\(\int (d+e x) (a+b x+c x^2)^{5/2} \, dx\) [2357]

Optimal result

Integrand size = 20, antiderivative size = 207 \[ \int (d+e x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {5 \left (b^2-4 a c\right )^2 (2 c d-b e) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^4}-\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{24 c^2}+\frac {e \left (a+b x+c x^2\right )^{7/2}}{7 c}-\frac {5 \left (b^2-4 a c\right )^3 (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{9/2}} \]

[Out]

-5/384*(-4*a*c+b^2)*(-b*e+2*c*d)*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c^3+1/24*(-b*e+2*c*d)*(2*c*x+b)*(c*x^2+b*x+a)^(
5/2)/c^2+1/7*e*(c*x^2+b*x+a)^(7/2)/c-5/2048*(-4*a*c+b^2)^3*(-b*e+2*c*d)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b
*x+a)^(1/2))/c^(9/2)+5/1024*(-4*a*c+b^2)^2*(-b*e+2*c*d)*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^4

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {654, 626, 635, 212} \[ \int (d+e x) \left (a+b x+c x^2\right )^{5/2} \, dx=-\frac {5 \left (b^2-4 a c\right )^3 (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{9/2}}+\frac {5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2} (2 c d-b e)}{1024 c^4}-\frac {5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{384 c^3}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{24 c^2}+\frac {e \left (a+b x+c x^2\right )^{7/2}}{7 c} \]

[In]

Int[(d + e*x)*(a + b*x + c*x^2)^(5/2),x]

[Out]

(5*(b^2 - 4*a*c)^2*(2*c*d - b*e)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(1024*c^4) - (5*(b^2 - 4*a*c)*(2*c*d - b*e
)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(384*c^3) + ((2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(24*c^2
) + (e*(a + b*x + c*x^2)^(7/2))/(7*c) - (5*(b^2 - 4*a*c)^3*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a
+ b*x + c*x^2])])/(2048*c^(9/2))

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 626

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] - Dist[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]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 654

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] + Dist[(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[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {e \left (a+b x+c x^2\right )^{7/2}}{7 c}+\frac {(2 c d-b e) \int \left (a+b x+c x^2\right )^{5/2} \, dx}{2 c} \\ & = \frac {(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{24 c^2}+\frac {e \left (a+b x+c x^2\right )^{7/2}}{7 c}-\frac {\left (5 \left (b^2-4 a c\right ) (2 c d-b e)\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{48 c^2} \\ & = -\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{24 c^2}+\frac {e \left (a+b x+c x^2\right )^{7/2}}{7 c}+\frac {\left (5 \left (b^2-4 a c\right )^2 (2 c d-b e)\right ) \int \sqrt {a+b x+c x^2} \, dx}{256 c^3} \\ & = \frac {5 \left (b^2-4 a c\right )^2 (2 c d-b e) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^4}-\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{24 c^2}+\frac {e \left (a+b x+c x^2\right )^{7/2}}{7 c}-\frac {\left (5 \left (b^2-4 a c\right )^3 (2 c d-b e)\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2048 c^4} \\ & = \frac {5 \left (b^2-4 a c\right )^2 (2 c d-b e) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^4}-\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{24 c^2}+\frac {e \left (a+b x+c x^2\right )^{7/2}}{7 c}-\frac {\left (5 \left (b^2-4 a c\right )^3 (2 c d-b e)\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{1024 c^4} \\ & = \frac {5 \left (b^2-4 a c\right )^2 (2 c d-b e) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^4}-\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{24 c^2}+\frac {e \left (a+b x+c x^2\right )^{7/2}}{7 c}-\frac {5 \left (b^2-4 a c\right )^3 (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.80 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.50 \[ \int (d+e x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {\sqrt {c} \sqrt {a+x (b+c x)} \left (-105 b^6 e+70 b^5 c (3 d+e x)+28 b^4 c (40 a e-c x (5 d+2 e x))+16 b^3 c^2 \left (c x^2 (7 d+3 e x)-14 a (10 d+3 e x)\right )+64 c^3 \left (48 a^3 e+8 c^3 x^5 (7 d+6 e x)+3 a^2 c x (77 d+48 e x)+2 a c^2 x^3 (91 d+72 e x)\right )+16 b^2 c^2 \left (-231 a^2 e+6 a c x (14 d+5 e x)+2 c^2 x^3 (189 d+148 e x)\right )+32 b c^3 \left (3 a^2 (77 d+19 e x)+8 c^2 x^4 (35 d+29 e x)+2 a c x^2 (273 d+197 e x)\right )\right )+105 \left (b^2-4 a c\right )^3 (-2 c d+b e) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{21504 c^{9/2}} \]

[In]

Integrate[(d + e*x)*(a + b*x + c*x^2)^(5/2),x]

[Out]

(Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-105*b^6*e + 70*b^5*c*(3*d + e*x) + 28*b^4*c*(40*a*e - c*x*(5*d + 2*e*x)) + 16
*b^3*c^2*(c*x^2*(7*d + 3*e*x) - 14*a*(10*d + 3*e*x)) + 64*c^3*(48*a^3*e + 8*c^3*x^5*(7*d + 6*e*x) + 3*a^2*c*x*
(77*d + 48*e*x) + 2*a*c^2*x^3*(91*d + 72*e*x)) + 16*b^2*c^2*(-231*a^2*e + 6*a*c*x*(14*d + 5*e*x) + 2*c^2*x^3*(
189*d + 148*e*x)) + 32*b*c^3*(3*a^2*(77*d + 19*e*x) + 8*c^2*x^4*(35*d + 29*e*x) + 2*a*c*x^2*(273*d + 197*e*x))
) + 105*(b^2 - 4*a*c)^3*(-2*c*d + b*e)*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a + x*(b + c*x)])])/(21504*c^(9/2)
)

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.52

method	result	size
default	\(d \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )+e \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )\)	\(314\)
risch	\(\frac {\left (3072 c^{6} e \,x^{6}+7424 b \,c^{5} e \,x^{5}+3584 c^{6} d \,x^{5}+9216 a \,c^{5} e \,x^{4}+4736 b^{2} c^{4} e \,x^{4}+8960 b \,c^{5} d \,x^{4}+12608 a b \,c^{4} e \,x^{3}+11648 a \,c^{5} d \,x^{3}+48 b^{3} c^{3} e \,x^{3}+6048 b^{2} c^{4} d \,x^{3}+9216 a^{2} c^{4} e \,x^{2}+480 a \,b^{2} c^{3} e \,x^{2}+17472 a b \,c^{4} d \,x^{2}-56 b^{4} c^{2} e \,x^{2}+112 b^{3} c^{3} d \,x^{2}+1824 a^{2} b \,c^{3} e x +14784 a^{2} c^{4} d x -672 a \,b^{3} c^{2} e x +1344 a \,b^{2} c^{3} d x +70 b^{5} c e x -140 b^{4} c^{2} d x +3072 a^{3} c^{3} e -3696 a^{2} b^{2} c^{2} e +7392 a^{2} b \,c^{3} d +1120 a \,b^{4} c e -2240 a \,b^{3} c^{2} d -105 b^{6} e +210 b^{5} c d \right ) \sqrt {c \,x^{2}+b x +a}}{21504 c^{4}}-\frac {5 \left (64 a^{3} b \,c^{3} e -128 a^{3} c^{4} d -48 a^{2} b^{3} c^{2} e +96 a^{2} b^{2} c^{3} d +12 a \,b^{5} c e -24 a \,b^{4} c^{2} d -b^{7} e +2 b^{6} c d \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2048 c^{\frac {9}{2}}}\)	\(413\)

[In]

int((e*x+d)*(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

d*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^
2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))
))+e*(1/7*(c*x^2+b*x+a)^(7/2)/c-1/2*b/c*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x+b
)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/
2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (181) = 362\).

Time = 0.39 (sec) , antiderivative size = 857, normalized size of antiderivative = 4.14 \[ \int (d+e x) \left (a+b x+c x^2\right )^{5/2} \, dx=\left [\frac {105 \, {\left (2 \, {\left (b^{6} c - 12 \, a b^{4} c^{2} + 48 \, a^{2} b^{2} c^{3} - 64 \, a^{3} c^{4}\right )} d - {\left (b^{7} - 12 \, a b^{5} c + 48 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} e\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (3072 \, c^{7} e x^{6} + 256 \, {\left (14 \, c^{7} d + 29 \, b c^{6} e\right )} x^{5} + 128 \, {\left (70 \, b c^{6} d + {\left (37 \, b^{2} c^{5} + 72 \, a c^{6}\right )} e\right )} x^{4} + 16 \, {\left (14 \, {\left (27 \, b^{2} c^{5} + 52 \, a c^{6}\right )} d + {\left (3 \, b^{3} c^{4} + 788 \, a b c^{5}\right )} e\right )} x^{3} + 8 \, {\left (14 \, {\left (b^{3} c^{4} + 156 \, a b c^{5}\right )} d - {\left (7 \, b^{4} c^{3} - 60 \, a b^{2} c^{4} - 1152 \, a^{2} c^{5}\right )} e\right )} x^{2} + 14 \, {\left (15 \, b^{5} c^{2} - 160 \, a b^{3} c^{3} + 528 \, a^{2} b c^{4}\right )} d - {\left (105 \, b^{6} c - 1120 \, a b^{4} c^{2} + 3696 \, a^{2} b^{2} c^{3} - 3072 \, a^{3} c^{4}\right )} e - 2 \, {\left (14 \, {\left (5 \, b^{4} c^{3} - 48 \, a b^{2} c^{4} - 528 \, a^{2} c^{5}\right )} d - {\left (35 \, b^{5} c^{2} - 336 \, a b^{3} c^{3} + 912 \, a^{2} b c^{4}\right )} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{86016 \, c^{5}}, \frac {105 \, {\left (2 \, {\left (b^{6} c - 12 \, a b^{4} c^{2} + 48 \, a^{2} b^{2} c^{3} - 64 \, a^{3} c^{4}\right )} d - {\left (b^{7} - 12 \, a b^{5} c + 48 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (3072 \, c^{7} e x^{6} + 256 \, {\left (14 \, c^{7} d + 29 \, b c^{6} e\right )} x^{5} + 128 \, {\left (70 \, b c^{6} d + {\left (37 \, b^{2} c^{5} + 72 \, a c^{6}\right )} e\right )} x^{4} + 16 \, {\left (14 \, {\left (27 \, b^{2} c^{5} + 52 \, a c^{6}\right )} d + {\left (3 \, b^{3} c^{4} + 788 \, a b c^{5}\right )} e\right )} x^{3} + 8 \, {\left (14 \, {\left (b^{3} c^{4} + 156 \, a b c^{5}\right )} d - {\left (7 \, b^{4} c^{3} - 60 \, a b^{2} c^{4} - 1152 \, a^{2} c^{5}\right )} e\right )} x^{2} + 14 \, {\left (15 \, b^{5} c^{2} - 160 \, a b^{3} c^{3} + 528 \, a^{2} b c^{4}\right )} d - {\left (105 \, b^{6} c - 1120 \, a b^{4} c^{2} + 3696 \, a^{2} b^{2} c^{3} - 3072 \, a^{3} c^{4}\right )} e - 2 \, {\left (14 \, {\left (5 \, b^{4} c^{3} - 48 \, a b^{2} c^{4} - 528 \, a^{2} c^{5}\right )} d - {\left (35 \, b^{5} c^{2} - 336 \, a b^{3} c^{3} + 912 \, a^{2} b c^{4}\right )} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{43008 \, c^{5}}\right ] \]

[In]

integrate((e*x+d)*(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[1/86016*(105*(2*(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 -
64*a^3*b*c^3)*e)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 + 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c
) + 4*(3072*c^7*e*x^6 + 256*(14*c^7*d + 29*b*c^6*e)*x^5 + 128*(70*b*c^6*d + (37*b^2*c^5 + 72*a*c^6)*e)*x^4 + 1
6*(14*(27*b^2*c^5 + 52*a*c^6)*d + (3*b^3*c^4 + 788*a*b*c^5)*e)*x^3 + 8*(14*(b^3*c^4 + 156*a*b*c^5)*d - (7*b^4*
c^3 - 60*a*b^2*c^4 - 1152*a^2*c^5)*e)*x^2 + 14*(15*b^5*c^2 - 160*a*b^3*c^3 + 528*a^2*b*c^4)*d - (105*b^6*c - 1
120*a*b^4*c^2 + 3696*a^2*b^2*c^3 - 3072*a^3*c^4)*e - 2*(14*(5*b^4*c^3 - 48*a*b^2*c^4 - 528*a^2*c^5)*d - (35*b^
5*c^2 - 336*a*b^3*c^3 + 912*a^2*b*c^4)*e)*x)*sqrt(c*x^2 + b*x + a))/c^5, 1/43008*(105*(2*(b^6*c - 12*a*b^4*c^2
+ 48*a^2*b^2*c^3 - 64*a^3*c^4)*d - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*e)*sqrt(-c)*arctan(1/2*
sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) + 2*(3072*c^7*e*x^6 + 256*(14*c^7*d + 29*b
*c^6*e)*x^5 + 128*(70*b*c^6*d + (37*b^2*c^5 + 72*a*c^6)*e)*x^4 + 16*(14*(27*b^2*c^5 + 52*a*c^6)*d + (3*b^3*c^4
+ 788*a*b*c^5)*e)*x^3 + 8*(14*(b^3*c^4 + 156*a*b*c^5)*d - (7*b^4*c^3 - 60*a*b^2*c^4 - 1152*a^2*c^5)*e)*x^2 +
14*(15*b^5*c^2 - 160*a*b^3*c^3 + 528*a^2*b*c^4)*d - (105*b^6*c - 1120*a*b^4*c^2 + 3696*a^2*b^2*c^3 - 3072*a^3*
c^4)*e - 2*(14*(5*b^4*c^3 - 48*a*b^2*c^4 - 528*a^2*c^5)*d - (35*b^5*c^2 - 336*a*b^3*c^3 + 912*a^2*b*c^4)*e)*x)
*sqrt(c*x^2 + b*x + a))/c^5]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2428 vs. \(2 (197) = 394\).

Time = 0.67 (sec) , antiderivative size = 2428, normalized size of antiderivative = 11.73 \[ \int (d+e x) \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)*(c*x**2+b*x+a)**(5/2),x)

[Out]

Piecewise((sqrt(a + b*x + c*x**2)*(c**2*e*x**6/7 + x**5*(29*b*c**2*e/14 + c**3*d)/(6*c) + x**4*(15*a*c**2*e/7
+ 3*b**2*c*e + 3*b*c**2*d - 11*b*(29*b*c**2*e/14 + c**3*d)/(12*c))/(5*c) + x**3*(6*a*b*c*e + 3*a*c**2*d - 5*a*
(29*b*c**2*e/14 + c**3*d)/(6*c) + b**3*e + 3*b**2*c*d - 9*b*(15*a*c**2*e/7 + 3*b**2*c*e + 3*b*c**2*d - 11*b*(2
9*b*c**2*e/14 + c**3*d)/(12*c))/(10*c))/(4*c) + x**2*(3*a**2*c*e + 3*a*b**2*e + 6*a*b*c*d - 4*a*(15*a*c**2*e/7
+ 3*b**2*c*e + 3*b*c**2*d - 11*b*(29*b*c**2*e/14 + c**3*d)/(12*c))/(5*c) + b**3*d - 7*b*(6*a*b*c*e + 3*a*c**2
*d - 5*a*(29*b*c**2*e/14 + c**3*d)/(6*c) + b**3*e + 3*b**2*c*d - 9*b*(15*a*c**2*e/7 + 3*b**2*c*e + 3*b*c**2*d
- 11*b*(29*b*c**2*e/14 + c**3*d)/(12*c))/(10*c))/(8*c))/(3*c) + x*(3*a**2*b*e + 3*a**2*c*d + 3*a*b**2*d - 3*a*
(6*a*b*c*e + 3*a*c**2*d - 5*a*(29*b*c**2*e/14 + c**3*d)/(6*c) + b**3*e + 3*b**2*c*d - 9*b*(15*a*c**2*e/7 + 3*b
**2*c*e + 3*b*c**2*d - 11*b*(29*b*c**2*e/14 + c**3*d)/(12*c))/(10*c))/(4*c) - 5*b*(3*a**2*c*e + 3*a*b**2*e + 6
*a*b*c*d - 4*a*(15*a*c**2*e/7 + 3*b**2*c*e + 3*b*c**2*d - 11*b*(29*b*c**2*e/14 + c**3*d)/(12*c))/(5*c) + b**3*
d - 7*b*(6*a*b*c*e + 3*a*c**2*d - 5*a*(29*b*c**2*e/14 + c**3*d)/(6*c) + b**3*e + 3*b**2*c*d - 9*b*(15*a*c**2*e
/7 + 3*b**2*c*e + 3*b*c**2*d - 11*b*(29*b*c**2*e/14 + c**3*d)/(12*c))/(10*c))/(8*c))/(6*c))/(2*c) + (a**3*e +
3*a**2*b*d - 2*a*(3*a**2*c*e + 3*a*b**2*e + 6*a*b*c*d - 4*a*(15*a*c**2*e/7 + 3*b**2*c*e + 3*b*c**2*d - 11*b*(2
9*b*c**2*e/14 + c**3*d)/(12*c))/(5*c) + b**3*d - 7*b*(6*a*b*c*e + 3*a*c**2*d - 5*a*(29*b*c**2*e/14 + c**3*d)/(
6*c) + b**3*e + 3*b**2*c*d - 9*b*(15*a*c**2*e/7 + 3*b**2*c*e + 3*b*c**2*d - 11*b*(29*b*c**2*e/14 + c**3*d)/(12
*c))/(10*c))/(8*c))/(3*c) - 3*b*(3*a**2*b*e + 3*a**2*c*d + 3*a*b**2*d - 3*a*(6*a*b*c*e + 3*a*c**2*d - 5*a*(29*
b*c**2*e/14 + c**3*d)/(6*c) + b**3*e + 3*b**2*c*d - 9*b*(15*a*c**2*e/7 + 3*b**2*c*e + 3*b*c**2*d - 11*b*(29*b*
c**2*e/14 + c**3*d)/(12*c))/(10*c))/(4*c) - 5*b*(3*a**2*c*e + 3*a*b**2*e + 6*a*b*c*d - 4*a*(15*a*c**2*e/7 + 3*
b**2*c*e + 3*b*c**2*d - 11*b*(29*b*c**2*e/14 + c**3*d)/(12*c))/(5*c) + b**3*d - 7*b*(6*a*b*c*e + 3*a*c**2*d -
5*a*(29*b*c**2*e/14 + c**3*d)/(6*c) + b**3*e + 3*b**2*c*d - 9*b*(15*a*c**2*e/7 + 3*b**2*c*e + 3*b*c**2*d - 11*
b*(29*b*c**2*e/14 + c**3*d)/(12*c))/(10*c))/(8*c))/(6*c))/(4*c))/c) + (a**3*d - a*(3*a**2*b*e + 3*a**2*c*d + 3
*a*b**2*d - 3*a*(6*a*b*c*e + 3*a*c**2*d - 5*a*(29*b*c**2*e/14 + c**3*d)/(6*c) + b**3*e + 3*b**2*c*d - 9*b*(15*
a*c**2*e/7 + 3*b**2*c*e + 3*b*c**2*d - 11*b*(29*b*c**2*e/14 + c**3*d)/(12*c))/(10*c))/(4*c) - 5*b*(3*a**2*c*e
+ 3*a*b**2*e + 6*a*b*c*d - 4*a*(15*a*c**2*e/7 + 3*b**2*c*e + 3*b*c**2*d - 11*b*(29*b*c**2*e/14 + c**3*d)/(12*c
))/(5*c) + b**3*d - 7*b*(6*a*b*c*e + 3*a*c**2*d - 5*a*(29*b*c**2*e/14 + c**3*d)/(6*c) + b**3*e + 3*b**2*c*d -
9*b*(15*a*c**2*e/7 + 3*b**2*c*e + 3*b*c**2*d - 11*b*(29*b*c**2*e/14 + c**3*d)/(12*c))/(10*c))/(8*c))/(6*c))/(2
*c) - b*(a**3*e + 3*a**2*b*d - 2*a*(3*a**2*c*e + 3*a*b**2*e + 6*a*b*c*d - 4*a*(15*a*c**2*e/7 + 3*b**2*c*e + 3*
b*c**2*d - 11*b*(29*b*c**2*e/14 + c**3*d)/(12*c))/(5*c) + b**3*d - 7*b*(6*a*b*c*e + 3*a*c**2*d - 5*a*(29*b*c**
2*e/14 + c**3*d)/(6*c) + b**3*e + 3*b**2*c*d - 9*b*(15*a*c**2*e/7 + 3*b**2*c*e + 3*b*c**2*d - 11*b*(29*b*c**2*
e/14 + c**3*d)/(12*c))/(10*c))/(8*c))/(3*c) - 3*b*(3*a**2*b*e + 3*a**2*c*d + 3*a*b**2*d - 3*a*(6*a*b*c*e + 3*a
*c**2*d - 5*a*(29*b*c**2*e/14 + c**3*d)/(6*c) + b**3*e + 3*b**2*c*d - 9*b*(15*a*c**2*e/7 + 3*b**2*c*e + 3*b*c*
*2*d - 11*b*(29*b*c**2*e/14 + c**3*d)/(12*c))/(10*c))/(4*c) - 5*b*(3*a**2*c*e + 3*a*b**2*e + 6*a*b*c*d - 4*a*(
15*a*c**2*e/7 + 3*b**2*c*e + 3*b*c**2*d - 11*b*(29*b*c**2*e/14 + c**3*d)/(12*c))/(5*c) + b**3*d - 7*b*(6*a*b*c
*e + 3*a*c**2*d - 5*a*(29*b*c**2*e/14 + c**3*d)/(6*c) + b**3*e + 3*b**2*c*d - 9*b*(15*a*c**2*e/7 + 3*b**2*c*e
+ 3*b*c**2*d - 11*b*(29*b*c**2*e/14 + c**3*d)/(12*c))/(10*c))/(8*c))/(6*c))/(4*c))/(2*c))*Piecewise((log(b + 2
*sqrt(c)*sqrt(a + b*x + c*x**2) + 2*c*x)/sqrt(c), Ne(a - b**2/(4*c), 0)), ((b/(2*c) + x)*log(b/(2*c) + x)/sqrt
(c*(b/(2*c) + x)**2), True)), Ne(c, 0)), (2*(e*(a + b*x)**(9/2)/(9*b) + (a + b*x)**(7/2)*(-a*e + b*d)/(7*b))/b
, Ne(b, 0)), (a**(5/2)*(d*x + e*x**2/2), True))

Maxima [F(-2)]

Exception generated. \[ \int (d+e x) \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((e*x+d)*(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (181) = 362\).

Time = 0.29 (sec) , antiderivative size = 423, normalized size of antiderivative = 2.04 \[ \int (d+e x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {1}{21504} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (12 \, c^{2} e x + \frac {14 \, c^{8} d + 29 \, b c^{7} e}{c^{6}}\right )} x + \frac {70 \, b c^{7} d + 37 \, b^{2} c^{6} e + 72 \, a c^{7} e}{c^{6}}\right )} x + \frac {378 \, b^{2} c^{6} d + 728 \, a c^{7} d + 3 \, b^{3} c^{5} e + 788 \, a b c^{6} e}{c^{6}}\right )} x + \frac {14 \, b^{3} c^{5} d + 2184 \, a b c^{6} d - 7 \, b^{4} c^{4} e + 60 \, a b^{2} c^{5} e + 1152 \, a^{2} c^{6} e}{c^{6}}\right )} x - \frac {70 \, b^{4} c^{4} d - 672 \, a b^{2} c^{5} d - 7392 \, a^{2} c^{6} d - 35 \, b^{5} c^{3} e + 336 \, a b^{3} c^{4} e - 912 \, a^{2} b c^{5} e}{c^{6}}\right )} x + \frac {210 \, b^{5} c^{3} d - 2240 \, a b^{3} c^{4} d + 7392 \, a^{2} b c^{5} d - 105 \, b^{6} c^{2} e + 1120 \, a b^{4} c^{3} e - 3696 \, a^{2} b^{2} c^{4} e + 3072 \, a^{3} c^{5} e}{c^{6}}\right )} + \frac {5 \, {\left (2 \, b^{6} c d - 24 \, a b^{4} c^{2} d + 96 \, a^{2} b^{2} c^{3} d - 128 \, a^{3} c^{4} d - b^{7} e + 12 \, a b^{5} c e - 48 \, a^{2} b^{3} c^{2} e + 64 \, a^{3} b c^{3} e\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{2048 \, c^{\frac {9}{2}}} \]

[In]

integrate((e*x+d)*(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

1/21504*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(12*c^2*e*x + (14*c^8*d + 29*b*c^7*e)/c^6)*x + (70*b*c^7*d + 37*b
^2*c^6*e + 72*a*c^7*e)/c^6)*x + (378*b^2*c^6*d + 728*a*c^7*d + 3*b^3*c^5*e + 788*a*b*c^6*e)/c^6)*x + (14*b^3*c
^5*d + 2184*a*b*c^6*d - 7*b^4*c^4*e + 60*a*b^2*c^5*e + 1152*a^2*c^6*e)/c^6)*x - (70*b^4*c^4*d - 672*a*b^2*c^5*
d - 7392*a^2*c^6*d - 35*b^5*c^3*e + 336*a*b^3*c^4*e - 912*a^2*b*c^5*e)/c^6)*x + (210*b^5*c^3*d - 2240*a*b^3*c^
4*d + 7392*a^2*b*c^5*d - 105*b^6*c^2*e + 1120*a*b^4*c^3*e - 3696*a^2*b^2*c^4*e + 3072*a^3*c^5*e)/c^6) + 5/2048
*(2*b^6*c*d - 24*a*b^4*c^2*d + 96*a^2*b^2*c^3*d - 128*a^3*c^4*d - b^7*e + 12*a*b^5*c*e - 48*a^2*b^3*c^2*e + 64
*a^3*b*c^3*e)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(9/2)

Mupad [F(-1)]

Timed out. \[ \int (d+e x) \left (a+b x+c x^2\right )^{5/2} \, dx=\int \left (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x \]

[In]

int((d + e*x)*(a + b*x + c*x^2)^(5/2),x)

[Out]

int((d + e*x)*(a + b*x + c*x^2)^(5/2), x)