reduce [F]

\[ \int \left (a+b x^2\right )^{7/4} \left (c+d x^2\right )^3 \, dx=\frac {840 \left (b \,x^{2}+a \right )^{\frac {3}{4}} a^{4} d^{3} x -5292 \left (b \,x^{2}+a \right )^{\frac {3}{4}} a^{3} b c \,d^{2} x -700 \left (b \,x^{2}+a \right )^{\frac {3}{4}} a^{3} b \,d^{3} x^{3}+14994 \left (b \,x^{2}+a \right )^{\frac {3}{4}} a^{2} b^{2} c^{2} d x +4410 \left (b \,x^{2}+a \right )^{\frac {3}{4}} a^{2} b^{2} c \,d^{2} x^{3}+630 \left (b \,x^{2}+a \right )^{\frac {3}{4}} a^{2} b^{2} d^{3} x^{5}+37128 \left (b \,x^{2}+a \right )^{\frac {3}{4}} a \,b^{3} c^{3} x +57120 \left (b \,x^{2}+a \right )^{\frac {3}{4}} a \,b^{3} c^{2} d \,x^{3}+37800 \left (b \,x^{2}+a \right )^{\frac {3}{4}} a \,b^{3} c \,d^{2} x^{5}+9360 \left (b \,x^{2}+a \right )^{\frac {3}{4}} a \,b^{3} d^{3} x^{7}+15470 \left (b \,x^{2}+a \right )^{\frac {3}{4}} b^{4} c^{3} x^{3}+32130 \left (b \,x^{2}+a \right )^{\frac {3}{4}} b^{4} c^{2} d \,x^{5}+24570 \left (b \,x^{2}+a \right )^{\frac {3}{4}} b^{4} c \,d^{2} x^{7}+6630 \left (b \,x^{2}+a \right )^{\frac {3}{4}} b^{4} d^{3} x^{9}-840 \left (\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {1}{4}}}d x \right ) a^{5} d^{3}+5292 \left (\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {1}{4}}}d x \right ) a^{4} b c \,d^{2}-14994 \left (\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {1}{4}}}d x \right ) a^{3} b^{2} c^{2} d +32487 \left (\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {1}{4}}}d x \right ) a^{2} b^{3} c^{3}}{69615 b^{3}} \] Input:

int((b*x^2+a)^(7/4)*(d*x^2+c)^3,x)
 

Output:

(840*(a + b*x**2)**(3/4)*a**4*d**3*x - 5292*(a + b*x**2)**(3/4)*a**3*b*c*d 
**2*x - 700*(a + b*x**2)**(3/4)*a**3*b*d**3*x**3 + 14994*(a + b*x**2)**(3/ 
4)*a**2*b**2*c**2*d*x + 4410*(a + b*x**2)**(3/4)*a**2*b**2*c*d**2*x**3 + 6 
30*(a + b*x**2)**(3/4)*a**2*b**2*d**3*x**5 + 37128*(a + b*x**2)**(3/4)*a*b 
**3*c**3*x + 57120*(a + b*x**2)**(3/4)*a*b**3*c**2*d*x**3 + 37800*(a + b*x 
**2)**(3/4)*a*b**3*c*d**2*x**5 + 9360*(a + b*x**2)**(3/4)*a*b**3*d**3*x**7 
 + 15470*(a + b*x**2)**(3/4)*b**4*c**3*x**3 + 32130*(a + b*x**2)**(3/4)*b* 
*4*c**2*d*x**5 + 24570*(a + b*x**2)**(3/4)*b**4*c*d**2*x**7 + 6630*(a + b* 
x**2)**(3/4)*b**4*d**3*x**9 - 840*int((a + b*x**2)**(3/4)/(a + b*x**2),x)* 
a**5*d**3 + 5292*int((a + b*x**2)**(3/4)/(a + b*x**2),x)*a**4*b*c*d**2 - 1 
4994*int((a + b*x**2)**(3/4)/(a + b*x**2),x)*a**3*b**2*c**2*d + 32487*int( 
(a + b*x**2)**(3/4)/(a + b*x**2),x)*a**2*b**3*c**3)/(69615*b**3)