3.3 \(\int \frac {(c+d x+e x^2)^3}{\sqrt {a+b x}} \, dx\)

Optimal. Leaf size=274 \[ -\frac {2 e (a+b x)^{9/2} \left (-5 a^2 e^2+5 a b d e-\left (b^2 \left (c e+d^2\right )\right )\right )}{3 b^7}-\frac {2 (a+b x)^{7/2} (b d-2 a e) \left (-10 a^2 e^2+10 a b d e-\left (b^2 \left (6 c e+d^2\right )\right )\right )}{7 b^7}-\frac {6 (a+b x)^{5/2} \left (a^2 e-a b d+b^2 c\right ) \left (-5 a^2 e^2+5 a b d e-\left (b^2 \left (c e+d^2\right )\right )\right )}{5 b^7}+\frac {2 (a+b x)^{3/2} (b d-2 a e) \left (a^2 e-a b d+b^2 c\right )^2}{b^7}+\frac {2 \sqrt {a+b x} \left (a^2 e-a b d+b^2 c\right )^3}{b^7}+\frac {6 e^2 (a+b x)^{11/2} (b d-2 a e)}{11 b^7}+\frac {2 e^3 (a+b x)^{13/2}}{13 b^7} \]

[Out]

2*(-2*a*e+b*d)*(a^2*e-a*b*d+b^2*c)^2*(b*x+a)^(3/2)/b^7-6/5*(a^2*e-a*b*d+b^2*c)*(5*a*b*d*e-5*a^2*e^2-b^2*(c*e+d
^2))*(b*x+a)^(5/2)/b^7-2/7*(-2*a*e+b*d)*(10*a*b*d*e-10*a^2*e^2-b^2*(6*c*e+d^2))*(b*x+a)^(7/2)/b^7-2/3*e*(5*a*b
*d*e-5*a^2*e^2-b^2*(c*e+d^2))*(b*x+a)^(9/2)/b^7+6/11*e^2*(-2*a*e+b*d)*(b*x+a)^(11/2)/b^7+2/13*e^3*(b*x+a)^(13/
2)/b^7+2*(a^2*e-a*b*d+b^2*c)^3*(b*x+a)^(1/2)/b^7

________________________________________________________________________________________

Rubi [A]  time = 0.19, antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {698} \[ -\frac {2 e (a+b x)^{9/2} \left (-5 a^2 e^2+5 a b d e+b^2 \left (-\left (c e+d^2\right )\right )\right )}{3 b^7}-\frac {2 (a+b x)^{7/2} (b d-2 a e) \left (-10 a^2 e^2+10 a b d e+b^2 \left (-\left (6 c e+d^2\right )\right )\right )}{7 b^7}-\frac {6 (a+b x)^{5/2} \left (a^2 e-a b d+b^2 c\right ) \left (-5 a^2 e^2+5 a b d e+b^2 \left (-\left (c e+d^2\right )\right )\right )}{5 b^7}+\frac {2 (a+b x)^{3/2} (b d-2 a e) \left (a^2 e-a b d+b^2 c\right )^2}{b^7}+\frac {2 \sqrt {a+b x} \left (a^2 e-a b d+b^2 c\right )^3}{b^7}+\frac {6 e^2 (a+b x)^{11/2} (b d-2 a e)}{11 b^7}+\frac {2 e^3 (a+b x)^{13/2}}{13 b^7} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x + e*x^2)^3/Sqrt[a + b*x],x]

[Out]

(2*(b^2*c - a*b*d + a^2*e)^3*Sqrt[a + b*x])/b^7 + (2*(b*d - 2*a*e)*(b^2*c - a*b*d + a^2*e)^2*(a + b*x)^(3/2))/
b^7 - (6*(b^2*c - a*b*d + a^2*e)*(5*a*b*d*e - 5*a^2*e^2 - b^2*(d^2 + c*e))*(a + b*x)^(5/2))/(5*b^7) - (2*(b*d
- 2*a*e)*(10*a*b*d*e - 10*a^2*e^2 - b^2*(d^2 + 6*c*e))*(a + b*x)^(7/2))/(7*b^7) - (2*e*(5*a*b*d*e - 5*a^2*e^2
- b^2*(d^2 + c*e))*(a + b*x)^(9/2))/(3*b^7) + (6*e^2*(b*d - 2*a*e)*(a + b*x)^(11/2))/(11*b^7) + (2*e^3*(a + b*
x)^(13/2))/(13*b^7)

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int \frac {\left (c+d x+e x^2\right )^3}{\sqrt {a+b x}} \, dx &=\int \left (\frac {\left (b^2 c-a b d+a^2 e\right )^3}{b^6 \sqrt {a+b x}}+\frac {3 (b d-2 a e) \left (b^2 c-a b d+a^2 e\right )^2 \sqrt {a+b x}}{b^6}+\frac {3 \left (b^2 c-a b d+a^2 e\right ) \left (b^2 d^2+b^2 c e-5 a b d e+5 a^2 e^2\right ) (a+b x)^{3/2}}{b^6}+\frac {(b d-2 a e) \left (-10 a b d e+10 a^2 e^2+b^2 \left (d^2+6 c e\right )\right ) (a+b x)^{5/2}}{b^6}+\frac {3 e \left (-5 a b d e+5 a^2 e^2+b^2 \left (d^2+c e\right )\right ) (a+b x)^{7/2}}{b^6}+\frac {3 e^2 (b d-2 a e) (a+b x)^{9/2}}{b^6}+\frac {e^3 (a+b x)^{11/2}}{b^6}\right ) \, dx\\ &=\frac {2 \left (b^2 c-a b d+a^2 e\right )^3 \sqrt {a+b x}}{b^7}+\frac {2 (b d-2 a e) \left (b^2 c-a b d+a^2 e\right )^2 (a+b x)^{3/2}}{b^7}-\frac {6 \left (b^2 c-a b d+a^2 e\right ) \left (5 a b d e-5 a^2 e^2-b^2 \left (d^2+c e\right )\right ) (a+b x)^{5/2}}{5 b^7}-\frac {2 (b d-2 a e) \left (10 a b d e-10 a^2 e^2-b^2 \left (d^2+6 c e\right )\right ) (a+b x)^{7/2}}{7 b^7}-\frac {2 e \left (5 a b d e-5 a^2 e^2-b^2 \left (d^2+c e\right )\right ) (a+b x)^{9/2}}{3 b^7}+\frac {6 e^2 (b d-2 a e) (a+b x)^{11/2}}{11 b^7}+\frac {2 e^3 (a+b x)^{13/2}}{13 b^7}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 1.03, size = 294, normalized size = 1.07 \[ \frac {2 \sqrt {a+b x} (c+x (d+e x))^3}{b}-\frac {4 (a+b x)^{3/2} \left (-2560 a^5 e^3+640 a^4 b e^2 (13 d+6 e x)-64 a^3 b^2 e \left (e \left (143 c+75 e x^2\right )+143 d^2+195 d e x\right )+8 a^2 b^3 \left (78 d e \left (33 c+25 e x^2\right )+4 e^2 x \left (429 c+175 e x^2\right )+429 d^3+1716 d^2 e x\right )-4 a b^4 \left (3003 c^2 e+429 c \left (7 d^2+18 d e x+10 e^2 x^2\right )+x \left (1287 d^3+4290 d^2 e x+4550 d e^2 x^2+1575 e^3 x^3\right )\right )+b^5 \left (3003 c^2 (5 d+6 e x)+286 c x \left (63 d^2+135 d e x+70 e^2 x^2\right )+5 x^2 \left (1287 d^3+4004 d^2 e x+4095 d e^2 x^2+1386 e^3 x^3\right )\right )\right )}{15015 b^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x + e*x^2)^3/Sqrt[a + b*x],x]

[Out]

(2*Sqrt[a + b*x]*(c + x*(d + e*x))^3)/b - (4*(a + b*x)^(3/2)*(-2560*a^5*e^3 + 640*a^4*b*e^2*(13*d + 6*e*x) - 6
4*a^3*b^2*e*(143*d^2 + 195*d*e*x + e*(143*c + 75*e*x^2)) + 8*a^2*b^3*(429*d^3 + 1716*d^2*e*x + 78*d*e*(33*c +
25*e*x^2) + 4*e^2*x*(429*c + 175*e*x^2)) + b^5*(3003*c^2*(5*d + 6*e*x) + 286*c*x*(63*d^2 + 135*d*e*x + 70*e^2*
x^2) + 5*x^2*(1287*d^3 + 4004*d^2*e*x + 4095*d*e^2*x^2 + 1386*e^3*x^3)) - 4*a*b^4*(3003*c^2*e + 429*c*(7*d^2 +
18*d*e*x + 10*e^2*x^2) + x*(1287*d^3 + 4290*d^2*e*x + 4550*d*e^2*x^2 + 1575*e^3*x^3))))/(15015*b^7)

________________________________________________________________________________________

fricas [A]  time = 0.58, size = 457, normalized size = 1.67 \[ \frac {2 \, {\left (1155 \, b^{6} e^{3} x^{6} + 15015 \, b^{6} c^{3} - 30030 \, a b^{5} c^{2} d + 24024 \, a^{2} b^{4} c d^{2} - 6864 \, a^{3} b^{3} d^{3} + 5120 \, a^{6} e^{3} + 315 \, {\left (13 \, b^{6} d e^{2} - 4 \, a b^{5} e^{3}\right )} x^{5} + 35 \, {\left (143 \, b^{6} d^{2} e + 40 \, a^{2} b^{4} e^{3} + 13 \, {\left (11 \, b^{6} c - 10 \, a b^{5} d\right )} e^{2}\right )} x^{4} + 5 \, {\left (429 \, b^{6} d^{3} - 320 \, a^{3} b^{3} e^{3} - 104 \, {\left (11 \, a b^{5} c - 10 \, a^{2} b^{4} d\right )} e^{2} + 286 \, {\left (9 \, b^{6} c d - 4 \, a b^{5} d^{2}\right )} e\right )} x^{3} + 1664 \, {\left (11 \, a^{4} b^{2} c - 10 \, a^{5} b d\right )} e^{2} + 3 \, {\left (3003 \, b^{6} c d^{2} - 858 \, a b^{5} d^{3} + 640 \, a^{4} b^{2} e^{3} + 208 \, {\left (11 \, a^{2} b^{4} c - 10 \, a^{3} b^{3} d\right )} e^{2} + 143 \, {\left (21 \, b^{6} c^{2} - 36 \, a b^{5} c d + 16 \, a^{2} b^{4} d^{2}\right )} e\right )} x^{2} + 1144 \, {\left (21 \, a^{2} b^{4} c^{2} - 36 \, a^{3} b^{3} c d + 16 \, a^{4} b^{2} d^{2}\right )} e + {\left (15015 \, b^{6} c^{2} d - 12012 \, a b^{5} c d^{2} + 3432 \, a^{2} b^{4} d^{3} - 2560 \, a^{5} b e^{3} - 832 \, {\left (11 \, a^{3} b^{3} c - 10 \, a^{4} b^{2} d\right )} e^{2} - 572 \, {\left (21 \, a b^{5} c^{2} - 36 \, a^{2} b^{4} c d + 16 \, a^{3} b^{3} d^{2}\right )} e\right )} x\right )} \sqrt {b x + a}}{15015 \, b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d*x+c)^3/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

2/15015*(1155*b^6*e^3*x^6 + 15015*b^6*c^3 - 30030*a*b^5*c^2*d + 24024*a^2*b^4*c*d^2 - 6864*a^3*b^3*d^3 + 5120*
a^6*e^3 + 315*(13*b^6*d*e^2 - 4*a*b^5*e^3)*x^5 + 35*(143*b^6*d^2*e + 40*a^2*b^4*e^3 + 13*(11*b^6*c - 10*a*b^5*
d)*e^2)*x^4 + 5*(429*b^6*d^3 - 320*a^3*b^3*e^3 - 104*(11*a*b^5*c - 10*a^2*b^4*d)*e^2 + 286*(9*b^6*c*d - 4*a*b^
5*d^2)*e)*x^3 + 1664*(11*a^4*b^2*c - 10*a^5*b*d)*e^2 + 3*(3003*b^6*c*d^2 - 858*a*b^5*d^3 + 640*a^4*b^2*e^3 + 2
08*(11*a^2*b^4*c - 10*a^3*b^3*d)*e^2 + 143*(21*b^6*c^2 - 36*a*b^5*c*d + 16*a^2*b^4*d^2)*e)*x^2 + 1144*(21*a^2*
b^4*c^2 - 36*a^3*b^3*c*d + 16*a^4*b^2*d^2)*e + (15015*b^6*c^2*d - 12012*a*b^5*c*d^2 + 3432*a^2*b^4*d^3 - 2560*
a^5*b*e^3 - 832*(11*a^3*b^3*c - 10*a^4*b^2*d)*e^2 - 572*(21*a*b^5*c^2 - 36*a^2*b^4*c*d + 16*a^3*b^3*d^2)*e)*x)
*sqrt(b*x + a)/b^7

________________________________________________________________________________________

giac [B]  time = 0.23, size = 526, normalized size = 1.92 \[ \frac {2 \, {\left (15015 \, \sqrt {b x + a} c^{3} + \frac {15015 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} c^{2} d}{b} + \frac {3003 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} c d^{2}}{b^{2}} + \frac {3003 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} c^{2} e}{b^{2}} + \frac {429 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} d^{3}}{b^{3}} + \frac {2574 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} c d e}{b^{3}} + \frac {143 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} d^{2} e}{b^{4}} + \frac {143 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} c e^{2}}{b^{4}} + \frac {65 \, {\left (63 \, {\left (b x + a\right )}^{\frac {11}{2}} - 385 \, {\left (b x + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} - 693 \, \sqrt {b x + a} a^{5}\right )} d e^{2}}{b^{5}} + \frac {5 \, {\left (231 \, {\left (b x + a\right )}^{\frac {13}{2}} - 1638 \, {\left (b x + a\right )}^{\frac {11}{2}} a + 5005 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{2} - 8580 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{3} + 9009 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{4} - 6006 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{5} + 3003 \, \sqrt {b x + a} a^{6}\right )} e^{3}}{b^{6}}\right )}}{15015 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d*x+c)^3/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

2/15015*(15015*sqrt(b*x + a)*c^3 + 15015*((b*x + a)^(3/2) - 3*sqrt(b*x + a)*a)*c^2*d/b + 3003*(3*(b*x + a)^(5/
2) - 10*(b*x + a)^(3/2)*a + 15*sqrt(b*x + a)*a^2)*c*d^2/b^2 + 3003*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a +
15*sqrt(b*x + a)*a^2)*c^2*e/b^2 + 429*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2 - 35
*sqrt(b*x + a)*a^3)*d^3/b^3 + 2574*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2 - 35*sqr
t(b*x + a)*a^3)*c*d*e/b^3 + 143*(35*(b*x + a)^(9/2) - 180*(b*x + a)^(7/2)*a + 378*(b*x + a)^(5/2)*a^2 - 420*(b
*x + a)^(3/2)*a^3 + 315*sqrt(b*x + a)*a^4)*d^2*e/b^4 + 143*(35*(b*x + a)^(9/2) - 180*(b*x + a)^(7/2)*a + 378*(
b*x + a)^(5/2)*a^2 - 420*(b*x + a)^(3/2)*a^3 + 315*sqrt(b*x + a)*a^4)*c*e^2/b^4 + 65*(63*(b*x + a)^(11/2) - 38
5*(b*x + a)^(9/2)*a + 990*(b*x + a)^(7/2)*a^2 - 1386*(b*x + a)^(5/2)*a^3 + 1155*(b*x + a)^(3/2)*a^4 - 693*sqrt
(b*x + a)*a^5)*d*e^2/b^5 + 5*(231*(b*x + a)^(13/2) - 1638*(b*x + a)^(11/2)*a + 5005*(b*x + a)^(9/2)*a^2 - 8580
*(b*x + a)^(7/2)*a^3 + 9009*(b*x + a)^(5/2)*a^4 - 6006*(b*x + a)^(3/2)*a^5 + 3003*sqrt(b*x + a)*a^6)*e^3/b^6)/
b

________________________________________________________________________________________

maple [A]  time = 0.05, size = 495, normalized size = 1.81 \[ \frac {2 \sqrt {b x +a}\, \left (1155 e^{3} x^{6} b^{6}-1260 a \,b^{5} e^{3} x^{5}+4095 b^{6} d \,e^{2} x^{5}+1400 a^{2} b^{4} e^{3} x^{4}-4550 a \,b^{5} d \,e^{2} x^{4}+5005 b^{6} c \,e^{2} x^{4}+5005 b^{6} d^{2} e \,x^{4}-1600 a^{3} b^{3} e^{3} x^{3}+5200 a^{2} b^{4} d \,e^{2} x^{3}-5720 a \,b^{5} c \,e^{2} x^{3}-5720 a \,b^{5} d^{2} e \,x^{3}+12870 b^{6} c d e \,x^{3}+2145 b^{6} d^{3} x^{3}+1920 a^{4} b^{2} e^{3} x^{2}-6240 a^{3} b^{3} d \,e^{2} x^{2}+6864 a^{2} b^{4} c \,e^{2} x^{2}+6864 a^{2} b^{4} d^{2} e \,x^{2}-15444 a \,b^{5} c d e \,x^{2}-2574 a \,b^{5} d^{3} x^{2}+9009 b^{6} c^{2} e \,x^{2}+9009 b^{6} c \,d^{2} x^{2}-2560 a^{5} b \,e^{3} x +8320 a^{4} b^{2} d \,e^{2} x -9152 a^{3} b^{3} c \,e^{2} x -9152 a^{3} b^{3} d^{2} e x +20592 a^{2} b^{4} c d e x +3432 a^{2} b^{4} d^{3} x -12012 a \,b^{5} c^{2} e x -12012 a \,b^{5} c \,d^{2} x +15015 b^{6} c^{2} d x +5120 a^{6} e^{3}-16640 a^{5} b d \,e^{2}+18304 a^{4} b^{2} c \,e^{2}+18304 a^{4} b^{2} d^{2} e -41184 a^{3} b^{3} c d e -6864 a^{3} b^{3} d^{3}+24024 a^{2} b^{4} c^{2} e +24024 a^{2} b^{4} c \,d^{2}-30030 a \,b^{5} c^{2} d +15015 c^{3} b^{6}\right )}{15015 b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d*x+c)^3/(b*x+a)^(1/2),x)

[Out]

2/15015*(b*x+a)^(1/2)*(1155*b^6*e^3*x^6-1260*a*b^5*e^3*x^5+4095*b^6*d*e^2*x^5+1400*a^2*b^4*e^3*x^4-4550*a*b^5*
d*e^2*x^4+5005*b^6*c*e^2*x^4+5005*b^6*d^2*e*x^4-1600*a^3*b^3*e^3*x^3+5200*a^2*b^4*d*e^2*x^3-5720*a*b^5*c*e^2*x
^3-5720*a*b^5*d^2*e*x^3+12870*b^6*c*d*e*x^3+2145*b^6*d^3*x^3+1920*a^4*b^2*e^3*x^2-6240*a^3*b^3*d*e^2*x^2+6864*
a^2*b^4*c*e^2*x^2+6864*a^2*b^4*d^2*e*x^2-15444*a*b^5*c*d*e*x^2-2574*a*b^5*d^3*x^2+9009*b^6*c^2*e*x^2+9009*b^6*
c*d^2*x^2-2560*a^5*b*e^3*x+8320*a^4*b^2*d*e^2*x-9152*a^3*b^3*c*e^2*x-9152*a^3*b^3*d^2*e*x+20592*a^2*b^4*c*d*e*
x+3432*a^2*b^4*d^3*x-12012*a*b^5*c^2*e*x-12012*a*b^5*c*d^2*x+15015*b^6*c^2*d*x+5120*a^6*e^3-16640*a^5*b*d*e^2+
18304*a^4*b^2*c*e^2+18304*a^4*b^2*d^2*e-41184*a^3*b^3*c*d*e-6864*a^3*b^3*d^3+24024*a^2*b^4*c^2*e+24024*a^2*b^4
*c*d^2-30030*a*b^5*c^2*d+15015*b^6*c^3)/b^7

________________________________________________________________________________________

maxima [B]  time = 0.98, size = 525, normalized size = 1.92 \[ \frac {2 \, {\left (15015 \, \sqrt {b x + a} c^{3} + 3003 \, c^{2} {\left (\frac {5 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} d}{b} + \frac {{\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} e}{b^{2}}\right )} + 143 \, c {\left (\frac {21 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} d^{2}}{b^{2}} + \frac {18 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} d e}{b^{3}} + \frac {{\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} e^{2}}{b^{4}}\right )} + \frac {429 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} d^{3}}{b^{3}} + \frac {143 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} d^{2} e}{b^{4}} + \frac {65 \, {\left (63 \, {\left (b x + a\right )}^{\frac {11}{2}} - 385 \, {\left (b x + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} - 693 \, \sqrt {b x + a} a^{5}\right )} d e^{2}}{b^{5}} + \frac {5 \, {\left (231 \, {\left (b x + a\right )}^{\frac {13}{2}} - 1638 \, {\left (b x + a\right )}^{\frac {11}{2}} a + 5005 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{2} - 8580 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{3} + 9009 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{4} - 6006 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{5} + 3003 \, \sqrt {b x + a} a^{6}\right )} e^{3}}{b^{6}}\right )}}{15015 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d*x+c)^3/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

2/15015*(15015*sqrt(b*x + a)*c^3 + 3003*c^2*(5*((b*x + a)^(3/2) - 3*sqrt(b*x + a)*a)*d/b + (3*(b*x + a)^(5/2)
- 10*(b*x + a)^(3/2)*a + 15*sqrt(b*x + a)*a^2)*e/b^2) + 143*c*(21*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a +
15*sqrt(b*x + a)*a^2)*d^2/b^2 + 18*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2 - 35*sqr
t(b*x + a)*a^3)*d*e/b^3 + (35*(b*x + a)^(9/2) - 180*(b*x + a)^(7/2)*a + 378*(b*x + a)^(5/2)*a^2 - 420*(b*x + a
)^(3/2)*a^3 + 315*sqrt(b*x + a)*a^4)*e^2/b^4) + 429*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a)^(
3/2)*a^2 - 35*sqrt(b*x + a)*a^3)*d^3/b^3 + 143*(35*(b*x + a)^(9/2) - 180*(b*x + a)^(7/2)*a + 378*(b*x + a)^(5/
2)*a^2 - 420*(b*x + a)^(3/2)*a^3 + 315*sqrt(b*x + a)*a^4)*d^2*e/b^4 + 65*(63*(b*x + a)^(11/2) - 385*(b*x + a)^
(9/2)*a + 990*(b*x + a)^(7/2)*a^2 - 1386*(b*x + a)^(5/2)*a^3 + 1155*(b*x + a)^(3/2)*a^4 - 693*sqrt(b*x + a)*a^
5)*d*e^2/b^5 + 5*(231*(b*x + a)^(13/2) - 1638*(b*x + a)^(11/2)*a + 5005*(b*x + a)^(9/2)*a^2 - 8580*(b*x + a)^(
7/2)*a^3 + 9009*(b*x + a)^(5/2)*a^4 - 6006*(b*x + a)^(3/2)*a^5 + 3003*sqrt(b*x + a)*a^6)*e^3/b^6)/b

________________________________________________________________________________________

mupad [B]  time = 0.10, size = 299, normalized size = 1.09 \[ \frac {2\,e^3\,{\left (a+b\,x\right )}^{13/2}}{13\,b^7}-\frac {\left (12\,a\,e^3-6\,b\,d\,e^2\right )\,{\left (a+b\,x\right )}^{11/2}}{11\,b^7}+\frac {{\left (a+b\,x\right )}^{9/2}\,\left (30\,a^2\,e^3-30\,a\,b\,d\,e^2+6\,b^2\,d^2\,e+6\,c\,b^2\,e^2\right )}{9\,b^7}+\frac {2\,\sqrt {a+b\,x}\,{\left (e\,a^2-d\,a\,b+c\,b^2\right )}^3}{b^7}+\frac {{\left (a+b\,x\right )}^{5/2}\,\left (30\,a^4\,e^3-60\,a^3\,b\,d\,e^2+36\,a^2\,b^2\,c\,e^2+36\,a^2\,b^2\,d^2\,e-36\,a\,b^3\,c\,d\,e-6\,a\,b^3\,d^3+6\,b^4\,c^2\,e+6\,b^4\,c\,d^2\right )}{5\,b^7}-\frac {2\,\left (2\,a\,e-b\,d\right )\,{\left (a+b\,x\right )}^{7/2}\,\left (10\,a^2\,e^2-10\,a\,b\,d\,e+b^2\,d^2+6\,c\,b^2\,e\right )}{7\,b^7}-\frac {2\,\left (2\,a\,e-b\,d\right )\,{\left (a+b\,x\right )}^{3/2}\,{\left (e\,a^2-d\,a\,b+c\,b^2\right )}^2}{b^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x + e*x^2)^3/(a + b*x)^(1/2),x)

[Out]

(2*e^3*(a + b*x)^(13/2))/(13*b^7) - ((12*a*e^3 - 6*b*d*e^2)*(a + b*x)^(11/2))/(11*b^7) + ((a + b*x)^(9/2)*(30*
a^2*e^3 + 6*b^2*c*e^2 + 6*b^2*d^2*e - 30*a*b*d*e^2))/(9*b^7) + (2*(a + b*x)^(1/2)*(b^2*c + a^2*e - a*b*d)^3)/b
^7 + ((a + b*x)^(5/2)*(30*a^4*e^3 - 6*a*b^3*d^3 + 6*b^4*c*d^2 + 6*b^4*c^2*e + 36*a^2*b^2*c*e^2 + 36*a^2*b^2*d^
2*e - 60*a^3*b*d*e^2 - 36*a*b^3*c*d*e))/(5*b^7) - (2*(2*a*e - b*d)*(a + b*x)^(7/2)*(10*a^2*e^2 + b^2*d^2 + 6*b
^2*c*e - 10*a*b*d*e))/(7*b^7) - (2*(2*a*e - b*d)*(a + b*x)^(3/2)*(b^2*c + a^2*e - a*b*d)^2)/b^7

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d*x+c)**3/(b*x+a)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________