Integrand size = 21, antiderivative size = 275 \[ \int (c+d x)^3 \sqrt {a x+b x^2} \, dx=\frac {a (2 b c-a d) \left (16 b^2 c^2-16 a b c d+7 a^2 d^2\right ) \sqrt {a x+b x^2}}{128 b^4}+\frac {(2 b c-a d) \left (16 b^2 c^2-16 a b c d+7 a^2 d^2\right ) x \sqrt {a x+b x^2}}{64 b^3}+\frac {d \left (48 b^2 c^2-30 a b c d+7 a^2 d^2\right ) \left (a x+b x^2\right )^{3/2}}{48 b^3}+\frac {d^2 (30 b c-7 a d) x \left (a x+b x^2\right )^{3/2}}{40 b^2}+\frac {d^3 x^2 \left (a x+b x^2\right )^{3/2}}{5 b}-\frac {a^2 (2 b c-a d) \left (16 b^2 c^2-16 a b c d+7 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{128 b^{9/2}} \] Output:
1/128*a*(-a*d+2*b*c)*(7*a^2*d^2-16*a*b*c*d+16*b^2*c^2)*(b*x^2+a*x)^(1/2)/b ^4+1/64*(-a*d+2*b*c)*(7*a^2*d^2-16*a*b*c*d+16*b^2*c^2)*x*(b*x^2+a*x)^(1/2) /b^3+1/48*d*(7*a^2*d^2-30*a*b*c*d+48*b^2*c^2)*(b*x^2+a*x)^(3/2)/b^3+1/40*d ^2*(-7*a*d+30*b*c)*x*(b*x^2+a*x)^(3/2)/b^2+1/5*d^3*x^2*(b*x^2+a*x)^(3/2)/b -1/128*a^2*(-a*d+2*b*c)*(7*a^2*d^2-16*a*b*c*d+16*b^2*c^2)*arctanh(b^(1/2)* x/(b*x^2+a*x)^(1/2))/b^(9/2)
Time = 1.29 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.02 \[ \int (c+d x)^3 \sqrt {a x+b x^2} \, dx=\frac {\sqrt {x (a+b x)} \left (480 a b^3 c^3-720 a^2 b^2 c^2 d+450 a^3 b c d^2-105 a^4 d^3+960 b^4 c^3 x+480 a b^3 c^2 d x-300 a^2 b^2 c d^2 x+70 a^3 b d^3 x+1920 b^4 c^2 d x^2+240 a b^3 c d^2 x^2-56 a^2 b^2 d^3 x^2+1440 b^4 c d^2 x^3+48 a b^3 d^3 x^3+384 b^4 d^3 x^4\right )}{1920 b^4}+\frac {a^2 \left (-32 b^3 c^3+48 a b^2 c^2 d-30 a^2 b c d^2+7 a^3 d^3\right ) \sqrt {x (a+b x)} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )}{64 b^{9/2} \sqrt {x} \sqrt {a+b x}} \] Input:
Integrate[(c + d*x)^3*Sqrt[a*x + b*x^2],x]
Output:
(Sqrt[x*(a + b*x)]*(480*a*b^3*c^3 - 720*a^2*b^2*c^2*d + 450*a^3*b*c*d^2 - 105*a^4*d^3 + 960*b^4*c^3*x + 480*a*b^3*c^2*d*x - 300*a^2*b^2*c*d^2*x + 70 *a^3*b*d^3*x + 1920*b^4*c^2*d*x^2 + 240*a*b^3*c*d^2*x^2 - 56*a^2*b^2*d^3*x ^2 + 1440*b^4*c*d^2*x^3 + 48*a*b^3*d^3*x^3 + 384*b^4*d^3*x^4))/(1920*b^4) + (a^2*(-32*b^3*c^3 + 48*a*b^2*c^2*d - 30*a^2*b*c*d^2 + 7*a^3*d^3)*Sqrt[x* (a + b*x)]*ArcTanh[(Sqrt[b]*Sqrt[x])/(-Sqrt[a] + Sqrt[a + b*x])])/(64*b^(9 /2)*Sqrt[x]*Sqrt[a + b*x])
Time = 0.65 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.71, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1166, 27, 1225, 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.
| \(\displaystyle \int \sqrt {a x+b x^2} (c+d x)^3 \, dx\) |
| \(\Big \downarrow \) 1166 |
| \(\displaystyle \frac {\int \frac {1}{2} (c+d x) (c (10 b c-3 a d)+7 d (2 b c-a d) x) \sqrt {b x^2+a x}dx}{5 b}+\frac {d \left (a x+b x^2\right )^{3/2} (c+d x)^2}{5 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (c+d x) (c (10 b c-3 a d)+7 d (2 b c-a d) x) \sqrt {b x^2+a x}dx}{10 b}+\frac {d \left (a x+b x^2\right )^{3/2} (c+d x)^2}{5 b}\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {\frac {5 (2 b c-a d) \left (7 a^2 d^2-16 a b c d+16 b^2 c^2\right ) \int \sqrt {b x^2+a x}dx}{16 b^2}+\frac {d \left (a x+b x^2\right )^{3/2} \left (35 a^2 d^2+42 b d x (2 b c-a d)-150 a b c d+192 b^2 c^2\right )}{24 b^2}}{10 b}+\frac {d \left (a x+b x^2\right )^{3/2} (c+d x)^2}{5 b}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {\frac {5 (2 b c-a d) \left (7 a^2 d^2-16 a b c d+16 b^2 c^2\right ) \left (\frac {(a+2 b x) \sqrt {a x+b x^2}}{4 b}-\frac {a^2 \int \frac {1}{\sqrt {b x^2+a x}}dx}{8 b}\right )}{16 b^2}+\frac {d \left (a x+b x^2\right )^{3/2} \left (35 a^2 d^2+42 b d x (2 b c-a d)-150 a b c d+192 b^2 c^2\right )}{24 b^2}}{10 b}+\frac {d \left (a x+b x^2\right )^{3/2} (c+d x)^2}{5 b}\) |
| \(\Big \downarrow \) 1091 |
| \(\displaystyle \frac {\frac {5 (2 b c-a d) \left (7 a^2 d^2-16 a b c d+16 b^2 c^2\right ) \left (\frac {(a+2 b x) \sqrt {a x+b x^2}}{4 b}-\frac {a^2 \int \frac {1}{1-\frac {b x^2}{b x^2+a x}}d\frac {x}{\sqrt {b x^2+a x}}}{4 b}\right )}{16 b^2}+\frac {d \left (a x+b x^2\right )^{3/2} \left (35 a^2 d^2+42 b d x (2 b c-a d)-150 a b c d+192 b^2 c^2\right )}{24 b^2}}{10 b}+\frac {d \left (a x+b x^2\right )^{3/2} (c+d x)^2}{5 b}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {5 \left (\frac {(a+2 b x) \sqrt {a x+b x^2}}{4 b}-\frac {a^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{4 b^{3/2}}\right ) (2 b c-a d) \left (7 a^2 d^2-16 a b c d+16 b^2 c^2\right )}{16 b^2}+\frac {d \left (a x+b x^2\right )^{3/2} \left (35 a^2 d^2+42 b d x (2 b c-a d)-150 a b c d+192 b^2 c^2\right )}{24 b^2}}{10 b}+\frac {d \left (a x+b x^2\right )^{3/2} (c+d x)^2}{5 b}\) |
Input:
Int[(c + d*x)^3*Sqrt[a*x + b*x^2],x]
Output:
(d*(c + d*x)^2*(a*x + b*x^2)^(3/2))/(5*b) + ((d*(192*b^2*c^2 - 150*a*b*c*d + 35*a^2*d^2 + 42*b*d*(2*b*c - a*d)*x)*(a*x + b*x^2)^(3/2))/(24*b^2) + (5 *(2*b*c - a*d)*(16*b^2*c^2 - 16*a*b*c*d + 7*a^2*d^2)*(((a + 2*b*x)*Sqrt[a* x + b*x^2])/(4*b) - (a^2*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(4*b^(3/2 ))))/(16*b^2))/(10*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[1/(c*(m + 2*p + 1)) Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* (a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat icQ[a, b, c, d, e, m, p, x]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c , d, e, f, g, p}, x] && !LeQ[p, -1]
Time = 0.57 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.68
| method | result | size |
| pseudoelliptic | \(\frac {\frac {7 \left (a^{2} d^{2}-\frac {16}{7} a b c d +\frac {16}{7} b^{2} c^{2}\right ) \left (a d -2 b c \right ) a^{2} \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right )}{128}-\frac {7 \left (-\frac {32 \left (\frac {1}{10} d^{3} x^{3}+\frac {1}{2} c \,d^{2} x^{2}+c^{2} d x +c^{3}\right ) a \,b^{\frac {7}{2}}}{7}-\frac {64 x \left (\frac {2}{5} d^{3} x^{3}+\frac {3}{2} c \,d^{2} x^{2}+2 c^{2} d x +c^{3}\right ) b^{\frac {9}{2}}}{7}+d \left (\left (\frac {8}{15} d^{2} x^{2}+\frac {20}{7} c d x +\frac {48}{7} c^{2}\right ) b^{\frac {5}{2}}+d \left (\left (-\frac {2 d x}{3}-\frac {30 c}{7}\right ) b^{\frac {3}{2}}+\sqrt {b}\, a d \right ) a \right ) a^{2}\right ) \sqrt {x \left (b x +a \right )}}{128}}{b^{\frac {9}{2}}}\) | \(188\) |
| risch | \(-\frac {\left (-384 b^{4} d^{3} x^{4}-48 a \,b^{3} d^{3} x^{3}-1440 b^{4} c \,d^{2} x^{3}+56 a^{2} b^{2} d^{3} x^{2}-240 a \,b^{3} c \,d^{2} x^{2}-1920 b^{4} c^{2} d \,x^{2}-70 a^{3} b \,d^{3} x +300 a^{2} b^{2} c \,d^{2} x -480 a \,b^{3} c^{2} d x -960 b^{4} c^{3} x +105 a^{4} d^{3}-450 a^{3} b c \,d^{2}+720 a^{2} b^{2} c^{2} d -480 a \,b^{3} c^{3}\right ) x \left (b x +a \right )}{1920 b^{4} \sqrt {x \left (b x +a \right )}}+\frac {a^{2} \left (7 a^{3} d^{3}-30 a^{2} b c \,d^{2}+48 a \,b^{2} c^{2} d -32 b^{3} c^{3}\right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{256 b^{\frac {9}{2}}}\) | \(248\) |
| default | \(c^{3} \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )+d^{3} \left (\frac {x^{2} \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{5 b}-\frac {7 a \left (\frac {x \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{4 b}-\frac {5 a \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{3 b}-\frac {a \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{2 b}\right )}{8 b}\right )}{10 b}\right )+3 c \,d^{2} \left (\frac {x \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{4 b}-\frac {5 a \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{3 b}-\frac {a \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{2 b}\right )}{8 b}\right )+3 c^{2} d \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{3 b}-\frac {a \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{2 b}\right )\) | \(385\) |
Input:
int((d*x+c)^3*(b*x^2+a*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
7/128/b^(9/2)*((a^2*d^2-16/7*a*b*c*d+16/7*b^2*c^2)*(a*d-2*b*c)*a^2*arctanh ((x*(b*x+a))^(1/2)/x/b^(1/2))-(-32/7*(1/10*d^3*x^3+1/2*c*d^2*x^2+c^2*d*x+c ^3)*a*b^(7/2)-64/7*x*(2/5*d^3*x^3+3/2*c*d^2*x^2+2*c^2*d*x+c^3)*b^(9/2)+d*( (8/15*d^2*x^2+20/7*c*d*x+48/7*c^2)*b^(5/2)+d*((-2/3*d*x-30/7*c)*b^(3/2)+b^ (1/2)*a*d)*a)*a^2)*(x*(b*x+a))^(1/2))
Time = 0.09 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.81 \[ \int (c+d x)^3 \sqrt {a x+b x^2} \, dx=\left [-\frac {15 \, {\left (32 \, a^{2} b^{3} c^{3} - 48 \, a^{3} b^{2} c^{2} d + 30 \, a^{4} b c d^{2} - 7 \, a^{5} d^{3}\right )} \sqrt {b} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) - 2 \, {\left (384 \, b^{5} d^{3} x^{4} + 480 \, a b^{4} c^{3} - 720 \, a^{2} b^{3} c^{2} d + 450 \, a^{3} b^{2} c d^{2} - 105 \, a^{4} b d^{3} + 48 \, {\left (30 \, b^{5} c d^{2} + a b^{4} d^{3}\right )} x^{3} + 8 \, {\left (240 \, b^{5} c^{2} d + 30 \, a b^{4} c d^{2} - 7 \, a^{2} b^{3} d^{3}\right )} x^{2} + 10 \, {\left (96 \, b^{5} c^{3} + 48 \, a b^{4} c^{2} d - 30 \, a^{2} b^{3} c d^{2} + 7 \, a^{3} b^{2} d^{3}\right )} x\right )} \sqrt {b x^{2} + a x}}{3840 \, b^{5}}, \frac {15 \, {\left (32 \, a^{2} b^{3} c^{3} - 48 \, a^{3} b^{2} c^{2} d + 30 \, a^{4} b c d^{2} - 7 \, a^{5} d^{3}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x + a}\right ) + {\left (384 \, b^{5} d^{3} x^{4} + 480 \, a b^{4} c^{3} - 720 \, a^{2} b^{3} c^{2} d + 450 \, a^{3} b^{2} c d^{2} - 105 \, a^{4} b d^{3} + 48 \, {\left (30 \, b^{5} c d^{2} + a b^{4} d^{3}\right )} x^{3} + 8 \, {\left (240 \, b^{5} c^{2} d + 30 \, a b^{4} c d^{2} - 7 \, a^{2} b^{3} d^{3}\right )} x^{2} + 10 \, {\left (96 \, b^{5} c^{3} + 48 \, a b^{4} c^{2} d - 30 \, a^{2} b^{3} c d^{2} + 7 \, a^{3} b^{2} d^{3}\right )} x\right )} \sqrt {b x^{2} + a x}}{1920 \, b^{5}}\right ] \] Input:
integrate((d*x+c)^3*(b*x^2+a*x)^(1/2),x, algorithm="fricas")
Output:
[-1/3840*(15*(32*a^2*b^3*c^3 - 48*a^3*b^2*c^2*d + 30*a^4*b*c*d^2 - 7*a^5*d ^3)*sqrt(b)*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b)) - 2*(384*b^5*d^3* x^4 + 480*a*b^4*c^3 - 720*a^2*b^3*c^2*d + 450*a^3*b^2*c*d^2 - 105*a^4*b*d^ 3 + 48*(30*b^5*c*d^2 + a*b^4*d^3)*x^3 + 8*(240*b^5*c^2*d + 30*a*b^4*c*d^2 - 7*a^2*b^3*d^3)*x^2 + 10*(96*b^5*c^3 + 48*a*b^4*c^2*d - 30*a^2*b^3*c*d^2 + 7*a^3*b^2*d^3)*x)*sqrt(b*x^2 + a*x))/b^5, 1/1920*(15*(32*a^2*b^3*c^3 - 4 8*a^3*b^2*c^2*d + 30*a^4*b*c*d^2 - 7*a^5*d^3)*sqrt(-b)*arctan(sqrt(b*x^2 + a*x)*sqrt(-b)/(b*x + a)) + (384*b^5*d^3*x^4 + 480*a*b^4*c^3 - 720*a^2*b^3 *c^2*d + 450*a^3*b^2*c*d^2 - 105*a^4*b*d^3 + 48*(30*b^5*c*d^2 + a*b^4*d^3) *x^3 + 8*(240*b^5*c^2*d + 30*a*b^4*c*d^2 - 7*a^2*b^3*d^3)*x^2 + 10*(96*b^5 *c^3 + 48*a*b^4*c^2*d - 30*a^2*b^3*c*d^2 + 7*a^3*b^2*d^3)*x)*sqrt(b*x^2 + a*x))/b^5]
Time = 0.48 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.58 \[ \int (c+d x)^3 \sqrt {a x+b x^2} \, dx=\begin {cases} - \frac {a \left (a c^{3} - \frac {3 a \left (3 a c^{2} d - \frac {5 a \left (3 a c d^{2} - \frac {7 a \left (\frac {a d^{3}}{10} + 3 b c d^{2}\right )}{8 b} + 3 b c^{2} d\right )}{6 b} + b c^{3}\right )}{4 b}\right ) \left (\begin {cases} \frac {\log {\left (a + 2 \sqrt {b} \sqrt {a x + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: \frac {a^{2}}{b} \neq 0 \\\frac {\left (\frac {a}{2 b} + x\right ) \log {\left (\frac {a}{2 b} + x \right )}}{\sqrt {b \left (\frac {a}{2 b} + x\right )^{2}}} & \text {otherwise} \end {cases}\right )}{2 b} + \sqrt {a x + b x^{2}} \left (\frac {d^{3} x^{4}}{5} + \frac {x^{3} \left (\frac {a d^{3}}{10} + 3 b c d^{2}\right )}{4 b} + \frac {x^{2} \cdot \left (3 a c d^{2} - \frac {7 a \left (\frac {a d^{3}}{10} + 3 b c d^{2}\right )}{8 b} + 3 b c^{2} d\right )}{3 b} + \frac {x \left (3 a c^{2} d - \frac {5 a \left (3 a c d^{2} - \frac {7 a \left (\frac {a d^{3}}{10} + 3 b c d^{2}\right )}{8 b} + 3 b c^{2} d\right )}{6 b} + b c^{3}\right )}{2 b} + \frac {a c^{3} - \frac {3 a \left (3 a c^{2} d - \frac {5 a \left (3 a c d^{2} - \frac {7 a \left (\frac {a d^{3}}{10} + 3 b c d^{2}\right )}{8 b} + 3 b c^{2} d\right )}{6 b} + b c^{3}\right )}{4 b}}{b}\right ) & \text {for}\: b \neq 0 \\\frac {2 \left (\frac {c^{3} \left (a x\right )^{\frac {3}{2}}}{3} + \frac {3 c^{2} d \left (a x\right )^{\frac {5}{2}}}{5 a} + \frac {3 c d^{2} \left (a x\right )^{\frac {7}{2}}}{7 a^{2}} + \frac {d^{3} \left (a x\right )^{\frac {9}{2}}}{9 a^{3}}\right )}{a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate((d*x+c)**3*(b*x**2+a*x)**(1/2),x)
Output:
Piecewise((-a*(a*c**3 - 3*a*(3*a*c**2*d - 5*a*(3*a*c*d**2 - 7*a*(a*d**3/10 + 3*b*c*d**2)/(8*b) + 3*b*c**2*d)/(6*b) + b*c**3)/(4*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))/(2*b) + sqrt(a*x + b*x**2)*(d**3*x**4/5 + x**3*(a*d**3/10 + 3*b*c*d**2)/(4*b) + x**2*(3*a*c *d**2 - 7*a*(a*d**3/10 + 3*b*c*d**2)/(8*b) + 3*b*c**2*d)/(3*b) + x*(3*a*c* *2*d - 5*a*(3*a*c*d**2 - 7*a*(a*d**3/10 + 3*b*c*d**2)/(8*b) + 3*b*c**2*d)/ (6*b) + b*c**3)/(2*b) + (a*c**3 - 3*a*(3*a*c**2*d - 5*a*(3*a*c*d**2 - 7*a* (a*d**3/10 + 3*b*c*d**2)/(8*b) + 3*b*c**2*d)/(6*b) + b*c**3)/(4*b))/b), Ne (b, 0)), (2*(c**3*(a*x)**(3/2)/3 + 3*c**2*d*(a*x)**(5/2)/(5*a) + 3*c*d**2* (a*x)**(7/2)/(7*a**2) + d**3*(a*x)**(9/2)/(9*a**3))/a, Ne(a, 0)), (0, True ))
Time = 0.04 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.60 \[ \int (c+d x)^3 \sqrt {a x+b x^2} \, dx=\frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} d^{3} x^{2}}{5 \, b} + \frac {1}{2} \, \sqrt {b x^{2} + a x} c^{3} x - \frac {3 \, \sqrt {b x^{2} + a x} a c^{2} d x}{4 \, b} + \frac {15 \, \sqrt {b x^{2} + a x} a^{2} c d^{2} x}{32 \, b^{2}} + \frac {3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} c d^{2} x}{4 \, b} - \frac {7 \, \sqrt {b x^{2} + a x} a^{3} d^{3} x}{64 \, b^{3}} - \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a d^{3} x}{40 \, b^{2}} - \frac {a^{2} c^{3} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{8 \, b^{\frac {3}{2}}} + \frac {3 \, a^{3} c^{2} d \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{16 \, b^{\frac {5}{2}}} - \frac {15 \, a^{4} c d^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{128 \, b^{\frac {7}{2}}} + \frac {7 \, a^{5} d^{3} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{256 \, b^{\frac {9}{2}}} + \frac {\sqrt {b x^{2} + a x} a c^{3}}{4 \, b} - \frac {3 \, \sqrt {b x^{2} + a x} a^{2} c^{2} d}{8 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} c^{2} d}{b} + \frac {15 \, \sqrt {b x^{2} + a x} a^{3} c d^{2}}{64 \, b^{3}} - \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a c d^{2}}{8 \, b^{2}} - \frac {7 \, \sqrt {b x^{2} + a x} a^{4} d^{3}}{128 \, b^{4}} + \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2} d^{3}}{48 \, b^{3}} \] Input:
integrate((d*x+c)^3*(b*x^2+a*x)^(1/2),x, algorithm="maxima")
Output:
1/5*(b*x^2 + a*x)^(3/2)*d^3*x^2/b + 1/2*sqrt(b*x^2 + a*x)*c^3*x - 3/4*sqrt (b*x^2 + a*x)*a*c^2*d*x/b + 15/32*sqrt(b*x^2 + a*x)*a^2*c*d^2*x/b^2 + 3/4* (b*x^2 + a*x)^(3/2)*c*d^2*x/b - 7/64*sqrt(b*x^2 + a*x)*a^3*d^3*x/b^3 - 7/4 0*(b*x^2 + a*x)^(3/2)*a*d^3*x/b^2 - 1/8*a^2*c^3*log(2*b*x + a + 2*sqrt(b*x ^2 + a*x)*sqrt(b))/b^(3/2) + 3/16*a^3*c^2*d*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(5/2) - 15/128*a^4*c*d^2*log(2*b*x + a + 2*sqrt(b*x^2 + a *x)*sqrt(b))/b^(7/2) + 7/256*a^5*d^3*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*s qrt(b))/b^(9/2) + 1/4*sqrt(b*x^2 + a*x)*a*c^3/b - 3/8*sqrt(b*x^2 + a*x)*a^ 2*c^2*d/b^2 + (b*x^2 + a*x)^(3/2)*c^2*d/b + 15/64*sqrt(b*x^2 + a*x)*a^3*c* d^2/b^3 - 5/8*(b*x^2 + a*x)^(3/2)*a*c*d^2/b^2 - 7/128*sqrt(b*x^2 + a*x)*a^ 4*d^3/b^4 + 7/48*(b*x^2 + a*x)^(3/2)*a^2*d^3/b^3
Time = 0.13 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.93 \[ \int (c+d x)^3 \sqrt {a x+b x^2} \, dx=\frac {1}{1920} \, \sqrt {b x^{2} + a x} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, d^{3} x + \frac {30 \, b^{4} c d^{2} + a b^{3} d^{3}}{b^{4}}\right )} x + \frac {240 \, b^{4} c^{2} d + 30 \, a b^{3} c d^{2} - 7 \, a^{2} b^{2} d^{3}}{b^{4}}\right )} x + \frac {5 \, {\left (96 \, b^{4} c^{3} + 48 \, a b^{3} c^{2} d - 30 \, a^{2} b^{2} c d^{2} + 7 \, a^{3} b d^{3}\right )}}{b^{4}}\right )} x + \frac {15 \, {\left (32 \, a b^{3} c^{3} - 48 \, a^{2} b^{2} c^{2} d + 30 \, a^{3} b c d^{2} - 7 \, a^{4} d^{3}\right )}}{b^{4}}\right )} + \frac {{\left (32 \, a^{2} b^{3} c^{3} - 48 \, a^{3} b^{2} c^{2} d + 30 \, a^{4} b c d^{2} - 7 \, a^{5} d^{3}\right )} \log \left ({\left | 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} + a \right |}\right )}{256 \, b^{\frac {9}{2}}} \] Input:
integrate((d*x+c)^3*(b*x^2+a*x)^(1/2),x, algorithm="giac")
Output:
1/1920*sqrt(b*x^2 + a*x)*(2*(4*(6*(8*d^3*x + (30*b^4*c*d^2 + a*b^3*d^3)/b^ 4)*x + (240*b^4*c^2*d + 30*a*b^3*c*d^2 - 7*a^2*b^2*d^3)/b^4)*x + 5*(96*b^4 *c^3 + 48*a*b^3*c^2*d - 30*a^2*b^2*c*d^2 + 7*a^3*b*d^3)/b^4)*x + 15*(32*a* b^3*c^3 - 48*a^2*b^2*c^2*d + 30*a^3*b*c*d^2 - 7*a^4*d^3)/b^4) + 1/256*(32* a^2*b^3*c^3 - 48*a^3*b^2*c^2*d + 30*a^4*b*c*d^2 - 7*a^5*d^3)*log(abs(2*(sq rt(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) + a))/b^(9/2)
Time = 9.48 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.31 \[ \int (c+d x)^3 \sqrt {a x+b x^2} \, dx=c^3\,\sqrt {b\,x^2+a\,x}\,\left (\frac {x}{2}+\frac {a}{4\,b}\right )-\frac {7\,a\,d^3\,\left (\frac {x\,{\left (b\,x^2+a\,x\right )}^{3/2}}{4\,b}-\frac {5\,a\,\left (\frac {a^3\,\ln \left (\frac {a+2\,b\,x}{\sqrt {b}}+2\,\sqrt {b\,x^2+a\,x}\right )}{16\,b^{5/2}}+\frac {\sqrt {b\,x^2+a\,x}\,\left (-3\,a^2+2\,a\,b\,x+8\,b^2\,x^2\right )}{24\,b^2}\right )}{8\,b}\right )}{10\,b}+\frac {d^3\,x^2\,{\left (b\,x^2+a\,x\right )}^{3/2}}{5\,b}-\frac {a^2\,c^3\,\ln \left (\frac {\frac {a}{2}+b\,x}{\sqrt {b}}+\sqrt {b\,x^2+a\,x}\right )}{8\,b^{3/2}}+\frac {3\,c\,d^2\,x\,{\left (b\,x^2+a\,x\right )}^{3/2}}{4\,b}-\frac {15\,a\,c\,d^2\,\left (\frac {a^3\,\ln \left (\frac {a+2\,b\,x}{\sqrt {b}}+2\,\sqrt {b\,x^2+a\,x}\right )}{16\,b^{5/2}}+\frac {\sqrt {b\,x^2+a\,x}\,\left (-3\,a^2+2\,a\,b\,x+8\,b^2\,x^2\right )}{24\,b^2}\right )}{8\,b}+\frac {3\,a^3\,c^2\,d\,\ln \left (\frac {a+2\,b\,x}{\sqrt {b}}+2\,\sqrt {b\,x^2+a\,x}\right )}{16\,b^{5/2}}+\frac {c^2\,d\,\sqrt {b\,x^2+a\,x}\,\left (-3\,a^2+2\,a\,b\,x+8\,b^2\,x^2\right )}{8\,b^2} \] Input:
int((a*x + b*x^2)^(1/2)*(c + d*x)^3,x)
Output:
c^3*(a*x + b*x^2)^(1/2)*(x/2 + a/(4*b)) - (7*a*d^3*((x*(a*x + b*x^2)^(3/2) )/(4*b) - (5*a*((a^3*log((a + 2*b*x)/b^(1/2) + 2*(a*x + b*x^2)^(1/2)))/(16 *b^(5/2)) + ((a*x + b*x^2)^(1/2)*(8*b^2*x^2 - 3*a^2 + 2*a*b*x))/(24*b^2))) /(8*b)))/(10*b) + (d^3*x^2*(a*x + b*x^2)^(3/2))/(5*b) - (a^2*c^3*log((a/2 + b*x)/b^(1/2) + (a*x + b*x^2)^(1/2)))/(8*b^(3/2)) + (3*c*d^2*x*(a*x + b*x ^2)^(3/2))/(4*b) - (15*a*c*d^2*((a^3*log((a + 2*b*x)/b^(1/2) + 2*(a*x + b* x^2)^(1/2)))/(16*b^(5/2)) + ((a*x + b*x^2)^(1/2)*(8*b^2*x^2 - 3*a^2 + 2*a* b*x))/(24*b^2)))/(8*b) + (3*a^3*c^2*d*log((a + 2*b*x)/b^(1/2) + 2*(a*x + b *x^2)^(1/2)))/(16*b^(5/2)) + (c^2*d*(a*x + b*x^2)^(1/2)*(8*b^2*x^2 - 3*a^2 + 2*a*b*x))/(8*b^2)
Time = 1.67 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.45 \[ \int (c+d x)^3 \sqrt {a x+b x^2} \, dx=\frac {-105 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b \,d^{3}+450 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{2} c \,d^{2}+70 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{2} d^{3} x -720 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{3} c^{2} d -300 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{3} c \,d^{2} x -56 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{3} d^{3} x^{2}+480 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{4} c^{3}+480 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{4} c^{2} d x +240 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{4} c \,d^{2} x^{2}+48 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{4} d^{3} x^{3}+960 \sqrt {x}\, \sqrt {b x +a}\, b^{5} c^{3} x +1920 \sqrt {x}\, \sqrt {b x +a}\, b^{5} c^{2} d \,x^{2}+1440 \sqrt {x}\, \sqrt {b x +a}\, b^{5} c \,d^{2} x^{3}+384 \sqrt {x}\, \sqrt {b x +a}\, b^{5} d^{3} x^{4}+105 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{5} d^{3}-450 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{4} b c \,d^{2}+720 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{3} b^{2} c^{2} d -480 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{2} b^{3} c^{3}}{1920 b^{5}} \] Input:
int((d*x+c)^3*(b*x^2+a*x)^(1/2),x)
Output:
( - 105*sqrt(x)*sqrt(a + b*x)*a**4*b*d**3 + 450*sqrt(x)*sqrt(a + b*x)*a**3 *b**2*c*d**2 + 70*sqrt(x)*sqrt(a + b*x)*a**3*b**2*d**3*x - 720*sqrt(x)*sqr t(a + b*x)*a**2*b**3*c**2*d - 300*sqrt(x)*sqrt(a + b*x)*a**2*b**3*c*d**2*x - 56*sqrt(x)*sqrt(a + b*x)*a**2*b**3*d**3*x**2 + 480*sqrt(x)*sqrt(a + b*x )*a*b**4*c**3 + 480*sqrt(x)*sqrt(a + b*x)*a*b**4*c**2*d*x + 240*sqrt(x)*sq rt(a + b*x)*a*b**4*c*d**2*x**2 + 48*sqrt(x)*sqrt(a + b*x)*a*b**4*d**3*x**3 + 960*sqrt(x)*sqrt(a + b*x)*b**5*c**3*x + 1920*sqrt(x)*sqrt(a + b*x)*b**5 *c**2*d*x**2 + 1440*sqrt(x)*sqrt(a + b*x)*b**5*c*d**2*x**3 + 384*sqrt(x)*s qrt(a + b*x)*b**5*d**3*x**4 + 105*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqr t(b))/sqrt(a))*a**5*d**3 - 450*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b ))/sqrt(a))*a**4*b*c*d**2 + 720*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt( b))/sqrt(a))*a**3*b**2*c**2*d - 480*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*s qrt(b))/sqrt(a))*a**2*b**3*c**3)/(1920*b**5)