Integrand size = 22, antiderivative size = 170 \[ \int (A+B x) (c+d x) \left (a+b x^2\right )^{5/2} \, dx=\frac {5 a^2 (8 A b c-a B d) x \sqrt {a+b x^2}}{128 b}+\frac {5 a (8 A b c-a B d) x \left (a+b x^2\right )^{3/2}}{192 b}+\frac {(8 A b c-a B d) x \left (a+b x^2\right )^{5/2}}{48 b}+\frac {(8 (B c+A d)+7 B d x) \left (a+b x^2\right )^{7/2}}{56 b}+\frac {5 a^3 (8 A b c-a B d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{3/2}} \] Output:
5/128*a^2*(8*A*b*c-B*a*d)*x*(b*x^2+a)^(1/2)/b+5/192*a*(8*A*b*c-B*a*d)*x*(b *x^2+a)^(3/2)/b+1/48*(8*A*b*c-B*a*d)*x*(b*x^2+a)^(5/2)/b+1/56*(7*B*d*x+8*A *d+8*B*c)*(b*x^2+a)^(7/2)/b+5/128*a^3*(8*A*b*c-B*a*d)*arctanh(b^(1/2)*x/(b *x^2+a)^(1/2))/b^(3/2)
Time = 0.85 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.04 \[ \int (A+B x) (c+d x) \left (a+b x^2\right )^{5/2} \, dx=\frac {\sqrt {b} \sqrt {a+b x^2} \left (3 a^3 (128 B c+128 A d+35 B d x)+16 b^3 x^5 (4 A (7 c+6 d x)+3 B x (8 c+7 d x))+8 a b^2 x^3 (2 A (91 c+72 d x)+B x (144 c+119 d x))+2 a^2 b x (12 A (77 c+48 d x)+B x (576 c+413 d x))\right )+105 a^3 (-8 A b c+a B d) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{2688 b^{3/2}} \] Input:
Integrate[(A + B*x)*(c + d*x)*(a + b*x^2)^(5/2),x]
Output:
(Sqrt[b]*Sqrt[a + b*x^2]*(3*a^3*(128*B*c + 128*A*d + 35*B*d*x) + 16*b^3*x^ 5*(4*A*(7*c + 6*d*x) + 3*B*x*(8*c + 7*d*x)) + 8*a*b^2*x^3*(2*A*(91*c + 72* d*x) + B*x*(144*c + 119*d*x)) + 2*a^2*b*x*(12*A*(77*c + 48*d*x) + B*x*(576 *c + 413*d*x))) + 105*a^3*(-8*A*b*c + a*B*d)*Log[-(Sqrt[b]*x) + Sqrt[a + b *x^2]])/(2688*b^(3/2))
Time = 0.25 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.91, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {676, 211, 211, 211, 224, 219}
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.
| \(\displaystyle \int \left (a+b x^2\right )^{5/2} (A+B x) (c+d x) \, dx\) |
| \(\Big \downarrow \) 676 |
| \(\displaystyle \frac {(8 A b c-a B d) \int \left (b x^2+a\right )^{5/2}dx}{8 b}+\frac {\left (a+b x^2\right )^{7/2} (A d+B c)}{7 b}+\frac {B d x \left (a+b x^2\right )^{7/2}}{8 b}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {(8 A b c-a B d) \left (\frac {5}{6} a \int \left (b x^2+a\right )^{3/2}dx+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )}{8 b}+\frac {\left (a+b x^2\right )^{7/2} (A d+B c)}{7 b}+\frac {B d x \left (a+b x^2\right )^{7/2}}{8 b}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {(8 A b c-a B d) \left (\frac {5}{6} a \left (\frac {3}{4} a \int \sqrt {b x^2+a}dx+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )}{8 b}+\frac {\left (a+b x^2\right )^{7/2} (A d+B c)}{7 b}+\frac {B d x \left (a+b x^2\right )^{7/2}}{8 b}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {(8 A b c-a B d) \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )}{8 b}+\frac {\left (a+b x^2\right )^{7/2} (A d+B c)}{7 b}+\frac {B d x \left (a+b x^2\right )^{7/2}}{8 b}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {(8 A b c-a B d) \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )}{8 b}+\frac {\left (a+b x^2\right )^{7/2} (A d+B c)}{7 b}+\frac {B d x \left (a+b x^2\right )^{7/2}}{8 b}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right ) (8 A b c-a B d)}{8 b}+\frac {\left (a+b x^2\right )^{7/2} (A d+B c)}{7 b}+\frac {B d x \left (a+b x^2\right )^{7/2}}{8 b}\) |
Input:
Int[(A + B*x)*(c + d*x)*(a + b*x^2)^(5/2),x]
Output:
((B*c + A*d)*(a + b*x^2)^(7/2))/(7*b) + (B*d*x*(a + b*x^2)^(7/2))/(8*b) + ((8*A*b*c - a*B*d)*((x*(a + b*x^2)^(5/2))/6 + (5*a*((x*(a + b*x^2)^(3/2))/ 4 + (3*a*((x*Sqrt[a + b*x^2])/2 + (a*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]]) /(2*Sqrt[b])))/4))/6))/(8*b)
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])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(e*f + d*g)*((a + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + (Sim p[e*g*x*((a + c*x^2)^(p + 1)/(c*(2*p + 3))), x] - Simp[(a*e*g - c*d*f*(2*p + 3))/(c*(2*p + 3)) Int[(a + c*x^2)^p, x], x]) /; FreeQ[{a, c, d, e, f, g , p}, x] && !LeQ[p, -1]
Time = 0.88 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.09
| method | result | size |
| default | \(A c \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )+\frac {\left (A d +B c \right ) \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{7 b}+B d \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )\) | \(185\) |
| risch | \(\frac {\left (336 B \,b^{3} d \,x^{7}+384 A \,b^{3} d \,x^{6}+384 B \,b^{3} c \,x^{6}+448 A \,b^{3} c \,x^{5}+952 a B \,b^{2} d \,x^{5}+1152 A a \,b^{2} d \,x^{4}+1152 B a \,b^{2} c \,x^{4}+1456 A a \,b^{2} c \,x^{3}+826 B \,a^{2} b d \,x^{3}+1152 a^{2} x^{2} A b d +1152 a^{2} x^{2} B b c +1848 A \,a^{2} b c x +105 B \,a^{3} d x +384 A \,a^{3} d +384 B \,a^{3} c \right ) \sqrt {b \,x^{2}+a}}{2688 b}+\frac {5 a^{3} \left (8 A b c -B a d \right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{128 b^{\frac {3}{2}}}\) | \(200\) |
Input:
int((B*x+A)*(d*x+c)*(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
A*c*(1/6*x*(b*x^2+a)^(5/2)+5/6*a*(1/4*x*(b*x^2+a)^(3/2)+3/4*a*(1/2*x*(b*x^ 2+a)^(1/2)+1/2*a/b^(1/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2)))))+1/7*(A*d+B*c)*(b *x^2+a)^(7/2)/b+B*d*(1/8*x*(b*x^2+a)^(7/2)/b-1/8*a/b*(1/6*x*(b*x^2+a)^(5/2 )+5/6*a*(1/4*x*(b*x^2+a)^(3/2)+3/4*a*(1/2*x*(b*x^2+a)^(1/2)+1/2*a/b^(1/2)* ln(b^(1/2)*x+(b*x^2+a)^(1/2))))))
Time = 0.10 (sec) , antiderivative size = 434, normalized size of antiderivative = 2.55 \[ \int (A+B x) (c+d x) \left (a+b x^2\right )^{5/2} \, dx=\left [-\frac {105 \, {\left (8 \, A a^{3} b c - B a^{4} d\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (336 \, B b^{4} d x^{7} + 384 \, {\left (B b^{4} c + A b^{4} d\right )} x^{6} + 384 \, B a^{3} b c + 384 \, A a^{3} b d + 56 \, {\left (8 \, A b^{4} c + 17 \, B a b^{3} d\right )} x^{5} + 1152 \, {\left (B a b^{3} c + A a b^{3} d\right )} x^{4} + 14 \, {\left (104 \, A a b^{3} c + 59 \, B a^{2} b^{2} d\right )} x^{3} + 1152 \, {\left (B a^{2} b^{2} c + A a^{2} b^{2} d\right )} x^{2} + 21 \, {\left (88 \, A a^{2} b^{2} c + 5 \, B a^{3} b d\right )} x\right )} \sqrt {b x^{2} + a}}{5376 \, b^{2}}, -\frac {105 \, {\left (8 \, A a^{3} b c - B a^{4} d\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (336 \, B b^{4} d x^{7} + 384 \, {\left (B b^{4} c + A b^{4} d\right )} x^{6} + 384 \, B a^{3} b c + 384 \, A a^{3} b d + 56 \, {\left (8 \, A b^{4} c + 17 \, B a b^{3} d\right )} x^{5} + 1152 \, {\left (B a b^{3} c + A a b^{3} d\right )} x^{4} + 14 \, {\left (104 \, A a b^{3} c + 59 \, B a^{2} b^{2} d\right )} x^{3} + 1152 \, {\left (B a^{2} b^{2} c + A a^{2} b^{2} d\right )} x^{2} + 21 \, {\left (88 \, A a^{2} b^{2} c + 5 \, B a^{3} b d\right )} x\right )} \sqrt {b x^{2} + a}}{2688 \, b^{2}}\right ] \] Input:
integrate((B*x+A)*(d*x+c)*(b*x^2+a)^(5/2),x, algorithm="fricas")
Output:
[-1/5376*(105*(8*A*a^3*b*c - B*a^4*d)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(336*B*b^4*d*x^7 + 384*(B*b^4*c + A*b^4*d)*x^6 + 3 84*B*a^3*b*c + 384*A*a^3*b*d + 56*(8*A*b^4*c + 17*B*a*b^3*d)*x^5 + 1152*(B *a*b^3*c + A*a*b^3*d)*x^4 + 14*(104*A*a*b^3*c + 59*B*a^2*b^2*d)*x^3 + 1152 *(B*a^2*b^2*c + A*a^2*b^2*d)*x^2 + 21*(88*A*a^2*b^2*c + 5*B*a^3*b*d)*x)*sq rt(b*x^2 + a))/b^2, -1/2688*(105*(8*A*a^3*b*c - B*a^4*d)*sqrt(-b)*arctan(s qrt(-b)*x/sqrt(b*x^2 + a)) - (336*B*b^4*d*x^7 + 384*(B*b^4*c + A*b^4*d)*x^ 6 + 384*B*a^3*b*c + 384*A*a^3*b*d + 56*(8*A*b^4*c + 17*B*a*b^3*d)*x^5 + 11 52*(B*a*b^3*c + A*a*b^3*d)*x^4 + 14*(104*A*a*b^3*c + 59*B*a^2*b^2*d)*x^3 + 1152*(B*a^2*b^2*c + A*a^2*b^2*d)*x^2 + 21*(88*A*a^2*b^2*c + 5*B*a^3*b*d)* x)*sqrt(b*x^2 + a))/b^2]
Leaf count of result is larger than twice the leaf count of optimal. 561 vs. \(2 (160) = 320\).
Time = 0.89 (sec) , antiderivative size = 561, normalized size of antiderivative = 3.30 \[ \int (A+B x) (c+d x) \left (a+b x^2\right )^{5/2} \, dx=\begin {cases} \sqrt {a + b x^{2}} \left (\frac {B b^{2} d x^{7}}{8} + \frac {x^{6} \left (A b^{3} d + B b^{3} c\right )}{7 b} + \frac {x^{5} \left (A b^{3} c + \frac {17 B a b^{2} d}{8}\right )}{6 b} + \frac {x^{4} \cdot \left (3 A a b^{2} d + 3 B a b^{2} c - \frac {6 a \left (A b^{3} d + B b^{3} c\right )}{7 b}\right )}{5 b} + \frac {x^{3} \cdot \left (3 A a b^{2} c + 3 B a^{2} b d - \frac {5 a \left (A b^{3} c + \frac {17 B a b^{2} d}{8}\right )}{6 b}\right )}{4 b} + \frac {x^{2} \cdot \left (3 A a^{2} b d + 3 B a^{2} b c - \frac {4 a \left (3 A a b^{2} d + 3 B a b^{2} c - \frac {6 a \left (A b^{3} d + B b^{3} c\right )}{7 b}\right )}{5 b}\right )}{3 b} + \frac {x \left (3 A a^{2} b c + B a^{3} d - \frac {3 a \left (3 A a b^{2} c + 3 B a^{2} b d - \frac {5 a \left (A b^{3} c + \frac {17 B a b^{2} d}{8}\right )}{6 b}\right )}{4 b}\right )}{2 b} + \frac {A a^{3} d + B a^{3} c - \frac {2 a \left (3 A a^{2} b d + 3 B a^{2} b c - \frac {4 a \left (3 A a b^{2} d + 3 B a b^{2} c - \frac {6 a \left (A b^{3} d + B b^{3} c\right )}{7 b}\right )}{5 b}\right )}{3 b}}{b}\right ) + \left (A a^{3} c - \frac {a \left (3 A a^{2} b c + B a^{3} d - \frac {3 a \left (3 A a b^{2} c + 3 B a^{2} b d - \frac {5 a \left (A b^{3} c + \frac {17 B a b^{2} d}{8}\right )}{6 b}\right )}{4 b}\right )}{2 b}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: b \neq 0 \\a^{\frac {5}{2}} \left (A c x + \frac {B d x^{3}}{3} + \frac {x^{2} \left (A d + B c\right )}{2}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((B*x+A)*(d*x+c)*(b*x**2+a)**(5/2),x)
Output:
Piecewise((sqrt(a + b*x**2)*(B*b**2*d*x**7/8 + x**6*(A*b**3*d + B*b**3*c)/ (7*b) + x**5*(A*b**3*c + 17*B*a*b**2*d/8)/(6*b) + x**4*(3*A*a*b**2*d + 3*B *a*b**2*c - 6*a*(A*b**3*d + B*b**3*c)/(7*b))/(5*b) + x**3*(3*A*a*b**2*c + 3*B*a**2*b*d - 5*a*(A*b**3*c + 17*B*a*b**2*d/8)/(6*b))/(4*b) + x**2*(3*A*a **2*b*d + 3*B*a**2*b*c - 4*a*(3*A*a*b**2*d + 3*B*a*b**2*c - 6*a*(A*b**3*d + B*b**3*c)/(7*b))/(5*b))/(3*b) + x*(3*A*a**2*b*c + B*a**3*d - 3*a*(3*A*a* b**2*c + 3*B*a**2*b*d - 5*a*(A*b**3*c + 17*B*a*b**2*d/8)/(6*b))/(4*b))/(2* b) + (A*a**3*d + B*a**3*c - 2*a*(3*A*a**2*b*d + 3*B*a**2*b*c - 4*a*(3*A*a* b**2*d + 3*B*a*b**2*c - 6*a*(A*b**3*d + B*b**3*c)/(7*b))/(5*b))/(3*b))/b) + (A*a**3*c - a*(3*A*a**2*b*c + B*a**3*d - 3*a*(3*A*a*b**2*c + 3*B*a**2*b* d - 5*a*(A*b**3*c + 17*B*a*b**2*d/8)/(6*b))/(4*b))/(2*b))*Piecewise((log(2 *sqrt(b)*sqrt(a + b*x**2) + 2*b*x)/sqrt(b), Ne(a, 0)), (x*log(x)/sqrt(b*x* *2), True)), Ne(b, 0)), (a**(5/2)*(A*c*x + B*d*x**3/3 + x**2*(A*d + B*c)/2 ), True))
Time = 0.04 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.13 \[ \int (A+B x) (c+d x) \left (a+b x^2\right )^{5/2} \, dx=\frac {1}{6} \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A c x + \frac {5}{24} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A a c x + \frac {5}{16} \, \sqrt {b x^{2} + a} A a^{2} c x + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B d x}{8 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B a d x}{48 \, b} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{2} d x}{192 \, b} - \frac {5 \, \sqrt {b x^{2} + a} B a^{3} d x}{128 \, b} + \frac {5 \, A a^{3} c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, \sqrt {b}} - \frac {5 \, B a^{4} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {3}{2}}} + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B c}{7 \, b} + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A d}{7 \, b} \] Input:
integrate((B*x+A)*(d*x+c)*(b*x^2+a)^(5/2),x, algorithm="maxima")
Output:
1/6*(b*x^2 + a)^(5/2)*A*c*x + 5/24*(b*x^2 + a)^(3/2)*A*a*c*x + 5/16*sqrt(b *x^2 + a)*A*a^2*c*x + 1/8*(b*x^2 + a)^(7/2)*B*d*x/b - 1/48*(b*x^2 + a)^(5/ 2)*B*a*d*x/b - 5/192*(b*x^2 + a)^(3/2)*B*a^2*d*x/b - 5/128*sqrt(b*x^2 + a) *B*a^3*d*x/b + 5/16*A*a^3*c*arcsinh(b*x/sqrt(a*b))/sqrt(b) - 5/128*B*a^4*d *arcsinh(b*x/sqrt(a*b))/b^(3/2) + 1/7*(b*x^2 + a)^(7/2)*B*c/b + 1/7*(b*x^2 + a)^(7/2)*A*d/b
Time = 0.34 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.40 \[ \int (A+B x) (c+d x) \left (a+b x^2\right )^{5/2} \, dx=\frac {1}{2688} \, \sqrt {b x^{2} + a} {\left ({\left (2 \, {\left ({\left (4 \, {\left ({\left (6 \, {\left (7 \, B b^{2} d x + \frac {8 \, {\left (B b^{8} c + A b^{8} d\right )}}{b^{6}}\right )} x + \frac {7 \, {\left (8 \, A b^{8} c + 17 \, B a b^{7} d\right )}}{b^{6}}\right )} x + \frac {144 \, {\left (B a b^{7} c + A a b^{7} d\right )}}{b^{6}}\right )} x + \frac {7 \, {\left (104 \, A a b^{7} c + 59 \, B a^{2} b^{6} d\right )}}{b^{6}}\right )} x + \frac {576 \, {\left (B a^{2} b^{6} c + A a^{2} b^{6} d\right )}}{b^{6}}\right )} x + \frac {21 \, {\left (88 \, A a^{2} b^{6} c + 5 \, B a^{3} b^{5} d\right )}}{b^{6}}\right )} x + \frac {384 \, {\left (B a^{3} b^{5} c + A a^{3} b^{5} d\right )}}{b^{6}}\right )} - \frac {5 \, {\left (8 \, A a^{3} b c - B a^{4} d\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{128 \, b^{\frac {3}{2}}} \] Input:
integrate((B*x+A)*(d*x+c)*(b*x^2+a)^(5/2),x, algorithm="giac")
Output:
1/2688*sqrt(b*x^2 + a)*((2*((4*((6*(7*B*b^2*d*x + 8*(B*b^8*c + A*b^8*d)/b^ 6)*x + 7*(8*A*b^8*c + 17*B*a*b^7*d)/b^6)*x + 144*(B*a*b^7*c + A*a*b^7*d)/b ^6)*x + 7*(104*A*a*b^7*c + 59*B*a^2*b^6*d)/b^6)*x + 576*(B*a^2*b^6*c + A*a ^2*b^6*d)/b^6)*x + 21*(88*A*a^2*b^6*c + 5*B*a^3*b^5*d)/b^6)*x + 384*(B*a^3 *b^5*c + A*a^3*b^5*d)/b^6) - 5/128*(8*A*a^3*b*c - B*a^4*d)*log(abs(-sqrt(b )*x + sqrt(b*x^2 + a)))/b^(3/2)
Timed out. \[ \int (A+B x) (c+d x) \left (a+b x^2\right )^{5/2} \, dx=\int {\left (b\,x^2+a\right )}^{5/2}\,\left (A+B\,x\right )\,\left (c+d\,x\right ) \,d x \] Input:
int((a + b*x^2)^(5/2)*(A + B*x)*(c + d*x),x)
Output:
int((a + b*x^2)^(5/2)*(A + B*x)*(c + d*x), x)
Time = 0.26 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.91 \[ \int (A+B x) (c+d x) \left (a+b x^2\right )^{5/2} \, dx=\frac {384 \sqrt {b \,x^{2}+a}\, a^{4} d +1848 \sqrt {b \,x^{2}+a}\, a^{3} b c x +384 \sqrt {b \,x^{2}+a}\, a^{3} b c +1152 \sqrt {b \,x^{2}+a}\, a^{3} b d \,x^{2}+105 \sqrt {b \,x^{2}+a}\, a^{3} b d x +1456 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c \,x^{3}+1152 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c \,x^{2}+1152 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d \,x^{4}+826 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d \,x^{3}+448 \sqrt {b \,x^{2}+a}\, a \,b^{3} c \,x^{5}+1152 \sqrt {b \,x^{2}+a}\, a \,b^{3} c \,x^{4}+384 \sqrt {b \,x^{2}+a}\, a \,b^{3} d \,x^{6}+952 \sqrt {b \,x^{2}+a}\, a \,b^{3} d \,x^{5}+384 \sqrt {b \,x^{2}+a}\, b^{4} c \,x^{6}+336 \sqrt {b \,x^{2}+a}\, b^{4} d \,x^{7}+840 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} c -105 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} d}{2688 b} \] Input:
int((B*x+A)*(d*x+c)*(b*x^2+a)^(5/2),x)
Output:
(384*sqrt(a + b*x**2)*a**4*d + 1848*sqrt(a + b*x**2)*a**3*b*c*x + 384*sqrt (a + b*x**2)*a**3*b*c + 1152*sqrt(a + b*x**2)*a**3*b*d*x**2 + 105*sqrt(a + b*x**2)*a**3*b*d*x + 1456*sqrt(a + b*x**2)*a**2*b**2*c*x**3 + 1152*sqrt(a + b*x**2)*a**2*b**2*c*x**2 + 1152*sqrt(a + b*x**2)*a**2*b**2*d*x**4 + 826 *sqrt(a + b*x**2)*a**2*b**2*d*x**3 + 448*sqrt(a + b*x**2)*a*b**3*c*x**5 + 1152*sqrt(a + b*x**2)*a*b**3*c*x**4 + 384*sqrt(a + b*x**2)*a*b**3*d*x**6 + 952*sqrt(a + b*x**2)*a*b**3*d*x**5 + 384*sqrt(a + b*x**2)*b**4*c*x**6 + 3 36*sqrt(a + b*x**2)*b**4*d*x**7 + 840*sqrt(b)*log((sqrt(a + b*x**2) + sqrt (b)*x)/sqrt(a))*a**4*c - 105*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sq rt(a))*a**4*d)/(2688*b)