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

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.193835, antiderivative size = 274, normalized size of antiderivative = 1., 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 = 0.538488, size = 294, normalized size = 1.07 \[ \frac{2 \sqrt{a+b x} (c+x (d+e x))^3}{b}-\frac{4 (a+b x)^{3/2} \left (8 a^2 b^3 \left (78 d e \left (33 c+25 e x^2\right )+4 e^2 x \left (429 c+175 e x^2\right )+1716 d^2 e x+429 d^3\right )-64 a^3 b^2 e \left (e \left (143 c+75 e x^2\right )+143 d^2+195 d e x\right )+640 a^4 b e^2 (13 d+6 e x)-2560 a^5 e^3-4 a b^4 \left (3003 c^2 e+429 c \left (7 d^2+18 d e x+10 e^2 x^2\right )+x \left (4290 d^2 e x+1287 d^3+4550 d e^2 x^2+1575 e^3 x^3\right )\right )+b^5 \left (3003 c^2 (5 d+6 e x)+286 c x \left (63 d^2+135 d e x+70 e^2 x^2\right )+5 x^2 \left (4004 d^2 e x+1287 d^3+4095 d e^2 x^2+1386 e^3 x^3\right )\right )\right )}{15015 b^7} \]

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)

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Maple [A]  time = 0.006, size = 495, normalized size = 1.8 \begin{align*}{\frac{2310\,{e}^{3}{x}^{6}{b}^{6}-2520\,a{b}^{5}{e}^{3}{x}^{5}+8190\,{b}^{6}d{e}^{2}{x}^{5}+2800\,{a}^{2}{b}^{4}{e}^{3}{x}^{4}-9100\,a{b}^{5}d{e}^{2}{x}^{4}+10010\,{b}^{6}c{e}^{2}{x}^{4}+10010\,{b}^{6}{d}^{2}e{x}^{4}-3200\,{a}^{3}{b}^{3}{e}^{3}{x}^{3}+10400\,{a}^{2}{b}^{4}d{e}^{2}{x}^{3}-11440\,a{b}^{5}c{e}^{2}{x}^{3}-11440\,a{b}^{5}{d}^{2}e{x}^{3}+25740\,{b}^{6}cde{x}^{3}+4290\,{b}^{6}{d}^{3}{x}^{3}+3840\,{a}^{4}{b}^{2}{e}^{3}{x}^{2}-12480\,{a}^{3}{b}^{3}d{e}^{2}{x}^{2}+13728\,{a}^{2}{b}^{4}c{e}^{2}{x}^{2}+13728\,{a}^{2}{b}^{4}{d}^{2}e{x}^{2}-30888\,a{b}^{5}cde{x}^{2}-5148\,a{b}^{5}{d}^{3}{x}^{2}+18018\,{b}^{6}{c}^{2}e{x}^{2}+18018\,{b}^{6}c{d}^{2}{x}^{2}-5120\,{a}^{5}b{e}^{3}x+16640\,{a}^{4}{b}^{2}d{e}^{2}x-18304\,{a}^{3}{b}^{3}c{e}^{2}x-18304\,{a}^{3}{b}^{3}{d}^{2}ex+41184\,{a}^{2}{b}^{4}cdex+6864\,{a}^{2}{b}^{4}{d}^{3}x-24024\,a{b}^{5}{c}^{2}ex-24024\,a{b}^{5}c{d}^{2}x+30030\,{b}^{6}{c}^{2}dx+10240\,{a}^{6}{e}^{3}-33280\,{a}^{5}bd{e}^{2}+36608\,{a}^{4}{b}^{2}c{e}^{2}+36608\,{a}^{4}{b}^{2}{d}^{2}e-82368\,{a}^{3}{b}^{3}cde-13728\,{a}^{3}{b}^{3}{d}^{3}+48048\,{a}^{2}{b}^{4}{c}^{2}e+48048\,{a}^{2}{b}^{4}c{d}^{2}-60060\,a{b}^{5}{c}^{2}d+30030\,{c}^{3}{b}^{6}}{15015\,{b}^{7}}\sqrt{bx+a}} \end{align*}

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

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Maxima [B]  time = 0.957792, size = 709, normalized size = 2.59 \begin{align*} \text{result too large to display} \end{align*}

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

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Fricas [A]  time = 1.2812, size = 1067, normalized size = 3.89 \begin{align*} \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}} \end{align*}

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

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Sympy [A]  time = 116.817, size = 1406, normalized size = 5.13 \begin{align*} \text{result too large to display} \end{align*}

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]

Piecewise((-(2*a*c**3/sqrt(a + b*x) + 6*a*c**2*d*(-a/sqrt(a + b*x) - sqrt(a + b*x))/b + 6*a*c**2*e*(a**2/sqrt(
a + b*x) + 2*a*sqrt(a + b*x) - (a + b*x)**(3/2)/3)/b**2 + 6*a*c*d**2*(a**2/sqrt(a + b*x) + 2*a*sqrt(a + b*x) -
(a + b*x)**(3/2)/3)/b**2 + 12*a*c*d*e*(-a**3/sqrt(a + b*x) - 3*a**2*sqrt(a + b*x) + a*(a + b*x)**(3/2) - (a +
b*x)**(5/2)/5)/b**3 + 2*a*d**3*(-a**3/sqrt(a + b*x) - 3*a**2*sqrt(a + b*x) + a*(a + b*x)**(3/2) - (a + b*x)**
(5/2)/5)/b**3 + 6*a*c*e**2*(a**4/sqrt(a + b*x) + 4*a**3*sqrt(a + b*x) - 2*a**2*(a + b*x)**(3/2) + 4*a*(a + b*x
)**(5/2)/5 - (a + b*x)**(7/2)/7)/b**4 + 6*a*d**2*e*(a**4/sqrt(a + b*x) + 4*a**3*sqrt(a + b*x) - 2*a**2*(a + b*
x)**(3/2) + 4*a*(a + b*x)**(5/2)/5 - (a + b*x)**(7/2)/7)/b**4 + 6*a*d*e**2*(-a**5/sqrt(a + b*x) - 5*a**4*sqrt(
a + b*x) + 10*a**3*(a + b*x)**(3/2)/3 - 2*a**2*(a + b*x)**(5/2) + 5*a*(a + b*x)**(7/2)/7 - (a + b*x)**(9/2)/9)
/b**5 + 2*a*e**3*(a**6/sqrt(a + b*x) + 6*a**5*sqrt(a + b*x) - 5*a**4*(a + b*x)**(3/2) + 4*a**3*(a + b*x)**(5/2
) - 15*a**2*(a + b*x)**(7/2)/7 + 2*a*(a + b*x)**(9/2)/3 - (a + b*x)**(11/2)/11)/b**6 + 2*c**3*(-a/sqrt(a + b*x
) - sqrt(a + b*x)) + 6*c**2*d*(a**2/sqrt(a + b*x) + 2*a*sqrt(a + b*x) - (a + b*x)**(3/2)/3)/b + 6*c**2*e*(-a**
3/sqrt(a + b*x) - 3*a**2*sqrt(a + b*x) + a*(a + b*x)**(3/2) - (a + b*x)**(5/2)/5)/b**2 + 6*c*d**2*(-a**3/sqrt(
a + b*x) - 3*a**2*sqrt(a + b*x) + a*(a + b*x)**(3/2) - (a + b*x)**(5/2)/5)/b**2 + 12*c*d*e*(a**4/sqrt(a + b*x)
+ 4*a**3*sqrt(a + b*x) - 2*a**2*(a + b*x)**(3/2) + 4*a*(a + b*x)**(5/2)/5 - (a + b*x)**(7/2)/7)/b**3 + 2*d**3
*(a**4/sqrt(a + b*x) + 4*a**3*sqrt(a + b*x) - 2*a**2*(a + b*x)**(3/2) + 4*a*(a + b*x)**(5/2)/5 - (a + b*x)**(7
/2)/7)/b**3 + 6*c*e**2*(-a**5/sqrt(a + b*x) - 5*a**4*sqrt(a + b*x) + 10*a**3*(a + b*x)**(3/2)/3 - 2*a**2*(a +
b*x)**(5/2) + 5*a*(a + b*x)**(7/2)/7 - (a + b*x)**(9/2)/9)/b**4 + 6*d**2*e*(-a**5/sqrt(a + b*x) - 5*a**4*sqrt(
a + b*x) + 10*a**3*(a + b*x)**(3/2)/3 - 2*a**2*(a + b*x)**(5/2) + 5*a*(a + b*x)**(7/2)/7 - (a + b*x)**(9/2)/9)
/b**4 + 6*d*e**2*(a**6/sqrt(a + b*x) + 6*a**5*sqrt(a + b*x) - 5*a**4*(a + b*x)**(3/2) + 4*a**3*(a + b*x)**(5/2
) - 15*a**2*(a + b*x)**(7/2)/7 + 2*a*(a + b*x)**(9/2)/3 - (a + b*x)**(11/2)/11)/b**5 + 2*e**3*(-a**7/sqrt(a +
b*x) - 7*a**6*sqrt(a + b*x) + 7*a**5*(a + b*x)**(3/2) - 7*a**4*(a + b*x)**(5/2) + 5*a**3*(a + b*x)**(7/2) - 7*
a**2*(a + b*x)**(9/2)/3 + 7*a*(a + b*x)**(11/2)/11 - (a + b*x)**(13/2)/13)/b**6)/b, Ne(b, 0)), ((c**3*x + 3*c*
*2*d*x**2/2 + d*e**2*x**6/2 + e**3*x**7/7 + x**5*(3*c*e**2 + 3*d**2*e)/5 + x**4*(6*c*d*e + d**3)/4 + x**3*(3*c
**2*e + 3*c*d**2)/3)/sqrt(a), True))

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Giac [B]  time = 1.11144, size = 710, normalized size = 2.59 \begin{align*} \text{result too large to display} \end{align*}

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