Integrand size = 21, antiderivative size = 214 \[ \int (d+e x)^2 \left (b x+c x^2\right )^{3/2} \, dx=-\frac {b^2 \left (24 c^2 d^2-24 b c d e+7 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{512 c^4}+\frac {\left (24 c^2 d^2-24 b c d e+7 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}+\frac {7 e (2 c d-b e) \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e x) \left (b x+c x^2\right )^{5/2}}{6 c}+\frac {b^4 \left (24 c^2 d^2-24 b c d e+7 b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{512 c^{9/2}} \]
1/192*(7*b^2*e^2-24*b*c*d*e+24*c^2*d^2)*(2*c*x+b)*(c*x^2+b*x)^(3/2)/c^3+7/ 60*e*(-b*e+2*c*d)*(c*x^2+b*x)^(5/2)/c^2+1/6*e*(e*x+d)*(c*x^2+b*x)^(5/2)/c+ 1/512*b^4*(7*b^2*e^2-24*b*c*d*e+24*c^2*d^2)*arctanh(x*c^(1/2)/(c*x^2+b*x)^ (1/2))/c^(9/2)-1/512*b^2*(7*b^2*e^2-24*b*c*d*e+24*c^2*d^2)*(2*c*x+b)*(c*x^ 2+b*x)^(1/2)/c^4
Time = 1.82 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.31 \[ \int (d+e x)^2 \left (b x+c x^2\right )^{3/2} \, dx=\frac {\sqrt {x} \sqrt {b+c x} \left (\sqrt {c} \sqrt {x} \sqrt {b+c x} \left (-105 b^5 e^2+10 b^4 c e (36 d+7 e x)+48 b^2 c^3 x \left (5 d^2+4 d e x+e^2 x^2\right )-8 b^3 c^2 \left (45 d^2+30 d e x+7 e^2 x^2\right )+128 c^5 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )+64 b c^4 x^2 \left (45 d^2+66 d e x+26 e^2 x^2\right )\right )+720 b^5 c d e \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}-\sqrt {b+c x}}\right )+30 b^4 \left (24 c^2 d^2+7 b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{-\sqrt {b}+\sqrt {b+c x}}\right )\right )}{7680 c^{9/2} \sqrt {x (b+c x)}} \]
(Sqrt[x]*Sqrt[b + c*x]*(Sqrt[c]*Sqrt[x]*Sqrt[b + c*x]*(-105*b^5*e^2 + 10*b ^4*c*e*(36*d + 7*e*x) + 48*b^2*c^3*x*(5*d^2 + 4*d*e*x + e^2*x^2) - 8*b^3*c ^2*(45*d^2 + 30*d*e*x + 7*e^2*x^2) + 128*c^5*x^3*(15*d^2 + 24*d*e*x + 10*e ^2*x^2) + 64*b*c^4*x^2*(45*d^2 + 66*d*e*x + 26*e^2*x^2)) + 720*b^5*c*d*e*A rcTanh[(Sqrt[c]*Sqrt[x])/(Sqrt[b] - Sqrt[b + c*x])] + 30*b^4*(24*c^2*d^2 + 7*b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[x])/(-Sqrt[b] + Sqrt[b + c*x])]))/(7680* c^(9/2)*Sqrt[x*(b + c*x)])
Time = 0.35 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.90, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1166, 27, 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.
\(\displaystyle \int \left (b x+c x^2\right )^{3/2} (d+e x)^2 \, dx\) |
\(\Big \downarrow \) 1166 |
\(\displaystyle \frac {\int \frac {1}{2} (d (12 c d-5 b e)+7 e (2 c d-b e) x) \left (c x^2+b x\right )^{3/2}dx}{6 c}+\frac {e \left (b x+c x^2\right )^{5/2} (d+e x)}{6 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int (d (12 c d-5 b e)+7 e (2 c d-b e) x) \left (c x^2+b x\right )^{3/2}dx}{12 c}+\frac {e \left (b x+c x^2\right )^{5/2} (d+e x)}{6 c}\) |
\(\Big \downarrow \) 1160 |
\(\displaystyle \frac {\frac {\left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right ) \int \left (c x^2+b x\right )^{3/2}dx}{2 c}+\frac {7 e \left (b x+c x^2\right )^{5/2} (2 c d-b e)}{5 c}}{12 c}+\frac {e \left (b x+c x^2\right )^{5/2} (d+e x)}{6 c}\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {\frac {\left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right ) \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \int \sqrt {c x^2+b x}dx}{16 c}\right )}{2 c}+\frac {7 e \left (b x+c x^2\right )^{5/2} (2 c d-b e)}{5 c}}{12 c}+\frac {e \left (b x+c x^2\right )^{5/2} (d+e x)}{6 c}\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {\frac {\left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right ) \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \left (\frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c}-\frac {b^2 \int \frac {1}{\sqrt {c x^2+b x}}dx}{8 c}\right )}{16 c}\right )}{2 c}+\frac {7 e \left (b x+c x^2\right )^{5/2} (2 c d-b e)}{5 c}}{12 c}+\frac {e \left (b x+c x^2\right )^{5/2} (d+e x)}{6 c}\) |
\(\Big \downarrow \) 1091 |
\(\displaystyle \frac {\frac {\left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right ) \left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \left (\frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c}-\frac {b^2 \int \frac {1}{1-\frac {c x^2}{c x^2+b x}}d\frac {x}{\sqrt {c x^2+b x}}}{4 c}\right )}{16 c}\right )}{2 c}+\frac {7 e \left (b x+c x^2\right )^{5/2} (2 c d-b e)}{5 c}}{12 c}+\frac {e \left (b x+c x^2\right )^{5/2} (d+e x)}{6 c}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {\left (\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac {3 b^2 \left (\frac {(b+2 c x) \sqrt {b x+c x^2}}{4 c}-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 c^{3/2}}\right )}{16 c}\right ) \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right )}{2 c}+\frac {7 e \left (b x+c x^2\right )^{5/2} (2 c d-b e)}{5 c}}{12 c}+\frac {e \left (b x+c x^2\right )^{5/2} (d+e x)}{6 c}\) |
(e*(d + e*x)*(b*x + c*x^2)^(5/2))/(6*c) + ((7*e*(2*c*d - b*e)*(b*x + c*x^2 )^(5/2))/(5*c) + ((24*c^2*d^2 - 24*b*c*d*e + 7*b^2*e^2)*(((b + 2*c*x)*(b*x + c*x^2)^(3/2))/(8*c) - (3*b^2*(((b + 2*c*x)*Sqrt[b*x + c*x^2])/(4*c) - ( b^2*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(4*c^(3/2))))/(16*c)))/(2*c))/ (12*c)
3.3.96.3.1 Defintions of rubi rules used
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_))*((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_)*((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]
Time = 2.02 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.88
method | result | size |
pseudoelliptic | \(\frac {\frac {7 b^{4} \left (b^{2} e^{2}-\frac {24}{7} b c d e +\frac {24}{7} c^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )}{512}-\frac {7 \left (-\frac {128 x^{3} \left (\frac {2}{3} x^{2} e^{2}+\frac {8}{5} d e x +d^{2}\right ) c^{\frac {11}{2}}}{7}+\left (\frac {24 \left (\frac {7}{45} x^{2} e^{2}+\frac {2}{3} d e x +d^{2}\right ) b^{2} c^{\frac {5}{2}}}{7}-\frac {16 x \left (\frac {1}{5} x^{2} e^{2}+\frac {4}{5} d e x +d^{2}\right ) b \,c^{\frac {7}{2}}}{7}-\frac {192 x^{2} \left (\frac {26}{45} x^{2} e^{2}+\frac {22}{15} d e x +d^{2}\right ) c^{\frac {9}{2}}}{7}+\left (\left (-\frac {2 e x}{3}-\frac {24 d}{7}\right ) c^{\frac {3}{2}}+\sqrt {c}\, b e \right ) e \,b^{3}\right ) b \right ) \sqrt {x \left (c x +b \right )}}{512}}{c^{\frac {9}{2}}}\) | \(188\) |
risch | \(-\frac {\left (-1280 c^{5} e^{2} x^{5}-1664 b \,c^{4} e^{2} x^{4}-3072 c^{5} d e \,x^{4}-48 b^{2} c^{3} e^{2} x^{3}-4224 b \,c^{4} d e \,x^{3}-1920 c^{5} d^{2} x^{3}+56 b^{3} c^{2} e^{2} x^{2}-192 b^{2} c^{3} d e \,x^{2}-2880 b \,c^{4} d^{2} x^{2}-70 b^{4} c \,e^{2} x +240 b^{3} c^{2} d e x -240 b^{2} c^{3} d^{2} x +105 b^{5} e^{2}-360 b^{4} c d e +360 b^{3} c^{2} d^{2}\right ) x \left (c x +b \right )}{7680 c^{4} \sqrt {x \left (c x +b \right )}}+\frac {b^{4} \left (7 b^{2} e^{2}-24 b c d e +24 c^{2} d^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{1024 c^{\frac {9}{2}}}\) | \(246\) |
default | \(d^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )+e^{2} \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{6 c}-\frac {7 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )}{12 c}\right )+2 d e \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )\) | \(342\) |
7/512/c^(9/2)*(b^4*(b^2*e^2-24/7*b*c*d*e+24/7*c^2*d^2)*arctanh((x*(c*x+b)) ^(1/2)/x/c^(1/2))-(-128/7*x^3*(2/3*x^2*e^2+8/5*d*e*x+d^2)*c^(11/2)+(24/7*( 7/45*x^2*e^2+2/3*d*e*x+d^2)*b^2*c^(5/2)-16/7*x*(1/5*x^2*e^2+4/5*d*e*x+d^2) *b*c^(7/2)-192/7*x^2*(26/45*x^2*e^2+22/15*d*e*x+d^2)*c^(9/2)+((-2/3*e*x-24 /7*d)*c^(3/2)+c^(1/2)*b*e)*e*b^3)*b)*(x*(c*x+b))^(1/2))
Time = 0.29 (sec) , antiderivative size = 490, normalized size of antiderivative = 2.29 \[ \int (d+e x)^2 \left (b x+c x^2\right )^{3/2} \, dx=\left [\frac {15 \, {\left (24 \, b^{4} c^{2} d^{2} - 24 \, b^{5} c d e + 7 \, b^{6} e^{2}\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (1280 \, c^{6} e^{2} x^{5} - 360 \, b^{3} c^{3} d^{2} + 360 \, b^{4} c^{2} d e - 105 \, b^{5} c e^{2} + 128 \, {\left (24 \, c^{6} d e + 13 \, b c^{5} e^{2}\right )} x^{4} + 48 \, {\left (40 \, c^{6} d^{2} + 88 \, b c^{5} d e + b^{2} c^{4} e^{2}\right )} x^{3} + 8 \, {\left (360 \, b c^{5} d^{2} + 24 \, b^{2} c^{4} d e - 7 \, b^{3} c^{3} e^{2}\right )} x^{2} + 10 \, {\left (24 \, b^{2} c^{4} d^{2} - 24 \, b^{3} c^{3} d e + 7 \, b^{4} c^{2} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{15360 \, c^{5}}, -\frac {15 \, {\left (24 \, b^{4} c^{2} d^{2} - 24 \, b^{5} c d e + 7 \, b^{6} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (1280 \, c^{6} e^{2} x^{5} - 360 \, b^{3} c^{3} d^{2} + 360 \, b^{4} c^{2} d e - 105 \, b^{5} c e^{2} + 128 \, {\left (24 \, c^{6} d e + 13 \, b c^{5} e^{2}\right )} x^{4} + 48 \, {\left (40 \, c^{6} d^{2} + 88 \, b c^{5} d e + b^{2} c^{4} e^{2}\right )} x^{3} + 8 \, {\left (360 \, b c^{5} d^{2} + 24 \, b^{2} c^{4} d e - 7 \, b^{3} c^{3} e^{2}\right )} x^{2} + 10 \, {\left (24 \, b^{2} c^{4} d^{2} - 24 \, b^{3} c^{3} d e + 7 \, b^{4} c^{2} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{7680 \, c^{5}}\right ] \]
[1/15360*(15*(24*b^4*c^2*d^2 - 24*b^5*c*d*e + 7*b^6*e^2)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) + 2*(1280*c^6*e^2*x^5 - 360*b^3*c^3*d^ 2 + 360*b^4*c^2*d*e - 105*b^5*c*e^2 + 128*(24*c^6*d*e + 13*b*c^5*e^2)*x^4 + 48*(40*c^6*d^2 + 88*b*c^5*d*e + b^2*c^4*e^2)*x^3 + 8*(360*b*c^5*d^2 + 24 *b^2*c^4*d*e - 7*b^3*c^3*e^2)*x^2 + 10*(24*b^2*c^4*d^2 - 24*b^3*c^3*d*e + 7*b^4*c^2*e^2)*x)*sqrt(c*x^2 + b*x))/c^5, -1/7680*(15*(24*b^4*c^2*d^2 - 24 *b^5*c*d*e + 7*b^6*e^2)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - (1280*c^6*e^2*x^5 - 360*b^3*c^3*d^2 + 360*b^4*c^2*d*e - 105*b^5*c*e^2 + 128*(24*c^6*d*e + 13*b*c^5*e^2)*x^4 + 48*(40*c^6*d^2 + 88*b*c^5*d*e + b^2* c^4*e^2)*x^3 + 8*(360*b*c^5*d^2 + 24*b^2*c^4*d*e - 7*b^3*c^3*e^2)*x^2 + 10 *(24*b^2*c^4*d^2 - 24*b^3*c^3*d*e + 7*b^4*c^2*e^2)*x)*sqrt(c*x^2 + b*x))/c ^5]
Leaf count of result is larger than twice the leaf count of optimal. 570 vs. \(2 (207) = 414\).
Time = 0.49 (sec) , antiderivative size = 570, normalized size of antiderivative = 2.66 \[ \int (d+e x)^2 \left (b x+c x^2\right )^{3/2} \, dx=\begin {cases} \frac {3 b^{2} \left (b^{2} d^{2} - \frac {5 b \left (2 b^{2} d e + 2 b c d^{2} - \frac {7 b \left (b^{2} e^{2} + 4 b c d e - \frac {9 b \left (\frac {13 b c e^{2}}{12} + 2 c^{2} d e\right )}{10 c} + c^{2} d^{2}\right )}{8 c}\right )}{6 c}\right ) \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: \frac {b^{2}}{c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right )}{8 c^{2}} + \sqrt {b x + c x^{2}} \left (- \frac {3 b \left (b^{2} d^{2} - \frac {5 b \left (2 b^{2} d e + 2 b c d^{2} - \frac {7 b \left (b^{2} e^{2} + 4 b c d e - \frac {9 b \left (\frac {13 b c e^{2}}{12} + 2 c^{2} d e\right )}{10 c} + c^{2} d^{2}\right )}{8 c}\right )}{6 c}\right )}{4 c^{2}} + \frac {c e^{2} x^{5}}{6} + \frac {x^{4} \cdot \left (\frac {13 b c e^{2}}{12} + 2 c^{2} d e\right )}{5 c} + \frac {x^{3} \left (b^{2} e^{2} + 4 b c d e - \frac {9 b \left (\frac {13 b c e^{2}}{12} + 2 c^{2} d e\right )}{10 c} + c^{2} d^{2}\right )}{4 c} + \frac {x^{2} \cdot \left (2 b^{2} d e + 2 b c d^{2} - \frac {7 b \left (b^{2} e^{2} + 4 b c d e - \frac {9 b \left (\frac {13 b c e^{2}}{12} + 2 c^{2} d e\right )}{10 c} + c^{2} d^{2}\right )}{8 c}\right )}{3 c} + \frac {x \left (b^{2} d^{2} - \frac {5 b \left (2 b^{2} d e + 2 b c d^{2} - \frac {7 b \left (b^{2} e^{2} + 4 b c d e - \frac {9 b \left (\frac {13 b c e^{2}}{12} + 2 c^{2} d e\right )}{10 c} + c^{2} d^{2}\right )}{8 c}\right )}{6 c}\right )}{2 c}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (\frac {d^{2} \left (b x\right )^{\frac {5}{2}}}{5} + \frac {2 d e \left (b x\right )^{\frac {7}{2}}}{7 b} + \frac {e^{2} \left (b x\right )^{\frac {9}{2}}}{9 b^{2}}\right )}{b} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((3*b**2*(b**2*d**2 - 5*b*(2*b**2*d*e + 2*b*c*d**2 - 7*b*(b**2*e* *2 + 4*b*c*d*e - 9*b*(13*b*c*e**2/12 + 2*c**2*d*e)/(10*c) + c**2*d**2)/(8* c))/(6*c))*Piecewise((log(b + 2*sqrt(c)*sqrt(b*x + c*x**2) + 2*c*x)/sqrt(c ), Ne(b**2/c, 0)), ((b/(2*c) + x)*log(b/(2*c) + x)/sqrt(c*(b/(2*c) + x)**2 ), True))/(8*c**2) + sqrt(b*x + c*x**2)*(-3*b*(b**2*d**2 - 5*b*(2*b**2*d*e + 2*b*c*d**2 - 7*b*(b**2*e**2 + 4*b*c*d*e - 9*b*(13*b*c*e**2/12 + 2*c**2* d*e)/(10*c) + c**2*d**2)/(8*c))/(6*c))/(4*c**2) + c*e**2*x**5/6 + x**4*(13 *b*c*e**2/12 + 2*c**2*d*e)/(5*c) + x**3*(b**2*e**2 + 4*b*c*d*e - 9*b*(13*b *c*e**2/12 + 2*c**2*d*e)/(10*c) + c**2*d**2)/(4*c) + x**2*(2*b**2*d*e + 2* b*c*d**2 - 7*b*(b**2*e**2 + 4*b*c*d*e - 9*b*(13*b*c*e**2/12 + 2*c**2*d*e)/ (10*c) + c**2*d**2)/(8*c))/(3*c) + x*(b**2*d**2 - 5*b*(2*b**2*d*e + 2*b*c* d**2 - 7*b*(b**2*e**2 + 4*b*c*d*e - 9*b*(13*b*c*e**2/12 + 2*c**2*d*e)/(10* c) + c**2*d**2)/(8*c))/(6*c))/(2*c)), Ne(c, 0)), (2*(d**2*(b*x)**(5/2)/5 + 2*d*e*(b*x)**(7/2)/(7*b) + e**2*(b*x)**(9/2)/(9*b**2))/b, Ne(b, 0)), (0, True))
Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (190) = 380\).
Time = 0.20 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.94 \[ \int (d+e x)^2 \left (b x+c x^2\right )^{3/2} \, dx=\frac {1}{4} \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} d^{2} x - \frac {3 \, \sqrt {c x^{2} + b x} b^{2} d^{2} x}{32 \, c} + \frac {3 \, \sqrt {c x^{2} + b x} b^{3} d e x}{32 \, c^{2}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} b d e x}{4 \, c} - \frac {7 \, \sqrt {c x^{2} + b x} b^{4} e^{2} x}{256 \, c^{3}} + \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2} e^{2} x}{96 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} e^{2} x}{6 \, c} + \frac {3 \, b^{4} d^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {5}{2}}} - \frac {3 \, b^{5} d e \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {7}{2}}} + \frac {7 \, b^{6} e^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{1024 \, c^{\frac {9}{2}}} - \frac {3 \, \sqrt {c x^{2} + b x} b^{3} d^{2}}{64 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} b d^{2}}{8 \, c} + \frac {3 \, \sqrt {c x^{2} + b x} b^{4} d e}{64 \, c^{3}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2} d e}{8 \, c^{2}} + \frac {2 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} d e}{5 \, c} - \frac {7 \, \sqrt {c x^{2} + b x} b^{5} e^{2}}{512 \, c^{4}} + \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3} e^{2}}{192 \, c^{3}} - \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b e^{2}}{60 \, c^{2}} \]
1/4*(c*x^2 + b*x)^(3/2)*d^2*x - 3/32*sqrt(c*x^2 + b*x)*b^2*d^2*x/c + 3/32* sqrt(c*x^2 + b*x)*b^3*d*e*x/c^2 - 1/4*(c*x^2 + b*x)^(3/2)*b*d*e*x/c - 7/25 6*sqrt(c*x^2 + b*x)*b^4*e^2*x/c^3 + 7/96*(c*x^2 + b*x)^(3/2)*b^2*e^2*x/c^2 + 1/6*(c*x^2 + b*x)^(5/2)*e^2*x/c + 3/128*b^4*d^2*log(2*c*x + b + 2*sqrt( c*x^2 + b*x)*sqrt(c))/c^(5/2) - 3/128*b^5*d*e*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(7/2) + 7/1024*b^6*e^2*log(2*c*x + b + 2*sqrt(c*x^2 + b *x)*sqrt(c))/c^(9/2) - 3/64*sqrt(c*x^2 + b*x)*b^3*d^2/c^2 + 1/8*(c*x^2 + b *x)^(3/2)*b*d^2/c + 3/64*sqrt(c*x^2 + b*x)*b^4*d*e/c^3 - 1/8*(c*x^2 + b*x) ^(3/2)*b^2*d*e/c^2 + 2/5*(c*x^2 + b*x)^(5/2)*d*e/c - 7/512*sqrt(c*x^2 + b* x)*b^5*e^2/c^4 + 7/192*(c*x^2 + b*x)^(3/2)*b^3*e^2/c^3 - 7/60*(c*x^2 + b*x )^(5/2)*b*e^2/c^2
Time = 0.29 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.22 \[ \int (d+e x)^2 \left (b x+c x^2\right )^{3/2} \, dx=\frac {1}{7680} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, c e^{2} x + \frac {24 \, c^{6} d e + 13 \, b c^{5} e^{2}}{c^{5}}\right )} x + \frac {3 \, {\left (40 \, c^{6} d^{2} + 88 \, b c^{5} d e + b^{2} c^{4} e^{2}\right )}}{c^{5}}\right )} x + \frac {360 \, b c^{5} d^{2} + 24 \, b^{2} c^{4} d e - 7 \, b^{3} c^{3} e^{2}}{c^{5}}\right )} x + \frac {5 \, {\left (24 \, b^{2} c^{4} d^{2} - 24 \, b^{3} c^{3} d e + 7 \, b^{4} c^{2} e^{2}\right )}}{c^{5}}\right )} x - \frac {15 \, {\left (24 \, b^{3} c^{3} d^{2} - 24 \, b^{4} c^{2} d e + 7 \, b^{5} c e^{2}\right )}}{c^{5}}\right )} - \frac {{\left (24 \, b^{4} c^{2} d^{2} - 24 \, b^{5} c d e + 7 \, b^{6} e^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{1024 \, c^{\frac {9}{2}}} \]
1/7680*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(10*c*e^2*x + (24*c^6*d*e + 13*b*c^5* e^2)/c^5)*x + 3*(40*c^6*d^2 + 88*b*c^5*d*e + b^2*c^4*e^2)/c^5)*x + (360*b* c^5*d^2 + 24*b^2*c^4*d*e - 7*b^3*c^3*e^2)/c^5)*x + 5*(24*b^2*c^4*d^2 - 24* b^3*c^3*d*e + 7*b^4*c^2*e^2)/c^5)*x - 15*(24*b^3*c^3*d^2 - 24*b^4*c^2*d*e + 7*b^5*c*e^2)/c^5) - 1/1024*(24*b^4*c^2*d^2 - 24*b^5*c*d*e + 7*b^6*e^2)*l og(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) + b))/c^(9/2)
Timed out. \[ \int (d+e x)^2 \left (b x+c x^2\right )^{3/2} \, dx=\int {\left (c\,x^2+b\,x\right )}^{3/2}\,{\left (d+e\,x\right )}^2 \,d x \]