Integrand size = 19, antiderivative size = 382 \[ \int (d+e x)^2 \left (a+c x^4\right )^{3/2} \, dx=\frac {3}{8} a d e x^2 \sqrt {a+c x^4}+\frac {4 a^2 e^2 x \sqrt {a+c x^4}}{15 \sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {2}{105} a x \left (15 d^2+7 e^2 x^2\right ) \sqrt {a+c x^4}+\frac {1}{4} d e x^2 \left (a+c x^4\right )^{3/2}+\frac {1}{63} x \left (9 d^2+7 e^2 x^2\right ) \left (a+c x^4\right )^{3/2}+\frac {3 a^2 d e \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{8 \sqrt {c}}-\frac {4 a^{9/4} e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 c^{3/4} \sqrt {a+c x^4}}+\frac {2 a^{7/4} \left (15 \sqrt {c} d^2+7 \sqrt {a} e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{105 c^{3/4} \sqrt {a+c x^4}} \] Output:
3/8*a*d*e*x^2*(c*x^4+a)^(1/2)+4/15*a^2*e^2*x*(c*x^4+a)^(1/2)/c^(1/2)/(a^(1 /2)+c^(1/2)*x^2)+2/105*a*x*(7*e^2*x^2+15*d^2)*(c*x^4+a)^(1/2)+1/4*d*e*x^2* (c*x^4+a)^(3/2)+1/63*x*(7*e^2*x^2+9*d^2)*(c*x^4+a)^(3/2)+3/8*a^2*d*e*arcta nh(c^(1/2)*x^2/(c*x^4+a)^(1/2))/c^(1/2)-4/15*a^(9/4)*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)+2/105*a^(7/4)*(15*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*arctan(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 = 10.21 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.41 \[ \int (d+e x)^2 \left (a+c x^4\right )^{3/2} \, dx=\frac {\sqrt {a+c x^4} \left (24 a \sqrt {c} d^2 x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},-\frac {c x^4}{a}\right )+e \left (3 \sqrt {c} d x^2 \left (5 a+2 c x^4\right ) \sqrt {1+\frac {c x^4}{a}}+9 a^{3/2} d \text {arcsinh}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )+8 a \sqrt {c} e x^3 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^4}{a}\right )\right )\right )}{24 \sqrt {c} \sqrt {1+\frac {c x^4}{a}}} \] Input:
Integrate[(d + e*x)^2*(a + c*x^4)^(3/2),x]
Output:
(Sqrt[a + c*x^4]*(24*a*Sqrt[c]*d^2*x*Hypergeometric2F1[-3/2, 1/4, 5/4, -(( c*x^4)/a)] + e*(3*Sqrt[c]*d*x^2*(5*a + 2*c*x^4)*Sqrt[1 + (c*x^4)/a] + 9*a^ (3/2)*d*ArcSinh[(Sqrt[c]*x^2)/Sqrt[a]] + 8*a*Sqrt[c]*e*x^3*Hypergeometric2 F1[-3/2, 3/4, 7/4, -((c*x^4)/a)])))/(24*Sqrt[c]*Sqrt[1 + (c*x^4)/a])
Time = 0.83 (sec) , antiderivative size = 382, 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.
| \(\displaystyle \int \left (a+c x^4\right )^{3/2} (d+e x)^2 \, dx\) |
| \(\Big \downarrow \) 2424 |
| \(\displaystyle \int \left (\left (a+c x^4\right )^{3/2} \left (d^2+e^2 x^2\right )+2 d e x \left (a+c x^4\right )^{3/2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 a^{7/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (7 \sqrt {a} e^2+15 \sqrt {c} d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{105 c^{3/4} \sqrt {a+c x^4}}-\frac {4 a^{9/4} e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 c^{3/4} \sqrt {a+c x^4}}+\frac {3 a^2 d e \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{8 \sqrt {c}}+\frac {4 a^2 e^2 x \sqrt {a+c x^4}}{15 \sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {2}{105} a x \sqrt {a+c x^4} \left (15 d^2+7 e^2 x^2\right )+\frac {1}{63} x \left (a+c x^4\right )^{3/2} \left (9 d^2+7 e^2 x^2\right )+\frac {3}{8} a d e x^2 \sqrt {a+c x^4}+\frac {1}{4} d e x^2 \left (a+c x^4\right )^{3/2}\) |
Input:
Int[(d + e*x)^2*(a + c*x^4)^(3/2),x]
Output:
(3*a*d*e*x^2*Sqrt[a + c*x^4])/8 + (4*a^2*e^2*x*Sqrt[a + c*x^4])/(15*Sqrt[c ]*(Sqrt[a] + Sqrt[c]*x^2)) + (2*a*x*(15*d^2 + 7*e^2*x^2)*Sqrt[a + c*x^4])/ 105 + (d*e*x^2*(a + c*x^4)^(3/2))/4 + (x*(9*d^2 + 7*e^2*x^2)*(a + c*x^4)^( 3/2))/63 + (3*a^2*d*e*ArcTanh[(Sqrt[c]*x^2)/Sqrt[a + c*x^4]])/(8*Sqrt[c]) - (4*a^(9/4)*e^2*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[ c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(15*c^(3/4)*Sqrt [a + c*x^4]) + (2*a^(7/4)*(15*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])/(105*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 = 1.14 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.69
| method | result | size |
| risch | \(\frac {x \left (280 c \,e^{2} x^{6}+630 c d e \,x^{5}+360 c \,d^{2} x^{4}+616 a \,e^{2} x^{2}+1575 a d e x +1080 a \,d^{2}\right ) \sqrt {c \,x^{4}+a}}{2520}+\frac {a^{2} \left (\frac {240 d^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {112 i e^{2} \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}+\frac {315 d e \ln \left (\sqrt {c}\, x^{2}+\sqrt {c \,x^{4}+a}\right )}{2 \sqrt {c}}\right )}{420}\) | \(265\) |
| default | \(d^{2} \left (\frac {c \,x^{5} \sqrt {c \,x^{4}+a}}{7}+\frac {3 a x \sqrt {c \,x^{4}+a}}{7}+\frac {4 a^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+e^{2} \left (\frac {c \,x^{7} \sqrt {c \,x^{4}+a}}{9}+\frac {11 a \,x^{3} \sqrt {c \,x^{4}+a}}{45}+\frac {4 i a^{\frac {5}{2}} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\right )+2 d e \left (\frac {3 a^{2} \ln \left (\sqrt {c}\, x^{2}+\sqrt {c \,x^{4}+a}\right )}{16 \sqrt {c}}+\frac {c \,x^{6} \sqrt {c \,x^{4}+a}}{8}+\frac {5 a \,x^{2} \sqrt {c \,x^{4}+a}}{16}\right )\) | \(300\) |
| elliptic | \(\frac {c \,e^{2} x^{7} \sqrt {c \,x^{4}+a}}{9}+\frac {c d e \,x^{6} \sqrt {c \,x^{4}+a}}{4}+\frac {c \,d^{2} x^{5} \sqrt {c \,x^{4}+a}}{7}+\frac {11 a \,e^{2} x^{3} \sqrt {c \,x^{4}+a}}{45}+\frac {5 a d e \,x^{2} \sqrt {c \,x^{4}+a}}{8}+\frac {3 a \,d^{2} x \sqrt {c \,x^{4}+a}}{7}+\frac {4 a^{2} d^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {3 a^{2} d e \ln \left (2 \sqrt {c}\, x^{2}+2 \sqrt {c \,x^{4}+a}\right )}{8 \sqrt {c}}+\frac {4 i a^{\frac {5}{2}} e^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\) | \(312\) |
Input:
int((e*x+d)^2*(c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/2520*x*(280*c*e^2*x^6+630*c*d*e*x^5+360*c*d^2*x^4+616*a*e^2*x^2+1575*a*d *e*x+1080*a*d^2)*(c*x^4+a)^(1/2)+1/420*a^2*(240*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)+112*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)-EllipticE(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 = 0.17 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.55 \[ \int (d+e x)^2 \left (a+c x^4\right )^{3/2} \, dx=\frac {1344 \, a^{2} \sqrt {c} e^{2} x \left (-\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + 945 \, a^{2} \sqrt {c} d e x \log \left (-2 \, c x^{4} - 2 \, \sqrt {c x^{4} + a} \sqrt {c} x^{2} - a\right ) + 192 \, {\left (15 \, a c d^{2} - 7 \, a^{2} e^{2}\right )} \sqrt {c} x \left (-\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + 2 \, {\left (280 \, c^{2} e^{2} x^{8} + 630 \, c^{2} d e x^{7} + 360 \, c^{2} d^{2} x^{6} + 616 \, a c e^{2} x^{4} + 1575 \, a c d e x^{3} + 1080 \, a c d^{2} x^{2} + 672 \, a^{2} e^{2}\right )} \sqrt {c x^{4} + a}}{5040 \, c x} \] Input:
integrate((e*x+d)^2*(c*x^4+a)^(3/2),x, algorithm="fricas")
Output:
1/5040*(1344*a^2*sqrt(c)*e^2*x*(-a/c)^(3/4)*elliptic_e(arcsin((-a/c)^(1/4) /x), -1) + 945*a^2*sqrt(c)*d*e*x*log(-2*c*x^4 - 2*sqrt(c*x^4 + a)*sqrt(c)* x^2 - a) + 192*(15*a*c*d^2 - 7*a^2*e^2)*sqrt(c)*x*(-a/c)^(3/4)*elliptic_f( arcsin((-a/c)^(1/4)/x), -1) + 2*(280*c^2*e^2*x^8 + 630*c^2*d*e*x^7 + 360*c ^2*d^2*x^6 + 616*a*c*e^2*x^4 + 1575*a*c*d*e*x^3 + 1080*a*c*d^2*x^2 + 672*a ^2*e^2)*sqrt(c*x^4 + a))/(c*x)
Result contains complex when optimal does not.
Time = 5.13 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.83 \[ \int (d+e x)^2 \left (a+c x^4\right )^{3/2} \, dx=\frac {a^{\frac {3}{2}} d^{2} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {a^{\frac {3}{2}} d e x^{2} \sqrt {1 + \frac {c x^{4}}{a}}}{2} + \frac {a^{\frac {3}{2}} d e x^{2}}{8 \sqrt {1 + \frac {c x^{4}}{a}}} + \frac {a^{\frac {3}{2}} e^{2} x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {\sqrt {a} c d^{2} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {3 \sqrt {a} c d e x^{6}}{8 \sqrt {1 + \frac {c x^{4}}{a}}} + \frac {\sqrt {a} c e^{2} x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} + \frac {3 a^{2} d e \operatorname {asinh}{\left (\frac {\sqrt {c} x^{2}}{\sqrt {a}} \right )}}{8 \sqrt {c}} + \frac {c^{2} d e x^{10}}{4 \sqrt {a} \sqrt {1 + \frac {c x^{4}}{a}}} \] Input:
integrate((e*x+d)**2*(c*x**4+a)**(3/2),x)
Output:
a**(3/2)*d**2*x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), c*x**4*exp_polar(I*p i)/a)/(4*gamma(5/4)) + a**(3/2)*d*e*x**2*sqrt(1 + c*x**4/a)/2 + a**(3/2)*d *e*x**2/(8*sqrt(1 + c*x**4/a)) + a**(3/2)*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**2* x**5*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), c*x**4*exp_polar(I*pi)/a)/(4*ga mma(9/4)) + 3*sqrt(a)*c*d*e*x**6/(8*sqrt(1 + c*x**4/a)) + sqrt(a)*c*e**2*x **7*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), c*x**4*exp_polar(I*pi)/a)/(4*ga mma(11/4)) + 3*a**2*d*e*asinh(sqrt(c)*x**2/sqrt(a))/(8*sqrt(c)) + c**2*d*e *x**10/(4*sqrt(a)*sqrt(1 + c*x**4/a))
\[ \int (d+e x)^2 \left (a+c x^4\right )^{3/2} \, dx=\int { {\left (c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{2} \,d x } \] Input:
integrate((e*x+d)^2*(c*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate((c*x^4 + a)^(3/2)*(e*x + d)^2, x)
\[ \int (d+e x)^2 \left (a+c x^4\right )^{3/2} \, dx=\int { {\left (c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{2} \,d x } \] Input:
integrate((e*x+d)^2*(c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((c*x^4 + a)^(3/2)*(e*x + d)^2, x)
Timed out. \[ \int (d+e x)^2 \left (a+c x^4\right )^{3/2} \, dx=\int {\left (c\,x^4+a\right )}^{3/2}\,{\left (d+e\,x\right )}^2 \,d x \] Input:
int((a + c*x^4)^(3/2)*(d + e*x)^2,x)
Output:
int((a + c*x^4)^(3/2)*(d + e*x)^2, x)
\[ \int (d+e x)^2 \left (a+c x^4\right )^{3/2} \, dx=\frac {2160 \sqrt {c \,x^{4}+a}\, a c \,d^{2} x +3150 \sqrt {c \,x^{4}+a}\, a c d e \,x^{2}+1232 \sqrt {c \,x^{4}+a}\, a c \,e^{2} x^{3}+720 \sqrt {c \,x^{4}+a}\, c^{2} d^{2} x^{5}+1260 \sqrt {c \,x^{4}+a}\, c^{2} d e \,x^{6}+560 \sqrt {c \,x^{4}+a}\, c^{2} e^{2} x^{7}-945 \sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{4}+a}-\sqrt {c}\, x^{2}\right ) a^{2} d e +945 \sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{4}+a}+\sqrt {c}\, x^{2}\right ) a^{2} d e +2880 \left (\int \frac {\sqrt {c \,x^{4}+a}}{c \,x^{4}+a}d x \right ) a^{2} c \,d^{2}+1344 \left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c \,x^{4}+a}d x \right ) a^{2} c \,e^{2}}{5040 c} \] Input:
int((e*x+d)^2*(c*x^4+a)^(3/2),x)
Output:
(2160*sqrt(a + c*x**4)*a*c*d**2*x + 3150*sqrt(a + c*x**4)*a*c*d*e*x**2 + 1 232*sqrt(a + c*x**4)*a*c*e**2*x**3 + 720*sqrt(a + c*x**4)*c**2*d**2*x**5 + 1260*sqrt(a + c*x**4)*c**2*d*e*x**6 + 560*sqrt(a + c*x**4)*c**2*e**2*x**7 - 945*sqrt(c)*log(sqrt(a + c*x**4) - sqrt(c)*x**2)*a**2*d*e + 945*sqrt(c) *log(sqrt(a + c*x**4) + sqrt(c)*x**2)*a**2*d*e + 2880*int(sqrt(a + c*x**4) /(a + c*x**4),x)*a**2*c*d**2 + 1344*int((sqrt(a + c*x**4)*x**2)/(a + c*x** 4),x)*a**2*c*e**2)/(5040*c)