Integrand size = 19, antiderivative size = 414 \[ \int (d+e x)^3 \left (a+c x^4\right )^{3/2} \, dx=\frac {9}{16} a d^2 e x^2 \sqrt {a+c x^4}+\frac {4 a^2 d e^2 x \sqrt {a+c x^4}}{5 \sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {2}{35} a d x \left (5 d^2+7 e^2 x^2\right ) \sqrt {a+c x^4}+\frac {3}{8} d^2 e x^2 \left (a+c x^4\right )^{3/2}+\frac {1}{21} d x \left (3 d^2+7 e^2 x^2\right ) \left (a+c x^4\right )^{3/2}+\frac {e^3 \left (a+c x^4\right )^{5/2}}{10 c}+\frac {9 a^2 d^2 e \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{16 \sqrt {c}}-\frac {4 a^{9/4} d e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 c^{3/4} \sqrt {a+c x^4}}+\frac {2 a^{7/4} d \left (5 \sqrt {c} d^2+7 \sqrt {a} e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{35 c^{3/4} \sqrt {a+c x^4}} \] Output:
9/16*a*d^2*e*x^2*(c*x^4+a)^(1/2)+4/5*a^2*d*e^2*x*(c*x^4+a)^(1/2)/c^(1/2)/( a^(1/2)+c^(1/2)*x^2)+2/35*a*d*x*(7*e^2*x^2+5*d^2)*(c*x^4+a)^(1/2)+3/8*d^2* e*x^2*(c*x^4+a)^(3/2)+1/21*d*x*(7*e^2*x^2+3*d^2)*(c*x^4+a)^(3/2)+1/10*e^3* (c*x^4+a)^(5/2)/c+9/16*a^2*d^2*e*arctanh(c^(1/2)*x^2/(c*x^4+a)^(1/2))/c^(1 /2)-4/5*a^(9/4)*d*e^2*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^(1/2)*x^ 2)^2)^(1/2)*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*2^(1/2))/c^(3/4 )/(c*x^4+a)^(1/2)+2/35*a^(7/4)*d*(5*c^(1/2)*d^2+7*a^(1/2)*e^2)*(a^(1/2)+c^ (1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arc tan(c^(1/4)*x/a^(1/4)),1/2*2^(1/2))/c^(3/4)/(c*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.60 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.45 \[ \int (d+e x)^3 \left (a+c x^4\right )^{3/2} \, dx=\frac {1}{80} \sqrt {a+c x^4} \left (\frac {8 e^3 \left (a+c x^4\right )^2}{c}+15 d^2 e \left (5 a x^2+2 c x^6+\frac {3 a^{5/2} \sqrt {1+\frac {c x^4}{a}} \text {arcsinh}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{\sqrt {c} \left (a+c x^4\right )}\right )+\frac {80 a d^3 x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},-\frac {c x^4}{a}\right )}{\sqrt {1+\frac {c x^4}{a}}}+\frac {80 a d e^2 x^3 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^4}{a}\right )}{\sqrt {1+\frac {c x^4}{a}}}\right ) \] Input:
Integrate[(d + e*x)^3*(a + c*x^4)^(3/2),x]
Output:
(Sqrt[a + c*x^4]*((8*e^3*(a + c*x^4)^2)/c + 15*d^2*e*(5*a*x^2 + 2*c*x^6 + (3*a^(5/2)*Sqrt[1 + (c*x^4)/a]*ArcSinh[(Sqrt[c]*x^2)/Sqrt[a]])/(Sqrt[c]*(a + c*x^4))) + (80*a*d^3*x*Hypergeometric2F1[-3/2, 1/4, 5/4, -((c*x^4)/a)]) /Sqrt[1 + (c*x^4)/a] + (80*a*d*e^2*x^3*Hypergeometric2F1[-3/2, 3/4, 7/4, - ((c*x^4)/a)])/Sqrt[1 + (c*x^4)/a]))/80
Time = 0.89 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2424, 2009}
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.
| \(\displaystyle \int \left (a+c x^4\right )^{3/2} (d+e x)^3 \, dx\) |
| \(\Big \downarrow \) 2424 |
| \(\displaystyle \int \left (\left (a+c x^4\right )^{3/2} \left (d^3+3 d e^2 x^2\right )+x \left (a+c x^4\right )^{3/2} \left (3 d^2 e+e^3 x^2\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 a^{7/4} d \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (7 \sqrt {a} e^2+5 \sqrt {c} d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{35 c^{3/4} \sqrt {a+c x^4}}-\frac {4 a^{9/4} d e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 c^{3/4} \sqrt {a+c x^4}}+\frac {9 a^2 d^2 e \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{16 \sqrt {c}}+\frac {4 a^2 d e^2 x \sqrt {a+c x^4}}{5 \sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {1}{21} d x \left (a+c x^4\right )^{3/2} \left (3 d^2+7 e^2 x^2\right )+\frac {2}{35} a d x \sqrt {a+c x^4} \left (5 d^2+7 e^2 x^2\right )+\frac {3}{8} d^2 e x^2 \left (a+c x^4\right )^{3/2}+\frac {9}{16} a d^2 e x^2 \sqrt {a+c x^4}+\frac {e^3 \left (a+c x^4\right )^{5/2}}{10 c}\) |
Input:
Int[(d + e*x)^3*(a + c*x^4)^(3/2),x]
Output:
(9*a*d^2*e*x^2*Sqrt[a + c*x^4])/16 + (4*a^2*d*e^2*x*Sqrt[a + c*x^4])/(5*Sq rt[c]*(Sqrt[a] + Sqrt[c]*x^2)) + (2*a*d*x*(5*d^2 + 7*e^2*x^2)*Sqrt[a + c*x ^4])/35 + (3*d^2*e*x^2*(a + c*x^4)^(3/2))/8 + (d*x*(3*d^2 + 7*e^2*x^2)*(a + c*x^4)^(3/2))/21 + (e^3*(a + c*x^4)^(5/2))/(10*c) + (9*a^2*d^2*e*ArcTanh [(Sqrt[c]*x^2)/Sqrt[a + c*x^4]])/(16*Sqrt[c]) - (4*a^(9/4)*d*e^2*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*Arc Tan[(c^(1/4)*x)/a^(1/4)], 1/2])/(5*c^(3/4)*Sqrt[a + c*x^4]) + (2*a^(7/4)*d *(5*Sqrt[c]*d^2 + 7*Sqrt[a]*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/ (Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/ (35*c^(3/4)*Sqrt[a + c*x^4])
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], j, k}, Int[Sum[x^j*Sum[Coeff[Pq, x, j + k*(n/2)]*x^(k*(n/2)), {k, 0, 2 *((q - j)/n) + 1}]*(a + b*x^n)^p, {j, 0, n/2 - 1}], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && !PolyQ[Pq, x^(n/2)]
Result contains complex when optimal does not.
Time = 1.28 (sec) , antiderivative size = 315, normalized size of antiderivative = 0.76
| method | result | size |
| risch | \(\frac {\left (168 x^{8} e^{3} c^{2}+560 d \,e^{2} x^{7} c^{2}+630 d^{2} e \,x^{6} c^{2}+240 d^{3} x^{5} c^{2}+336 x^{4} a \,e^{3} c +1232 a \,x^{3} c d \,e^{2}+1575 a \,d^{2} e \,x^{2} c +720 a \,d^{3} x c +168 a^{2} e^{3}\right ) \sqrt {c \,x^{4}+a}}{1680 c}+\frac {a^{2} d \left (\frac {160 d^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {224 i e^{2} \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}+\frac {315 d e \ln \left (\sqrt {c}\, x^{2}+\sqrt {c \,x^{4}+a}\right )}{2 \sqrt {c}}\right )}{280}\) | \(315\) |
| default | \(d^{3} \left (\frac {c \,x^{5} \sqrt {c \,x^{4}+a}}{7}+\frac {3 a x \sqrt {c \,x^{4}+a}}{7}+\frac {4 a^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+\frac {e^{3} \left (c \,x^{4}+a \right )^{\frac {5}{2}}}{10 c}+3 d \,e^{2} \left (\frac {c \,x^{7} \sqrt {c \,x^{4}+a}}{9}+\frac {11 a \,x^{3} \sqrt {c \,x^{4}+a}}{45}+\frac {4 i a^{\frac {5}{2}} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\right )+3 d^{2} e \left (\frac {3 a^{2} \ln \left (\sqrt {c}\, x^{2}+\sqrt {c \,x^{4}+a}\right )}{16 \sqrt {c}}+\frac {c \,x^{6} \sqrt {c \,x^{4}+a}}{8}+\frac {5 a \,x^{2} \sqrt {c \,x^{4}+a}}{16}\right )\) | \(321\) |
| elliptic | \(\frac {e^{3} c \,x^{8} \sqrt {c \,x^{4}+a}}{10}+\frac {c d \,e^{2} x^{7} \sqrt {c \,x^{4}+a}}{3}+\frac {3 d^{2} e c \,x^{6} \sqrt {c \,x^{4}+a}}{8}+\frac {c \,d^{3} x^{5} \sqrt {c \,x^{4}+a}}{7}+\frac {e^{3} a \,x^{4} \sqrt {c \,x^{4}+a}}{5}+\frac {11 a d \,e^{2} x^{3} \sqrt {c \,x^{4}+a}}{15}+\frac {15 a \,d^{2} e \,x^{2} \sqrt {c \,x^{4}+a}}{16}+\frac {3 d^{3} a x \sqrt {c \,x^{4}+a}}{7}+\frac {a^{2} e^{3} \sqrt {c \,x^{4}+a}}{10 c}+\frac {4 d^{3} a^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {9 d^{2} e \,a^{2} \ln \left (2 \sqrt {c}\, x^{2}+2 \sqrt {c \,x^{4}+a}\right )}{16 \sqrt {c}}+\frac {4 i a^{\frac {5}{2}} d \,e^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\) | \(377\) |
Input:
int((e*x+d)^3*(c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/1680*(168*c^2*e^3*x^8+560*c^2*d*e^2*x^7+630*c^2*d^2*e*x^6+240*c^2*d^3*x^ 5+336*a*c*e^3*x^4+1232*a*c*d*e^2*x^3+1575*a*c*d^2*e*x^2+720*a*c*d^3*x+168* a^2*e^3)/c*(c*x^4+a)^(1/2)+1/280*a^2*d*(160*d^2/(I*c^(1/2)/a^(1/2))^(1/2)* (1-I*c^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^ (1/2)*EllipticF(x*(I*c^(1/2)/a^(1/2))^(1/2),I)+224*I*e^2*a^(1/2)/(I*c^(1/2 )/a^(1/2))^(1/2)*(1-I*c^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a^(1/2)) ^(1/2)/(c*x^4+a)^(1/2)/c^(1/2)*(EllipticF(x*(I*c^(1/2)/a^(1/2))^(1/2),I)-E llipticE(x*(I*c^(1/2)/a^(1/2))^(1/2),I))+315/2*d*e*ln(c^(1/2)*x^2+(c*x^4+a )^(1/2))/c^(1/2))
Time = 0.18 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.60 \[ \int (d+e x)^3 \left (a+c x^4\right )^{3/2} \, dx=\frac {2688 \, a^{2} \sqrt {c} d e^{2} x \left (-\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + 945 \, a^{2} \sqrt {c} d^{2} e x \log \left (-2 \, c x^{4} - 2 \, \sqrt {c x^{4} + a} \sqrt {c} x^{2} - a\right ) + 384 \, {\left (5 \, a c d^{3} - 7 \, a^{2} d e^{2}\right )} \sqrt {c} x \left (-\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + 2 \, {\left (168 \, c^{2} e^{3} x^{9} + 560 \, c^{2} d e^{2} x^{8} + 630 \, c^{2} d^{2} e x^{7} + 240 \, c^{2} d^{3} x^{6} + 336 \, a c e^{3} x^{5} + 1232 \, a c d e^{2} x^{4} + 1575 \, a c d^{2} e x^{3} + 720 \, a c d^{3} x^{2} + 168 \, a^{2} e^{3} x + 1344 \, a^{2} d e^{2}\right )} \sqrt {c x^{4} + a}}{3360 \, c x} \] Input:
integrate((e*x+d)^3*(c*x^4+a)^(3/2),x, algorithm="fricas")
Output:
1/3360*(2688*a^2*sqrt(c)*d*e^2*x*(-a/c)^(3/4)*elliptic_e(arcsin((-a/c)^(1/ 4)/x), -1) + 945*a^2*sqrt(c)*d^2*e*x*log(-2*c*x^4 - 2*sqrt(c*x^4 + a)*sqrt (c)*x^2 - a) + 384*(5*a*c*d^3 - 7*a^2*d*e^2)*sqrt(c)*x*(-a/c)^(3/4)*ellipt ic_f(arcsin((-a/c)^(1/4)/x), -1) + 2*(168*c^2*e^3*x^9 + 560*c^2*d*e^2*x^8 + 630*c^2*d^2*e*x^7 + 240*c^2*d^3*x^6 + 336*a*c*e^3*x^5 + 1232*a*c*d*e^2*x ^4 + 1575*a*c*d^2*e*x^3 + 720*a*c*d^3*x^2 + 168*a^2*e^3*x + 1344*a^2*d*e^2 )*sqrt(c*x^4 + a))/(c*x)
Time = 5.78 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.05 \[ \int (d+e x)^3 \left (a+c x^4\right )^{3/2} \, dx=\frac {a^{\frac {3}{2}} d^{3} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {3 a^{\frac {3}{2}} d^{2} e x^{2} \sqrt {1 + \frac {c x^{4}}{a}}}{4} + \frac {3 a^{\frac {3}{2}} d^{2} e x^{2}}{16 \sqrt {1 + \frac {c x^{4}}{a}}} + \frac {3 a^{\frac {3}{2}} d e^{2} x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {\sqrt {a} c d^{3} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {9 \sqrt {a} c d^{2} e x^{6}}{16 \sqrt {1 + \frac {c x^{4}}{a}}} + \frac {3 \sqrt {a} c d e^{2} x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} + \frac {9 a^{2} d^{2} e \operatorname {asinh}{\left (\frac {\sqrt {c} x^{2}}{\sqrt {a}} \right )}}{16 \sqrt {c}} + a e^{3} \left (\begin {cases} \frac {\sqrt {a} x^{4}}{4} & \text {for}\: c = 0 \\\frac {\left (a + c x^{4}\right )^{\frac {3}{2}}}{6 c} & \text {otherwise} \end {cases}\right ) + c e^{3} \left (\begin {cases} - \frac {a^{2} \sqrt {a + c x^{4}}}{15 c^{2}} + \frac {a x^{4} \sqrt {a + c x^{4}}}{30 c} + \frac {x^{8} \sqrt {a + c x^{4}}}{10} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{8}}{8} & \text {otherwise} \end {cases}\right ) + \frac {3 c^{2} d^{2} e x^{10}}{8 \sqrt {a} \sqrt {1 + \frac {c x^{4}}{a}}} \] Input:
integrate((e*x+d)**3*(c*x**4+a)**(3/2),x)
Output:
a**(3/2)*d**3*x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), c*x**4*exp_polar(I*p i)/a)/(4*gamma(5/4)) + 3*a**(3/2)*d**2*e*x**2*sqrt(1 + c*x**4/a)/4 + 3*a** (3/2)*d**2*e*x**2/(16*sqrt(1 + c*x**4/a)) + 3*a**(3/2)*d*e**2*x**3*gamma(3 /4)*hyper((-1/2, 3/4), (7/4,), c*x**4*exp_polar(I*pi)/a)/(4*gamma(7/4)) + sqrt(a)*c*d**3*x**5*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), c*x**4*exp_polar (I*pi)/a)/(4*gamma(9/4)) + 9*sqrt(a)*c*d**2*e*x**6/(16*sqrt(1 + c*x**4/a)) + 3*sqrt(a)*c*d*e**2*x**7*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), c*x**4*e xp_polar(I*pi)/a)/(4*gamma(11/4)) + 9*a**2*d**2*e*asinh(sqrt(c)*x**2/sqrt( a))/(16*sqrt(c)) + a*e**3*Piecewise((sqrt(a)*x**4/4, Eq(c, 0)), ((a + c*x* *4)**(3/2)/(6*c), True)) + c*e**3*Piecewise((-a**2*sqrt(a + c*x**4)/(15*c* *2) + a*x**4*sqrt(a + c*x**4)/(30*c) + x**8*sqrt(a + c*x**4)/10, Ne(c, 0)) , (sqrt(a)*x**8/8, True)) + 3*c**2*d**2*e*x**10/(8*sqrt(a)*sqrt(1 + c*x**4 /a))
\[ \int (d+e x)^3 \left (a+c x^4\right )^{3/2} \, dx=\int { {\left (c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{3} \,d x } \] Input:
integrate((e*x+d)^3*(c*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate((c*x^4 + a)^(3/2)*(e*x + d)^3, x)
\[ \int (d+e x)^3 \left (a+c x^4\right )^{3/2} \, dx=\int { {\left (c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{3} \,d x } \] Input:
integrate((e*x+d)^3*(c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((c*x^4 + a)^(3/2)*(e*x + d)^3, x)
Timed out. \[ \int (d+e x)^3 \left (a+c x^4\right )^{3/2} \, dx=\int {\left (c\,x^4+a\right )}^{3/2}\,{\left (d+e\,x\right )}^3 \,d x \] Input:
int((a + c*x^4)^(3/2)*(d + e*x)^3,x)
Output:
int((a + c*x^4)^(3/2)*(d + e*x)^3, x)
\[ \int (d+e x)^3 \left (a+c x^4\right )^{3/2} \, dx=\frac {336 \sqrt {c \,x^{4}+a}\, a^{2} e^{3}+1440 \sqrt {c \,x^{4}+a}\, a c \,d^{3} x +3150 \sqrt {c \,x^{4}+a}\, a c \,d^{2} e \,x^{2}+2464 \sqrt {c \,x^{4}+a}\, a c d \,e^{2} x^{3}+672 \sqrt {c \,x^{4}+a}\, a c \,e^{3} x^{4}+480 \sqrt {c \,x^{4}+a}\, c^{2} d^{3} x^{5}+1260 \sqrt {c \,x^{4}+a}\, c^{2} d^{2} e \,x^{6}+1120 \sqrt {c \,x^{4}+a}\, c^{2} d \,e^{2} x^{7}+336 \sqrt {c \,x^{4}+a}\, c^{2} e^{3} x^{8}-945 \sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{4}+a}-\sqrt {c}\, x^{2}\right ) a^{2} d^{2} e +945 \sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{4}+a}+\sqrt {c}\, x^{2}\right ) a^{2} d^{2} e +1920 \left (\int \frac {\sqrt {c \,x^{4}+a}}{c \,x^{4}+a}d x \right ) a^{2} c \,d^{3}+2688 \left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c \,x^{4}+a}d x \right ) a^{2} c d \,e^{2}}{3360 c} \] Input:
int((e*x+d)^3*(c*x^4+a)^(3/2),x)
Output:
(336*sqrt(a + c*x**4)*a**2*e**3 + 1440*sqrt(a + c*x**4)*a*c*d**3*x + 3150* sqrt(a + c*x**4)*a*c*d**2*e*x**2 + 2464*sqrt(a + c*x**4)*a*c*d*e**2*x**3 + 672*sqrt(a + c*x**4)*a*c*e**3*x**4 + 480*sqrt(a + c*x**4)*c**2*d**3*x**5 + 1260*sqrt(a + c*x**4)*c**2*d**2*e*x**6 + 1120*sqrt(a + c*x**4)*c**2*d*e* *2*x**7 + 336*sqrt(a + c*x**4)*c**2*e**3*x**8 - 945*sqrt(c)*log(sqrt(a + c *x**4) - sqrt(c)*x**2)*a**2*d**2*e + 945*sqrt(c)*log(sqrt(a + c*x**4) + sq rt(c)*x**2)*a**2*d**2*e + 1920*int(sqrt(a + c*x**4)/(a + c*x**4),x)*a**2*c *d**3 + 2688*int((sqrt(a + c*x**4)*x**2)/(a + c*x**4),x)*a**2*c*d*e**2)/(3 360*c)