Integrand size = 22, antiderivative size = 264 \[ \int x^2 (c+d x) \left (a x+b x^2\right )^{3/2} \, dx=-\frac {a^5 (14 b c-9 a d) \sqrt {a x+b x^2}}{1024 b^5}+\frac {a^4 (14 b c-9 a d) x \sqrt {a x+b x^2}}{1536 b^4}-\frac {a^3 (14 b c-9 a d) x^2 \sqrt {a x+b x^2}}{1920 b^3}+\frac {a^2 (14 b c-9 a d) x^3 \sqrt {a x+b x^2}}{2240 b^2}+\frac {13 a (14 b c-9 a d) x^4 \sqrt {a x+b x^2}}{840 b}+\frac {1}{84} (14 b c-9 a d) x^5 \sqrt {a x+b x^2}+\frac {d x^2 \left (a x+b x^2\right )^{5/2}}{7 b}+\frac {a^6 (14 b c-9 a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{1024 b^{11/2}} \] Output:
-1/1024*a^5*(-9*a*d+14*b*c)*(b*x^2+a*x)^(1/2)/b^5+1/1536*a^4*(-9*a*d+14*b* c)*x*(b*x^2+a*x)^(1/2)/b^4-1/1920*a^3*(-9*a*d+14*b*c)*x^2*(b*x^2+a*x)^(1/2 )/b^3+1/2240*a^2*(-9*a*d+14*b*c)*x^3*(b*x^2+a*x)^(1/2)/b^2+13/840*a*(-9*a* d+14*b*c)*x^4*(b*x^2+a*x)^(1/2)/b+1/84*(-9*a*d+14*b*c)*x^5*(b*x^2+a*x)^(1/ 2)+1/7*d*x^2*(b*x^2+a*x)^(5/2)/b+1/1024*a^6*(-9*a*d+14*b*c)*arctanh(b^(1/2 )*x/(b*x^2+a*x)^(1/2))/b^(11/2)
Time = 0.33 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.89 \[ \int x^2 (c+d x) \left (a x+b x^2\right )^{3/2} \, dx=\frac {(x (a+b x))^{3/2} \left (-1470 a^5 b c+945 a^6 d+980 a^4 b^2 c x-630 a^5 b d x-784 a^3 b^3 c x^2+504 a^4 b^2 d x^2+672 a^2 b^4 c x^3-432 a^3 b^3 d x^3+23296 a b^5 c x^4+384 a^2 b^4 d x^4+17920 b^6 c x^5+19200 a b^5 d x^5+15360 b^6 d x^6\right )}{107520 b^5 x (a+b x)}-\frac {a^6 (-14 b c+9 a d) (x (a+b x))^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )}{512 b^{11/2} x^{3/2} (a+b x)^{3/2}} \] Input:
Integrate[x^2*(c + d*x)*(a*x + b*x^2)^(3/2),x]
Output:
((x*(a + b*x))^(3/2)*(-1470*a^5*b*c + 945*a^6*d + 980*a^4*b^2*c*x - 630*a^ 5*b*d*x - 784*a^3*b^3*c*x^2 + 504*a^4*b^2*d*x^2 + 672*a^2*b^4*c*x^3 - 432* a^3*b^3*d*x^3 + 23296*a*b^5*c*x^4 + 384*a^2*b^4*d*x^4 + 17920*b^6*c*x^5 + 19200*a*b^5*d*x^5 + 15360*b^6*d*x^6))/(107520*b^5*x*(a + b*x)) - (a^6*(-14 *b*c + 9*a*d)*(x*(a + b*x))^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[x])/(-Sqrt[a] + Sq rt[a + b*x])])/(512*b^(11/2)*x^(3/2)*(a + b*x)^(3/2))
Time = 0.61 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.75, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1221, 1134, 1160, 1087, 1087, 1091, 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.
| \(\displaystyle \int x^2 \left (a x+b x^2\right )^{3/2} (c+d x) \, dx\) |
| \(\Big \downarrow \) 1221 |
| \(\displaystyle \frac {(14 b c-9 a d) \int x^2 \left (b x^2+a x\right )^{3/2}dx}{14 b}+\frac {d x^2 \left (a x+b x^2\right )^{5/2}}{7 b}\) |
| \(\Big \downarrow \) 1134 |
| \(\displaystyle \frac {(14 b c-9 a d) \left (\frac {x \left (a x+b x^2\right )^{5/2}}{6 b}-\frac {7 a \int x \left (b x^2+a x\right )^{3/2}dx}{12 b}\right )}{14 b}+\frac {d x^2 \left (a x+b x^2\right )^{5/2}}{7 b}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {(14 b c-9 a d) \left (\frac {x \left (a x+b x^2\right )^{5/2}}{6 b}-\frac {7 a \left (\frac {\left (a x+b x^2\right )^{5/2}}{5 b}-\frac {a \int \left (b x^2+a x\right )^{3/2}dx}{2 b}\right )}{12 b}\right )}{14 b}+\frac {d x^2 \left (a x+b x^2\right )^{5/2}}{7 b}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {(14 b c-9 a d) \left (\frac {x \left (a x+b x^2\right )^{5/2}}{6 b}-\frac {7 a \left (\frac {\left (a x+b x^2\right )^{5/2}}{5 b}-\frac {a \left (\frac {(a+2 b x) \left (a x+b x^2\right )^{3/2}}{8 b}-\frac {3 a^2 \int \sqrt {b x^2+a x}dx}{16 b}\right )}{2 b}\right )}{12 b}\right )}{14 b}+\frac {d x^2 \left (a x+b x^2\right )^{5/2}}{7 b}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {(14 b c-9 a d) \left (\frac {x \left (a x+b x^2\right )^{5/2}}{6 b}-\frac {7 a \left (\frac {\left (a x+b x^2\right )^{5/2}}{5 b}-\frac {a \left (\frac {(a+2 b x) \left (a x+b x^2\right )^{3/2}}{8 b}-\frac {3 a^2 \left (\frac {(a+2 b x) \sqrt {a x+b x^2}}{4 b}-\frac {a^2 \int \frac {1}{\sqrt {b x^2+a x}}dx}{8 b}\right )}{16 b}\right )}{2 b}\right )}{12 b}\right )}{14 b}+\frac {d x^2 \left (a x+b x^2\right )^{5/2}}{7 b}\) |
| \(\Big \downarrow \) 1091 |
| \(\displaystyle \frac {(14 b c-9 a d) \left (\frac {x \left (a x+b x^2\right )^{5/2}}{6 b}-\frac {7 a \left (\frac {\left (a x+b x^2\right )^{5/2}}{5 b}-\frac {a \left (\frac {(a+2 b x) \left (a x+b x^2\right )^{3/2}}{8 b}-\frac {3 a^2 \left (\frac {(a+2 b x) \sqrt {a x+b x^2}}{4 b}-\frac {a^2 \int \frac {1}{1-\frac {b x^2}{b x^2+a x}}d\frac {x}{\sqrt {b x^2+a x}}}{4 b}\right )}{16 b}\right )}{2 b}\right )}{12 b}\right )}{14 b}+\frac {d x^2 \left (a x+b x^2\right )^{5/2}}{7 b}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (\frac {x \left (a x+b x^2\right )^{5/2}}{6 b}-\frac {7 a \left (\frac {\left (a x+b x^2\right )^{5/2}}{5 b}-\frac {a \left (\frac {(a+2 b x) \left (a x+b x^2\right )^{3/2}}{8 b}-\frac {3 a^2 \left (\frac {(a+2 b x) \sqrt {a x+b x^2}}{4 b}-\frac {a^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{4 b^{3/2}}\right )}{16 b}\right )}{2 b}\right )}{12 b}\right ) (14 b c-9 a d)}{14 b}+\frac {d x^2 \left (a x+b x^2\right )^{5/2}}{7 b}\) |
Input:
Int[x^2*(c + d*x)*(a*x + b*x^2)^(3/2),x]
Output:
(d*x^2*(a*x + b*x^2)^(5/2))/(7*b) + ((14*b*c - 9*a*d)*((x*(a*x + b*x^2)^(5 /2))/(6*b) - (7*a*((a*x + b*x^2)^(5/2)/(5*b) - (a*(((a + 2*b*x)*(a*x + b*x ^2)^(3/2))/(8*b) - (3*a^2*(((a + 2*b*x)*Sqrt[a*x + b*x^2])/(4*b) - (a^2*Ar cTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(4*b^(3/2))))/(16*b)))/(2*b)))/(12*b )))/(14*b)
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[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{b, c}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[(m + p)*((2*c*d - b*e)/(c*(m + 2*p + 1))) Int[(d + e*x)^ (m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[ c*d^2 - b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2 *p]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1 )/(c*(m + 2*p + 2))), x] + Simp[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c *f - b*g))/(c*e*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x ] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[m + 2*p + 2, 0]
Time = 0.51 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.56
| method | result | size |
| pseudoelliptic | \(-\frac {9 \left (a^{6} \left (a d -\frac {14 b c}{9}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right )-\left (\frac {3328 x^{4} \left (\frac {75 d x}{91}+c \right ) a \,b^{\frac {11}{2}}}{135}+\frac {512 \left (\frac {6 d x}{7}+c \right ) x^{5} b^{\frac {13}{2}}}{27}+a^{2} \left (-\frac {14 \left (\frac {3 d x}{7}+c \right ) a^{3} b^{\frac {3}{2}}}{9}+\frac {28 x \left (\frac {18 d x}{35}+c \right ) a^{2} b^{\frac {5}{2}}}{27}-\frac {112 x^{2} \left (\frac {27 d x}{49}+c \right ) a \,b^{\frac {7}{2}}}{135}+\frac {32 x^{3} \left (\frac {4 d x}{7}+c \right ) b^{\frac {9}{2}}}{45}+\sqrt {b}\, a^{4} d \right )\right ) \sqrt {x \left (b x +a \right )}\right )}{1024 b^{\frac {11}{2}}}\) | \(148\) |
| risch | \(\frac {\left (15360 b^{6} d \,x^{6}+19200 a \,b^{5} d \,x^{5}+17920 b^{6} c \,x^{5}+384 a^{2} b^{4} d \,x^{4}+23296 a \,b^{5} c \,x^{4}-432 a^{3} b^{3} d \,x^{3}+672 a^{2} b^{4} c \,x^{3}+504 a^{4} b^{2} d \,x^{2}-784 a^{3} b^{3} c \,x^{2}-630 a^{5} b d x +980 a^{4} b^{2} c x +945 a^{6} d -1470 a^{5} b c \right ) x \left (b x +a \right )}{107520 b^{5} \sqrt {x \left (b x +a \right )}}-\frac {a^{6} \left (9 a d -14 b c \right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2048 b^{\frac {11}{2}}}\) | \(193\) |
| default | \(c \left (\frac {x \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{6 b}-\frac {7 a \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{5 b}-\frac {a \left (\frac {\left (2 b x +a \right ) \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{8 b}-\frac {3 a^{2} \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{16 b}\right )}{2 b}\right )}{12 b}\right )+d \left (\frac {x^{2} \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{7 b}-\frac {9 a \left (\frac {x \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{6 b}-\frac {7 a \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{5 b}-\frac {a \left (\frac {\left (2 b x +a \right ) \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{8 b}-\frac {3 a^{2} \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{16 b}\right )}{2 b}\right )}{12 b}\right )}{14 b}\right )\) | \(298\) |
Input:
int(x^2*(d*x+c)*(b*x^2+a*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
-9/1024*(a^6*(a*d-14/9*b*c)*arctanh((x*(b*x+a))^(1/2)/x/b^(1/2))-(3328/135 *x^4*(75/91*d*x+c)*a*b^(11/2)+512/27*(6/7*d*x+c)*x^5*b^(13/2)+a^2*(-14/9*( 3/7*d*x+c)*a^3*b^(3/2)+28/27*x*(18/35*d*x+c)*a^2*b^(5/2)-112/135*x^2*(27/4 9*d*x+c)*a*b^(7/2)+32/45*x^3*(4/7*d*x+c)*b^(9/2)+b^(1/2)*a^4*d))*(x*(b*x+a ))^(1/2))/b^(11/2)
Time = 0.11 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.52 \[ \int x^2 (c+d x) \left (a x+b x^2\right )^{3/2} \, dx=\left [-\frac {105 \, {\left (14 \, a^{6} b c - 9 \, a^{7} d\right )} \sqrt {b} \log \left (2 \, b x + a - 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) - 2 \, {\left (15360 \, b^{7} d x^{6} - 1470 \, a^{5} b^{2} c + 945 \, a^{6} b d + 1280 \, {\left (14 \, b^{7} c + 15 \, a b^{6} d\right )} x^{5} + 128 \, {\left (182 \, a b^{6} c + 3 \, a^{2} b^{5} d\right )} x^{4} + 48 \, {\left (14 \, a^{2} b^{5} c - 9 \, a^{3} b^{4} d\right )} x^{3} - 56 \, {\left (14 \, a^{3} b^{4} c - 9 \, a^{4} b^{3} d\right )} x^{2} + 70 \, {\left (14 \, a^{4} b^{3} c - 9 \, a^{5} b^{2} d\right )} x\right )} \sqrt {b x^{2} + a x}}{215040 \, b^{6}}, -\frac {105 \, {\left (14 \, a^{6} b c - 9 \, a^{7} d\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x + a}\right ) - {\left (15360 \, b^{7} d x^{6} - 1470 \, a^{5} b^{2} c + 945 \, a^{6} b d + 1280 \, {\left (14 \, b^{7} c + 15 \, a b^{6} d\right )} x^{5} + 128 \, {\left (182 \, a b^{6} c + 3 \, a^{2} b^{5} d\right )} x^{4} + 48 \, {\left (14 \, a^{2} b^{5} c - 9 \, a^{3} b^{4} d\right )} x^{3} - 56 \, {\left (14 \, a^{3} b^{4} c - 9 \, a^{4} b^{3} d\right )} x^{2} + 70 \, {\left (14 \, a^{4} b^{3} c - 9 \, a^{5} b^{2} d\right )} x\right )} \sqrt {b x^{2} + a x}}{107520 \, b^{6}}\right ] \] Input:
integrate(x^2*(d*x+c)*(b*x^2+a*x)^(3/2),x, algorithm="fricas")
Output:
[-1/215040*(105*(14*a^6*b*c - 9*a^7*d)*sqrt(b)*log(2*b*x + a - 2*sqrt(b*x^ 2 + a*x)*sqrt(b)) - 2*(15360*b^7*d*x^6 - 1470*a^5*b^2*c + 945*a^6*b*d + 12 80*(14*b^7*c + 15*a*b^6*d)*x^5 + 128*(182*a*b^6*c + 3*a^2*b^5*d)*x^4 + 48* (14*a^2*b^5*c - 9*a^3*b^4*d)*x^3 - 56*(14*a^3*b^4*c - 9*a^4*b^3*d)*x^2 + 7 0*(14*a^4*b^3*c - 9*a^5*b^2*d)*x)*sqrt(b*x^2 + a*x))/b^6, -1/107520*(105*( 14*a^6*b*c - 9*a^7*d)*sqrt(-b)*arctan(sqrt(b*x^2 + a*x)*sqrt(-b)/(b*x + a) ) - (15360*b^7*d*x^6 - 1470*a^5*b^2*c + 945*a^6*b*d + 1280*(14*b^7*c + 15* a*b^6*d)*x^5 + 128*(182*a*b^6*c + 3*a^2*b^5*d)*x^4 + 48*(14*a^2*b^5*c - 9* a^3*b^4*d)*x^3 - 56*(14*a^3*b^4*c - 9*a^4*b^3*d)*x^2 + 70*(14*a^4*b^3*c - 9*a^5*b^2*d)*x)*sqrt(b*x^2 + a*x))/b^6]
Time = 0.48 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.68 \[ \int x^2 (c+d x) \left (a x+b x^2\right )^{3/2} \, dx=\begin {cases} \frac {35 a^{4} \left (a^{2} c - \frac {9 a \left (a^{2} d + 2 a b c - \frac {11 a \left (\frac {15 a b d}{14} + b^{2} c\right )}{12 b}\right )}{10 b}\right ) \left (\begin {cases} \frac {\log {\left (a + 2 \sqrt {b} \sqrt {a x + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: \frac {a^{2}}{b} \neq 0 \\\frac {\left (\frac {a}{2 b} + x\right ) \log {\left (\frac {a}{2 b} + x \right )}}{\sqrt {b \left (\frac {a}{2 b} + x\right )^{2}}} & \text {otherwise} \end {cases}\right )}{128 b^{4}} + \sqrt {a x + b x^{2}} \left (- \frac {35 a^{3} \left (a^{2} c - \frac {9 a \left (a^{2} d + 2 a b c - \frac {11 a \left (\frac {15 a b d}{14} + b^{2} c\right )}{12 b}\right )}{10 b}\right )}{64 b^{4}} + \frac {35 a^{2} x \left (a^{2} c - \frac {9 a \left (a^{2} d + 2 a b c - \frac {11 a \left (\frac {15 a b d}{14} + b^{2} c\right )}{12 b}\right )}{10 b}\right )}{96 b^{3}} - \frac {7 a x^{2} \left (a^{2} c - \frac {9 a \left (a^{2} d + 2 a b c - \frac {11 a \left (\frac {15 a b d}{14} + b^{2} c\right )}{12 b}\right )}{10 b}\right )}{24 b^{2}} + \frac {b d x^{6}}{7} + \frac {x^{5} \cdot \left (\frac {15 a b d}{14} + b^{2} c\right )}{6 b} + \frac {x^{4} \left (a^{2} d + 2 a b c - \frac {11 a \left (\frac {15 a b d}{14} + b^{2} c\right )}{12 b}\right )}{5 b} + \frac {x^{3} \left (a^{2} c - \frac {9 a \left (a^{2} d + 2 a b c - \frac {11 a \left (\frac {15 a b d}{14} + b^{2} c\right )}{12 b}\right )}{10 b}\right )}{4 b}\right ) & \text {for}\: b \neq 0 \\\frac {2 \left (\frac {c \left (a x\right )^{\frac {9}{2}}}{9} + \frac {d \left (a x\right )^{\frac {11}{2}}}{11 a}\right )}{a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate(x**2*(d*x+c)*(b*x**2+a*x)**(3/2),x)
Output:
Piecewise((35*a**4*(a**2*c - 9*a*(a**2*d + 2*a*b*c - 11*a*(15*a*b*d/14 + b **2*c)/(12*b))/(10*b))*Piecewise((log(a + 2*sqrt(b)*sqrt(a*x + b*x**2) + 2 *b*x)/sqrt(b), Ne(a**2/b, 0)), ((a/(2*b) + x)*log(a/(2*b) + x)/sqrt(b*(a/( 2*b) + x)**2), True))/(128*b**4) + sqrt(a*x + b*x**2)*(-35*a**3*(a**2*c - 9*a*(a**2*d + 2*a*b*c - 11*a*(15*a*b*d/14 + b**2*c)/(12*b))/(10*b))/(64*b* *4) + 35*a**2*x*(a**2*c - 9*a*(a**2*d + 2*a*b*c - 11*a*(15*a*b*d/14 + b**2 *c)/(12*b))/(10*b))/(96*b**3) - 7*a*x**2*(a**2*c - 9*a*(a**2*d + 2*a*b*c - 11*a*(15*a*b*d/14 + b**2*c)/(12*b))/(10*b))/(24*b**2) + b*d*x**6/7 + x**5 *(15*a*b*d/14 + b**2*c)/(6*b) + x**4*(a**2*d + 2*a*b*c - 11*a*(15*a*b*d/14 + b**2*c)/(12*b))/(5*b) + x**3*(a**2*c - 9*a*(a**2*d + 2*a*b*c - 11*a*(15 *a*b*d/14 + b**2*c)/(12*b))/(10*b))/(4*b)), Ne(b, 0)), (2*(c*(a*x)**(9/2)/ 9 + d*(a*x)**(11/2)/(11*a))/a**3, Ne(a, 0)), (0, True))
Time = 0.04 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.23 \[ \int x^2 (c+d x) \left (a x+b x^2\right )^{3/2} \, dx=\frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} d x^{2}}{7 \, b} - \frac {7 \, \sqrt {b x^{2} + a x} a^{4} c x}{256 \, b^{3}} + \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2} c x}{96 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} c x}{6 \, b} + \frac {9 \, \sqrt {b x^{2} + a x} a^{5} d x}{512 \, b^{4}} - \frac {3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{3} d x}{64 \, b^{3}} - \frac {3 \, {\left (b x^{2} + a x\right )}^{\frac {5}{2}} a d x}{28 \, b^{2}} + \frac {7 \, a^{6} c \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{1024 \, b^{\frac {9}{2}}} - \frac {9 \, a^{7} d \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{2048 \, b^{\frac {11}{2}}} - \frac {7 \, \sqrt {b x^{2} + a x} a^{5} c}{512 \, b^{4}} + \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{3} c}{192 \, b^{3}} - \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {5}{2}} a c}{60 \, b^{2}} + \frac {9 \, \sqrt {b x^{2} + a x} a^{6} d}{1024 \, b^{5}} - \frac {3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{4} d}{128 \, b^{4}} + \frac {3 \, {\left (b x^{2} + a x\right )}^{\frac {5}{2}} a^{2} d}{40 \, b^{3}} \] Input:
integrate(x^2*(d*x+c)*(b*x^2+a*x)^(3/2),x, algorithm="maxima")
Output:
1/7*(b*x^2 + a*x)^(5/2)*d*x^2/b - 7/256*sqrt(b*x^2 + a*x)*a^4*c*x/b^3 + 7/ 96*(b*x^2 + a*x)^(3/2)*a^2*c*x/b^2 + 1/6*(b*x^2 + a*x)^(5/2)*c*x/b + 9/512 *sqrt(b*x^2 + a*x)*a^5*d*x/b^4 - 3/64*(b*x^2 + a*x)^(3/2)*a^3*d*x/b^3 - 3/ 28*(b*x^2 + a*x)^(5/2)*a*d*x/b^2 + 7/1024*a^6*c*log(2*b*x + a + 2*sqrt(b*x ^2 + a*x)*sqrt(b))/b^(9/2) - 9/2048*a^7*d*log(2*b*x + a + 2*sqrt(b*x^2 + a *x)*sqrt(b))/b^(11/2) - 7/512*sqrt(b*x^2 + a*x)*a^5*c/b^4 + 7/192*(b*x^2 + a*x)^(3/2)*a^3*c/b^3 - 7/60*(b*x^2 + a*x)^(5/2)*a*c/b^2 + 9/1024*sqrt(b*x ^2 + a*x)*a^6*d/b^5 - 3/128*(b*x^2 + a*x)^(3/2)*a^4*d/b^4 + 3/40*(b*x^2 + a*x)^(5/2)*a^2*d/b^3
Time = 0.14 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.83 \[ \int x^2 (c+d x) \left (a x+b x^2\right )^{3/2} \, dx=\frac {1}{107520} \, \sqrt {b x^{2} + a x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (12 \, b d x + \frac {14 \, b^{7} c + 15 \, a b^{6} d}{b^{6}}\right )} x + \frac {182 \, a b^{6} c + 3 \, a^{2} b^{5} d}{b^{6}}\right )} x + \frac {3 \, {\left (14 \, a^{2} b^{5} c - 9 \, a^{3} b^{4} d\right )}}{b^{6}}\right )} x - \frac {7 \, {\left (14 \, a^{3} b^{4} c - 9 \, a^{4} b^{3} d\right )}}{b^{6}}\right )} x + \frac {35 \, {\left (14 \, a^{4} b^{3} c - 9 \, a^{5} b^{2} d\right )}}{b^{6}}\right )} x - \frac {105 \, {\left (14 \, a^{5} b^{2} c - 9 \, a^{6} b d\right )}}{b^{6}}\right )} - \frac {{\left (14 \, a^{6} b c - 9 \, a^{7} d\right )} \log \left ({\left | 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} + a \right |}\right )}{2048 \, b^{\frac {11}{2}}} \] Input:
integrate(x^2*(d*x+c)*(b*x^2+a*x)^(3/2),x, algorithm="giac")
Output:
1/107520*sqrt(b*x^2 + a*x)*(2*(4*(2*(8*(10*(12*b*d*x + (14*b^7*c + 15*a*b^ 6*d)/b^6)*x + (182*a*b^6*c + 3*a^2*b^5*d)/b^6)*x + 3*(14*a^2*b^5*c - 9*a^3 *b^4*d)/b^6)*x - 7*(14*a^3*b^4*c - 9*a^4*b^3*d)/b^6)*x + 35*(14*a^4*b^3*c - 9*a^5*b^2*d)/b^6)*x - 105*(14*a^5*b^2*c - 9*a^6*b*d)/b^6) - 1/2048*(14*a ^6*b*c - 9*a^7*d)*log(abs(2*(sqrt(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) + a))/ b^(11/2)
Timed out. \[ \int x^2 (c+d x) \left (a x+b x^2\right )^{3/2} \, dx=\int x^2\,{\left (b\,x^2+a\,x\right )}^{3/2}\,\left (c+d\,x\right ) \,d x \] Input:
int(x^2*(a*x + b*x^2)^(3/2)*(c + d*x),x)
Output:
int(x^2*(a*x + b*x^2)^(3/2)*(c + d*x), x)
Time = 0.43 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.12 \[ \int x^2 (c+d x) \left (a x+b x^2\right )^{3/2} \, dx=\frac {945 \sqrt {x}\, \sqrt {b x +a}\, a^{6} b d -1470 \sqrt {x}\, \sqrt {b x +a}\, a^{5} b^{2} c -630 \sqrt {x}\, \sqrt {b x +a}\, a^{5} b^{2} d x +980 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{3} c x +504 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{3} d \,x^{2}-784 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{4} c \,x^{2}-432 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{4} d \,x^{3}+672 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{5} c \,x^{3}+384 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{5} d \,x^{4}+23296 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{6} c \,x^{4}+19200 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{6} d \,x^{5}+17920 \sqrt {x}\, \sqrt {b x +a}\, b^{7} c \,x^{5}+15360 \sqrt {x}\, \sqrt {b x +a}\, b^{7} d \,x^{6}-945 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{7} d +1470 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{6} b c}{107520 b^{6}} \] Input:
int(x^2*(d*x+c)*(b*x^2+a*x)^(3/2),x)
Output:
(945*sqrt(x)*sqrt(a + b*x)*a**6*b*d - 1470*sqrt(x)*sqrt(a + b*x)*a**5*b**2 *c - 630*sqrt(x)*sqrt(a + b*x)*a**5*b**2*d*x + 980*sqrt(x)*sqrt(a + b*x)*a **4*b**3*c*x + 504*sqrt(x)*sqrt(a + b*x)*a**4*b**3*d*x**2 - 784*sqrt(x)*sq rt(a + b*x)*a**3*b**4*c*x**2 - 432*sqrt(x)*sqrt(a + b*x)*a**3*b**4*d*x**3 + 672*sqrt(x)*sqrt(a + b*x)*a**2*b**5*c*x**3 + 384*sqrt(x)*sqrt(a + b*x)*a **2*b**5*d*x**4 + 23296*sqrt(x)*sqrt(a + b*x)*a*b**6*c*x**4 + 19200*sqrt(x )*sqrt(a + b*x)*a*b**6*d*x**5 + 17920*sqrt(x)*sqrt(a + b*x)*b**7*c*x**5 + 15360*sqrt(x)*sqrt(a + b*x)*b**7*d*x**6 - 945*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**7*d + 1470*sqrt(b)*log((sqrt(a + b*x) + sqrt (x)*sqrt(b))/sqrt(a))*a**6*b*c)/(107520*b**6)