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

Optimal. Leaf size=287 \[ -\frac {2 e (f+g x)^{7/2} \left (e g (-a e g-3 b d g+4 b e f)-c \left (3 d^2 g^2-12 d e f g+10 e^2 f^2\right )\right )}{7 g^6}+\frac {2 (f+g x)^{5/2} (e f-d g) \left (3 e g (-a e g-b d g+2 b e f)-c \left (d^2 g^2-8 d e f g+10 e^2 f^2\right )\right )}{5 g^6}-\frac {2 \sqrt {f+g x} (e f-d g)^3 \left (a g^2-b f g+c f^2\right )}{g^6}+\frac {2 (f+g x)^{3/2} (e f-d g)^2 (c f (5 e f-2 d g)-g (-3 a e g-b d g+4 b e f))}{3 g^6}-\frac {2 e^2 (f+g x)^{9/2} (-b e g-3 c d g+5 c e f)}{9 g^6}+\frac {2 c e^3 (f+g x)^{11/2}}{11 g^6} \]

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Rubi [A]  time = 0.50, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {897, 1153} \begin {gather*} -\frac {2 e (f+g x)^{7/2} \left (e g (-a e g-3 b d g+4 b e f)-c \left (3 d^2 g^2-12 d e f g+10 e^2 f^2\right )\right )}{7 g^6}+\frac {2 (f+g x)^{5/2} (e f-d g) \left (3 e g (-a e g-b d g+2 b e f)-c \left (d^2 g^2-8 d e f g+10 e^2 f^2\right )\right )}{5 g^6}-\frac {2 \sqrt {f+g x} (e f-d g)^3 \left (a g^2-b f g+c f^2\right )}{g^6}+\frac {2 (f+g x)^{3/2} (e f-d g)^2 (c f (5 e f-2 d g)-g (-3 a e g-b d g+4 b e f))}{3 g^6}-\frac {2 e^2 (f+g x)^{9/2} (-b e g-3 c d g+5 c e f)}{9 g^6}+\frac {2 c e^3 (f+g x)^{11/2}}{11 g^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(e*f - d*g)^3*(c*f^2 - b*f*g + a*g^2)*Sqrt[f + g*x])/g^6 + (2*(e*f - d*g)^2*(c*f*(5*e*f - 2*d*g) - g*(4*b*
e*f - b*d*g - 3*a*e*g))*(f + g*x)^(3/2))/(3*g^6) + (2*(e*f - d*g)*(3*e*g*(2*b*e*f - b*d*g - a*e*g) - c*(10*e^2
*f^2 - 8*d*e*f*g + d^2*g^2))*(f + g*x)^(5/2))/(5*g^6) - (2*e*(e*g*(4*b*e*f - 3*b*d*g - a*e*g) - c*(10*e^2*f^2
- 12*d*e*f*g + 3*d^2*g^2))*(f + g*x)^(7/2))/(7*g^6) - (2*e^2*(5*c*e*f - 3*c*d*g - b*e*g)*(f + g*x)^(9/2))/(9*g
^6) + (2*c*e^3*(f + g*x)^(11/2))/(11*g^6)

Rule 897

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - ((2*c*d - b*e)*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1153

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

Rubi steps

\begin {align*} \int \frac {(d+e x)^3 \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \left (\frac {-e f+d g}{g}+\frac {e x^2}{g}\right )^3 \left (\frac {c f^2-b f g+a g^2}{g^2}-\frac {(2 c f-b g) x^2}{g^2}+\frac {c x^4}{g^2}\right ) \, dx,x,\sqrt {f+g x}\right )}{g}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (\frac {(-e f+d g)^3 \left (c f^2-b f g+a g^2\right )}{g^5}+\frac {(e f-d g)^2 (c f (5 e f-2 d g)-g (4 b e f-b d g-3 a e g)) x^2}{g^5}+\frac {(e f-d g) \left (3 e g (2 b e f-b d g-a e g)-c \left (10 e^2 f^2-8 d e f g+d^2 g^2\right )\right ) x^4}{g^5}+\frac {e \left (-e g (4 b e f-3 b d g-a e g)+c \left (10 e^2 f^2-12 d e f g+3 d^2 g^2\right )\right ) x^6}{g^5}+\frac {e^2 (-5 c e f+3 c d g+b e g) x^8}{g^5}+\frac {c e^3 x^{10}}{g^5}\right ) \, dx,x,\sqrt {f+g x}\right )}{g}\\ &=-\frac {2 (e f-d g)^3 \left (c f^2-b f g+a g^2\right ) \sqrt {f+g x}}{g^6}+\frac {2 (e f-d g)^2 (c f (5 e f-2 d g)-g (4 b e f-b d g-3 a e g)) (f+g x)^{3/2}}{3 g^6}+\frac {2 (e f-d g) \left (3 e g (2 b e f-b d g-a e g)-c \left (10 e^2 f^2-8 d e f g+d^2 g^2\right )\right ) (f+g x)^{5/2}}{5 g^6}-\frac {2 e \left (e g (4 b e f-3 b d g-a e g)-c \left (10 e^2 f^2-12 d e f g+3 d^2 g^2\right )\right ) (f+g x)^{7/2}}{7 g^6}-\frac {2 e^2 (5 c e f-3 c d g-b e g) (f+g x)^{9/2}}{9 g^6}+\frac {2 c e^3 (f+g x)^{11/2}}{11 g^6}\\ \end {align*}

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Mathematica [A]  time = 0.41, size = 249, normalized size = 0.87 \begin {gather*} \frac {2 \sqrt {f+g x} \left (-495 e (f+g x)^3 \left (c \left (-3 d^2 g^2+12 d e f g-10 e^2 f^2\right )-e g (a e g+3 b d g-4 b e f)\right )+693 (f+g x)^2 (e f-d g) \left (-3 e g (a e g+b d g-2 b e f)-c \left (d^2 g^2-8 d e f g+10 e^2 f^2\right )\right )-3465 (e f-d g)^3 \left (g (a g-b f)+c f^2\right )+1155 (f+g x) (e f-d g)^2 (g (3 a e g+b d g-4 b e f)+c f (5 e f-2 d g))-385 e^2 (f+g x)^4 (-b e g-3 c d g+5 c e f)+315 c e^3 (f+g x)^5\right )}{3465 g^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[f + g*x]*(-3465*(e*f - d*g)^3*(c*f^2 + g*(-(b*f) + a*g)) + 1155*(e*f - d*g)^2*(c*f*(5*e*f - 2*d*g) + g
*(-4*b*e*f + b*d*g + 3*a*e*g))*(f + g*x) + 693*(e*f - d*g)*(-3*e*g*(-2*b*e*f + b*d*g + a*e*g) - c*(10*e^2*f^2
- 8*d*e*f*g + d^2*g^2))*(f + g*x)^2 - 495*e*(-(e*g*(-4*b*e*f + 3*b*d*g + a*e*g)) + c*(-10*e^2*f^2 + 12*d*e*f*g
- 3*d^2*g^2))*(f + g*x)^3 - 385*e^2*(5*c*e*f - 3*c*d*g - b*e*g)*(f + g*x)^4 + 315*c*e^3*(f + g*x)^5))/(3465*g
^6)

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IntegrateAlgebraic [B]  time = 0.35, size = 634, normalized size = 2.21 \begin {gather*} \frac {2 \sqrt {f+g x} \left (3465 a d^3 g^5+3465 a d^2 e g^4 (f+g x)-10395 a d^2 e f g^4+10395 a d e^2 f^2 g^3-6930 a d e^2 f g^3 (f+g x)+2079 a d e^2 g^3 (f+g x)^2-3465 a e^3 f^3 g^2+3465 a e^3 f^2 g^2 (f+g x)-2079 a e^3 f g^2 (f+g x)^2+495 a e^3 g^2 (f+g x)^3+1155 b d^3 g^4 (f+g x)-3465 b d^3 f g^4+10395 b d^2 e f^2 g^3-6930 b d^2 e f g^3 (f+g x)+2079 b d^2 e g^3 (f+g x)^2-10395 b d e^2 f^3 g^2+10395 b d e^2 f^2 g^2 (f+g x)-6237 b d e^2 f g^2 (f+g x)^2+1485 b d e^2 g^2 (f+g x)^3+3465 b e^3 f^4 g-4620 b e^3 f^3 g (f+g x)+4158 b e^3 f^2 g (f+g x)^2-1980 b e^3 f g (f+g x)^3+385 b e^3 g (f+g x)^4+3465 c d^3 f^2 g^3-2310 c d^3 f g^3 (f+g x)+693 c d^3 g^3 (f+g x)^2-10395 c d^2 e f^3 g^2+10395 c d^2 e f^2 g^2 (f+g x)-6237 c d^2 e f g^2 (f+g x)^2+1485 c d^2 e g^2 (f+g x)^3+10395 c d e^2 f^4 g-13860 c d e^2 f^3 g (f+g x)+12474 c d e^2 f^2 g (f+g x)^2-5940 c d e^2 f g (f+g x)^3+1155 c d e^2 g (f+g x)^4-3465 c e^3 f^5+5775 c e^3 f^4 (f+g x)-6930 c e^3 f^3 (f+g x)^2+4950 c e^3 f^2 (f+g x)^3-1925 c e^3 f (f+g x)^4+315 c e^3 (f+g x)^5\right )}{3465 g^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[f + g*x]*(-3465*c*e^3*f^5 + 10395*c*d*e^2*f^4*g + 3465*b*e^3*f^4*g - 10395*c*d^2*e*f^3*g^2 - 10395*b*d
*e^2*f^3*g^2 - 3465*a*e^3*f^3*g^2 + 3465*c*d^3*f^2*g^3 + 10395*b*d^2*e*f^2*g^3 + 10395*a*d*e^2*f^2*g^3 - 3465*
b*d^3*f*g^4 - 10395*a*d^2*e*f*g^4 + 3465*a*d^3*g^5 + 5775*c*e^3*f^4*(f + g*x) - 13860*c*d*e^2*f^3*g*(f + g*x)
- 4620*b*e^3*f^3*g*(f + g*x) + 10395*c*d^2*e*f^2*g^2*(f + g*x) + 10395*b*d*e^2*f^2*g^2*(f + g*x) + 3465*a*e^3*
f^2*g^2*(f + g*x) - 2310*c*d^3*f*g^3*(f + g*x) - 6930*b*d^2*e*f*g^3*(f + g*x) - 6930*a*d*e^2*f*g^3*(f + g*x) +
1155*b*d^3*g^4*(f + g*x) + 3465*a*d^2*e*g^4*(f + g*x) - 6930*c*e^3*f^3*(f + g*x)^2 + 12474*c*d*e^2*f^2*g*(f +
g*x)^2 + 4158*b*e^3*f^2*g*(f + g*x)^2 - 6237*c*d^2*e*f*g^2*(f + g*x)^2 - 6237*b*d*e^2*f*g^2*(f + g*x)^2 - 207
9*a*e^3*f*g^2*(f + g*x)^2 + 693*c*d^3*g^3*(f + g*x)^2 + 2079*b*d^2*e*g^3*(f + g*x)^2 + 2079*a*d*e^2*g^3*(f + g
*x)^2 + 4950*c*e^3*f^2*(f + g*x)^3 - 5940*c*d*e^2*f*g*(f + g*x)^3 - 1980*b*e^3*f*g*(f + g*x)^3 + 1485*c*d^2*e*
g^2*(f + g*x)^3 + 1485*b*d*e^2*g^2*(f + g*x)^3 + 495*a*e^3*g^2*(f + g*x)^3 - 1925*c*e^3*f*(f + g*x)^4 + 1155*c
*d*e^2*g*(f + g*x)^4 + 385*b*e^3*g*(f + g*x)^4 + 315*c*e^3*(f + g*x)^5))/(3465*g^6)

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fricas [A]  time = 0.56, size = 429, normalized size = 1.49 \begin {gather*} \frac {2 \, {\left (315 \, c e^{3} g^{5} x^{5} - 1280 \, c e^{3} f^{5} + 3465 \, a d^{3} g^{5} + 1408 \, {\left (3 \, c d e^{2} + b e^{3}\right )} f^{4} g - 1584 \, {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f^{3} g^{2} + 1848 \, {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} f^{2} g^{3} - 2310 \, {\left (b d^{3} + 3 \, a d^{2} e\right )} f g^{4} - 35 \, {\left (10 \, c e^{3} f g^{4} - 11 \, {\left (3 \, c d e^{2} + b e^{3}\right )} g^{5}\right )} x^{4} + 5 \, {\left (80 \, c e^{3} f^{2} g^{3} - 88 \, {\left (3 \, c d e^{2} + b e^{3}\right )} f g^{4} + 99 \, {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} g^{5}\right )} x^{3} - 3 \, {\left (160 \, c e^{3} f^{3} g^{2} - 176 \, {\left (3 \, c d e^{2} + b e^{3}\right )} f^{2} g^{3} + 198 \, {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f g^{4} - 231 \, {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} g^{5}\right )} x^{2} + {\left (640 \, c e^{3} f^{4} g - 704 \, {\left (3 \, c d e^{2} + b e^{3}\right )} f^{3} g^{2} + 792 \, {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f^{2} g^{3} - 924 \, {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} f g^{4} + 1155 \, {\left (b d^{3} + 3 \, a d^{2} e\right )} g^{5}\right )} x\right )} \sqrt {g x + f}}{3465 \, g^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(315*c*e^3*g^5*x^5 - 1280*c*e^3*f^5 + 3465*a*d^3*g^5 + 1408*(3*c*d*e^2 + b*e^3)*f^4*g - 1584*(3*c*d^2*e
+ 3*b*d*e^2 + a*e^3)*f^3*g^2 + 1848*(c*d^3 + 3*b*d^2*e + 3*a*d*e^2)*f^2*g^3 - 2310*(b*d^3 + 3*a*d^2*e)*f*g^4
- 35*(10*c*e^3*f*g^4 - 11*(3*c*d*e^2 + b*e^3)*g^5)*x^4 + 5*(80*c*e^3*f^2*g^3 - 88*(3*c*d*e^2 + b*e^3)*f*g^4 +
99*(3*c*d^2*e + 3*b*d*e^2 + a*e^3)*g^5)*x^3 - 3*(160*c*e^3*f^3*g^2 - 176*(3*c*d*e^2 + b*e^3)*f^2*g^3 + 198*(3*
c*d^2*e + 3*b*d*e^2 + a*e^3)*f*g^4 - 231*(c*d^3 + 3*b*d^2*e + 3*a*d*e^2)*g^5)*x^2 + (640*c*e^3*f^4*g - 704*(3*
c*d*e^2 + b*e^3)*f^3*g^2 + 792*(3*c*d^2*e + 3*b*d*e^2 + a*e^3)*f^2*g^3 - 924*(c*d^3 + 3*b*d^2*e + 3*a*d*e^2)*f
*g^4 + 1155*(b*d^3 + 3*a*d^2*e)*g^5)*x)*sqrt(g*x + f)/g^6

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giac [B]  time = 0.20, size = 565, normalized size = 1.97 \begin {gather*} \frac {2 \, {\left (3465 \, \sqrt {g x + f} a d^{3} + \frac {1155 \, {\left ({\left (g x + f\right )}^{\frac {3}{2}} - 3 \, \sqrt {g x + f} f\right )} b d^{3}}{g} + \frac {3465 \, {\left ({\left (g x + f\right )}^{\frac {3}{2}} - 3 \, \sqrt {g x + f} f\right )} a d^{2} e}{g} + \frac {231 \, {\left (3 \, {\left (g x + f\right )}^{\frac {5}{2}} - 10 \, {\left (g x + f\right )}^{\frac {3}{2}} f + 15 \, \sqrt {g x + f} f^{2}\right )} c d^{3}}{g^{2}} + \frac {693 \, {\left (3 \, {\left (g x + f\right )}^{\frac {5}{2}} - 10 \, {\left (g x + f\right )}^{\frac {3}{2}} f + 15 \, \sqrt {g x + f} f^{2}\right )} b d^{2} e}{g^{2}} + \frac {693 \, {\left (3 \, {\left (g x + f\right )}^{\frac {5}{2}} - 10 \, {\left (g x + f\right )}^{\frac {3}{2}} f + 15 \, \sqrt {g x + f} f^{2}\right )} a d e^{2}}{g^{2}} + \frac {297 \, {\left (5 \, {\left (g x + f\right )}^{\frac {7}{2}} - 21 \, {\left (g x + f\right )}^{\frac {5}{2}} f + 35 \, {\left (g x + f\right )}^{\frac {3}{2}} f^{2} - 35 \, \sqrt {g x + f} f^{3}\right )} c d^{2} e}{g^{3}} + \frac {297 \, {\left (5 \, {\left (g x + f\right )}^{\frac {7}{2}} - 21 \, {\left (g x + f\right )}^{\frac {5}{2}} f + 35 \, {\left (g x + f\right )}^{\frac {3}{2}} f^{2} - 35 \, \sqrt {g x + f} f^{3}\right )} b d e^{2}}{g^{3}} + \frac {99 \, {\left (5 \, {\left (g x + f\right )}^{\frac {7}{2}} - 21 \, {\left (g x + f\right )}^{\frac {5}{2}} f + 35 \, {\left (g x + f\right )}^{\frac {3}{2}} f^{2} - 35 \, \sqrt {g x + f} f^{3}\right )} a e^{3}}{g^{3}} + \frac {33 \, {\left (35 \, {\left (g x + f\right )}^{\frac {9}{2}} - 180 \, {\left (g x + f\right )}^{\frac {7}{2}} f + 378 \, {\left (g x + f\right )}^{\frac {5}{2}} f^{2} - 420 \, {\left (g x + f\right )}^{\frac {3}{2}} f^{3} + 315 \, \sqrt {g x + f} f^{4}\right )} c d e^{2}}{g^{4}} + \frac {11 \, {\left (35 \, {\left (g x + f\right )}^{\frac {9}{2}} - 180 \, {\left (g x + f\right )}^{\frac {7}{2}} f + 378 \, {\left (g x + f\right )}^{\frac {5}{2}} f^{2} - 420 \, {\left (g x + f\right )}^{\frac {3}{2}} f^{3} + 315 \, \sqrt {g x + f} f^{4}\right )} b e^{3}}{g^{4}} + \frac {5 \, {\left (63 \, {\left (g x + f\right )}^{\frac {11}{2}} - 385 \, {\left (g x + f\right )}^{\frac {9}{2}} f + 990 \, {\left (g x + f\right )}^{\frac {7}{2}} f^{2} - 1386 \, {\left (g x + f\right )}^{\frac {5}{2}} f^{3} + 1155 \, {\left (g x + f\right )}^{\frac {3}{2}} f^{4} - 693 \, \sqrt {g x + f} f^{5}\right )} c e^{3}}{g^{5}}\right )}}{3465 \, g} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(3465*sqrt(g*x + f)*a*d^3 + 1155*((g*x + f)^(3/2) - 3*sqrt(g*x + f)*f)*b*d^3/g + 3465*((g*x + f)^(3/2)
- 3*sqrt(g*x + f)*f)*a*d^2*e/g + 231*(3*(g*x + f)^(5/2) - 10*(g*x + f)^(3/2)*f + 15*sqrt(g*x + f)*f^2)*c*d^3/g
^2 + 693*(3*(g*x + f)^(5/2) - 10*(g*x + f)^(3/2)*f + 15*sqrt(g*x + f)*f^2)*b*d^2*e/g^2 + 693*(3*(g*x + f)^(5/2
) - 10*(g*x + f)^(3/2)*f + 15*sqrt(g*x + f)*f^2)*a*d*e^2/g^2 + 297*(5*(g*x + f)^(7/2) - 21*(g*x + f)^(5/2)*f +
35*(g*x + f)^(3/2)*f^2 - 35*sqrt(g*x + f)*f^3)*c*d^2*e/g^3 + 297*(5*(g*x + f)^(7/2) - 21*(g*x + f)^(5/2)*f +
35*(g*x + f)^(3/2)*f^2 - 35*sqrt(g*x + f)*f^3)*b*d*e^2/g^3 + 99*(5*(g*x + f)^(7/2) - 21*(g*x + f)^(5/2)*f + 35
*(g*x + f)^(3/2)*f^2 - 35*sqrt(g*x + f)*f^3)*a*e^3/g^3 + 33*(35*(g*x + f)^(9/2) - 180*(g*x + f)^(7/2)*f + 378*
(g*x + f)^(5/2)*f^2 - 420*(g*x + f)^(3/2)*f^3 + 315*sqrt(g*x + f)*f^4)*c*d*e^2/g^4 + 11*(35*(g*x + f)^(9/2) -
180*(g*x + f)^(7/2)*f + 378*(g*x + f)^(5/2)*f^2 - 420*(g*x + f)^(3/2)*f^3 + 315*sqrt(g*x + f)*f^4)*b*e^3/g^4 +
5*(63*(g*x + f)^(11/2) - 385*(g*x + f)^(9/2)*f + 990*(g*x + f)^(7/2)*f^2 - 1386*(g*x + f)^(5/2)*f^3 + 1155*(g
*x + f)^(3/2)*f^4 - 693*sqrt(g*x + f)*f^5)*c*e^3/g^5)/g

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maple [B]  time = 0.01, size = 540, normalized size = 1.88 \begin {gather*} \frac {2 \sqrt {g x +f}\, \left (315 e^{3} c \,x^{5} g^{5}+385 b \,e^{3} g^{5} x^{4}+1155 c d \,e^{2} g^{5} x^{4}-350 c \,e^{3} f \,g^{4} x^{4}+495 a \,e^{3} g^{5} x^{3}+1485 b d \,e^{2} g^{5} x^{3}-440 b \,e^{3} f \,g^{4} x^{3}+1485 c \,d^{2} e \,g^{5} x^{3}-1320 c d \,e^{2} f \,g^{4} x^{3}+400 c \,e^{3} f^{2} g^{3} x^{3}+2079 a d \,e^{2} g^{5} x^{2}-594 a \,e^{3} f \,g^{4} x^{2}+2079 b \,d^{2} e \,g^{5} x^{2}-1782 b d \,e^{2} f \,g^{4} x^{2}+528 b \,e^{3} f^{2} g^{3} x^{2}+693 c \,d^{3} g^{5} x^{2}-1782 c \,d^{2} e f \,g^{4} x^{2}+1584 c d \,e^{2} f^{2} g^{3} x^{2}-480 c \,e^{3} f^{3} g^{2} x^{2}+3465 a \,d^{2} e \,g^{5} x -2772 a d \,e^{2} f \,g^{4} x +792 a \,e^{3} f^{2} g^{3} x +1155 b \,d^{3} g^{5} x -2772 b \,d^{2} e f \,g^{4} x +2376 b d \,e^{2} f^{2} g^{3} x -704 b \,e^{3} f^{3} g^{2} x -924 c \,d^{3} f \,g^{4} x +2376 c \,d^{2} e \,f^{2} g^{3} x -2112 c d \,e^{2} f^{3} g^{2} x +640 c \,e^{3} f^{4} g x +3465 d^{3} a \,g^{5}-6930 a \,d^{2} e f \,g^{4}+5544 a d \,e^{2} f^{2} g^{3}-1584 a \,e^{3} f^{3} g^{2}-2310 b \,d^{3} f \,g^{4}+5544 b \,d^{2} e \,f^{2} g^{3}-4752 b d \,e^{2} f^{3} g^{2}+1408 b \,e^{3} f^{4} g +1848 c \,d^{3} f^{2} g^{3}-4752 c \,d^{2} e \,f^{3} g^{2}+4224 c d \,e^{2} f^{4} g -1280 c \,e^{3} f^{5}\right )}{3465 g^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(g*x+f)^(1/2)*(315*c*e^3*g^5*x^5+385*b*e^3*g^5*x^4+1155*c*d*e^2*g^5*x^4-350*c*e^3*f*g^4*x^4+495*a*e^3*g
^5*x^3+1485*b*d*e^2*g^5*x^3-440*b*e^3*f*g^4*x^3+1485*c*d^2*e*g^5*x^3-1320*c*d*e^2*f*g^4*x^3+400*c*e^3*f^2*g^3*
x^3+2079*a*d*e^2*g^5*x^2-594*a*e^3*f*g^4*x^2+2079*b*d^2*e*g^5*x^2-1782*b*d*e^2*f*g^4*x^2+528*b*e^3*f^2*g^3*x^2
+693*c*d^3*g^5*x^2-1782*c*d^2*e*f*g^4*x^2+1584*c*d*e^2*f^2*g^3*x^2-480*c*e^3*f^3*g^2*x^2+3465*a*d^2*e*g^5*x-27
72*a*d*e^2*f*g^4*x+792*a*e^3*f^2*g^3*x+1155*b*d^3*g^5*x-2772*b*d^2*e*f*g^4*x+2376*b*d*e^2*f^2*g^3*x-704*b*e^3*
f^3*g^2*x-924*c*d^3*f*g^4*x+2376*c*d^2*e*f^2*g^3*x-2112*c*d*e^2*f^3*g^2*x+640*c*e^3*f^4*g*x+3465*a*d^3*g^5-693
0*a*d^2*e*f*g^4+5544*a*d*e^2*f^2*g^3-1584*a*e^3*f^3*g^2-2310*b*d^3*f*g^4+5544*b*d^2*e*f^2*g^3-4752*b*d*e^2*f^3
*g^2+1408*b*e^3*f^4*g+1848*c*d^3*f^2*g^3-4752*c*d^2*e*f^3*g^2+4224*c*d*e^2*f^4*g-1280*c*e^3*f^5)/g^6

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maxima [A]  time = 0.46, size = 429, normalized size = 1.49 \begin {gather*} \frac {2 \, {\left (315 \, {\left (g x + f\right )}^{\frac {11}{2}} c e^{3} - 385 \, {\left (5 \, c e^{3} f - {\left (3 \, c d e^{2} + b e^{3}\right )} g\right )} {\left (g x + f\right )}^{\frac {9}{2}} + 495 \, {\left (10 \, c e^{3} f^{2} - 4 \, {\left (3 \, c d e^{2} + b e^{3}\right )} f g + {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} g^{2}\right )} {\left (g x + f\right )}^{\frac {7}{2}} - 693 \, {\left (10 \, c e^{3} f^{3} - 6 \, {\left (3 \, c d e^{2} + b e^{3}\right )} f^{2} g + 3 \, {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f g^{2} - {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} g^{3}\right )} {\left (g x + f\right )}^{\frac {5}{2}} + 1155 \, {\left (5 \, c e^{3} f^{4} - 4 \, {\left (3 \, c d e^{2} + b e^{3}\right )} f^{3} g + 3 \, {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f^{2} g^{2} - 2 \, {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} f g^{3} + {\left (b d^{3} + 3 \, a d^{2} e\right )} g^{4}\right )} {\left (g x + f\right )}^{\frac {3}{2}} - 3465 \, {\left (c e^{3} f^{5} - a d^{3} g^{5} - {\left (3 \, c d e^{2} + b e^{3}\right )} f^{4} g + {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f^{3} g^{2} - {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} f^{2} g^{3} + {\left (b d^{3} + 3 \, a d^{2} e\right )} f g^{4}\right )} \sqrt {g x + f}\right )}}{3465 \, g^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(315*(g*x + f)^(11/2)*c*e^3 - 385*(5*c*e^3*f - (3*c*d*e^2 + b*e^3)*g)*(g*x + f)^(9/2) + 495*(10*c*e^3*f
^2 - 4*(3*c*d*e^2 + b*e^3)*f*g + (3*c*d^2*e + 3*b*d*e^2 + a*e^3)*g^2)*(g*x + f)^(7/2) - 693*(10*c*e^3*f^3 - 6*
(3*c*d*e^2 + b*e^3)*f^2*g + 3*(3*c*d^2*e + 3*b*d*e^2 + a*e^3)*f*g^2 - (c*d^3 + 3*b*d^2*e + 3*a*d*e^2)*g^3)*(g*
x + f)^(5/2) + 1155*(5*c*e^3*f^4 - 4*(3*c*d*e^2 + b*e^3)*f^3*g + 3*(3*c*d^2*e + 3*b*d*e^2 + a*e^3)*f^2*g^2 - 2
*(c*d^3 + 3*b*d^2*e + 3*a*d*e^2)*f*g^3 + (b*d^3 + 3*a*d^2*e)*g^4)*(g*x + f)^(3/2) - 3465*(c*e^3*f^5 - a*d^3*g^
5 - (3*c*d*e^2 + b*e^3)*f^4*g + (3*c*d^2*e + 3*b*d*e^2 + a*e^3)*f^3*g^2 - (c*d^3 + 3*b*d^2*e + 3*a*d*e^2)*f^2*
g^3 + (b*d^3 + 3*a*d^2*e)*f*g^4)*sqrt(g*x + f))/g^6

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mupad [B]  time = 0.15, size = 283, normalized size = 0.99 \begin {gather*} \frac {{\left (f+g\,x\right )}^{9/2}\,\left (2\,b\,e^3\,g-10\,c\,e^3\,f+6\,c\,d\,e^2\,g\right )}{9\,g^6}+\frac {{\left (f+g\,x\right )}^{7/2}\,\left (6\,c\,d^2\,e\,g^2-24\,c\,d\,e^2\,f\,g+6\,b\,d\,e^2\,g^2+20\,c\,e^3\,f^2-8\,b\,e^3\,f\,g+2\,a\,e^3\,g^2\right )}{7\,g^6}+\frac {2\,{\left (f+g\,x\right )}^{5/2}\,\left (d\,g-e\,f\right )\,\left (c\,d^2\,g^2-8\,c\,d\,e\,f\,g+3\,b\,d\,e\,g^2+10\,c\,e^2\,f^2-6\,b\,e^2\,f\,g+3\,a\,e^2\,g^2\right )}{5\,g^6}+\frac {2\,\sqrt {f+g\,x}\,{\left (d\,g-e\,f\right )}^3\,\left (c\,f^2-b\,f\,g+a\,g^2\right )}{g^6}+\frac {2\,{\left (f+g\,x\right )}^{3/2}\,{\left (d\,g-e\,f\right )}^2\,\left (3\,a\,e\,g^2+b\,d\,g^2+5\,c\,e\,f^2-4\,b\,e\,f\,g-2\,c\,d\,f\,g\right )}{3\,g^6}+\frac {2\,c\,e^3\,{\left (f+g\,x\right )}^{11/2}}{11\,g^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((f + g*x)^(9/2)*(2*b*e^3*g - 10*c*e^3*f + 6*c*d*e^2*g))/(9*g^6) + ((f + g*x)^(7/2)*(2*a*e^3*g^2 + 20*c*e^3*f^
2 - 8*b*e^3*f*g + 6*b*d*e^2*g^2 + 6*c*d^2*e*g^2 - 24*c*d*e^2*f*g))/(7*g^6) + (2*(f + g*x)^(5/2)*(d*g - e*f)*(3
*a*e^2*g^2 + c*d^2*g^2 + 10*c*e^2*f^2 + 3*b*d*e*g^2 - 6*b*e^2*f*g - 8*c*d*e*f*g))/(5*g^6) + (2*(f + g*x)^(1/2)
*(d*g - e*f)^3*(a*g^2 + c*f^2 - b*f*g))/g^6 + (2*(f + g*x)^(3/2)*(d*g - e*f)^2*(3*a*e*g^2 + b*d*g^2 + 5*c*e*f^
2 - 4*b*e*f*g - 2*c*d*f*g))/(3*g^6) + (2*c*e^3*(f + g*x)^(11/2))/(11*g^6)

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sympy [A]  time = 164.70, size = 1544, normalized size = 5.38

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((-2*a*d**3*f/sqrt(f + g*x) - 2*a*d**3*(-f/sqrt(f + g*x) - sqrt(f + g*x)) - 6*a*d**2*e*f*(-f/sqrt(f
+ g*x) - sqrt(f + g*x))/g - 6*a*d**2*e*(f**2/sqrt(f + g*x) + 2*f*sqrt(f + g*x) - (f + g*x)**(3/2)/3)/g - 6*a*d
*e**2*f*(f**2/sqrt(f + g*x) + 2*f*sqrt(f + g*x) - (f + g*x)**(3/2)/3)/g**2 - 6*a*d*e**2*(-f**3/sqrt(f + g*x) -
3*f**2*sqrt(f + g*x) + f*(f + g*x)**(3/2) - (f + g*x)**(5/2)/5)/g**2 - 2*a*e**3*f*(-f**3/sqrt(f + g*x) - 3*f*
*2*sqrt(f + g*x) + f*(f + g*x)**(3/2) - (f + g*x)**(5/2)/5)/g**3 - 2*a*e**3*(f**4/sqrt(f + g*x) + 4*f**3*sqrt(
f + g*x) - 2*f**2*(f + g*x)**(3/2) + 4*f*(f + g*x)**(5/2)/5 - (f + g*x)**(7/2)/7)/g**3 - 2*b*d**3*f*(-f/sqrt(f
+ g*x) - sqrt(f + g*x))/g - 2*b*d**3*(f**2/sqrt(f + g*x) + 2*f*sqrt(f + g*x) - (f + g*x)**(3/2)/3)/g - 6*b*d*
*2*e*f*(f**2/sqrt(f + g*x) + 2*f*sqrt(f + g*x) - (f + g*x)**(3/2)/3)/g**2 - 6*b*d**2*e*(-f**3/sqrt(f + g*x) -
3*f**2*sqrt(f + g*x) + f*(f + g*x)**(3/2) - (f + g*x)**(5/2)/5)/g**2 - 6*b*d*e**2*f*(-f**3/sqrt(f + g*x) - 3*f
**2*sqrt(f + g*x) + f*(f + g*x)**(3/2) - (f + g*x)**(5/2)/5)/g**3 - 6*b*d*e**2*(f**4/sqrt(f + g*x) + 4*f**3*sq
rt(f + g*x) - 2*f**2*(f + g*x)**(3/2) + 4*f*(f + g*x)**(5/2)/5 - (f + g*x)**(7/2)/7)/g**3 - 2*b*e**3*f*(f**4/s
qrt(f + g*x) + 4*f**3*sqrt(f + g*x) - 2*f**2*(f + g*x)**(3/2) + 4*f*(f + g*x)**(5/2)/5 - (f + g*x)**(7/2)/7)/g
**4 - 2*b*e**3*(-f**5/sqrt(f + g*x) - 5*f**4*sqrt(f + g*x) + 10*f**3*(f + g*x)**(3/2)/3 - 2*f**2*(f + g*x)**(5
/2) + 5*f*(f + g*x)**(7/2)/7 - (f + g*x)**(9/2)/9)/g**4 - 2*c*d**3*f*(f**2/sqrt(f + g*x) + 2*f*sqrt(f + g*x) -
(f + g*x)**(3/2)/3)/g**2 - 2*c*d**3*(-f**3/sqrt(f + g*x) - 3*f**2*sqrt(f + g*x) + f*(f + g*x)**(3/2) - (f + g
*x)**(5/2)/5)/g**2 - 6*c*d**2*e*f*(-f**3/sqrt(f + g*x) - 3*f**2*sqrt(f + g*x) + f*(f + g*x)**(3/2) - (f + g*x)
**(5/2)/5)/g**3 - 6*c*d**2*e*(f**4/sqrt(f + g*x) + 4*f**3*sqrt(f + g*x) - 2*f**2*(f + g*x)**(3/2) + 4*f*(f + g
*x)**(5/2)/5 - (f + g*x)**(7/2)/7)/g**3 - 6*c*d*e**2*f*(f**4/sqrt(f + g*x) + 4*f**3*sqrt(f + g*x) - 2*f**2*(f
+ g*x)**(3/2) + 4*f*(f + g*x)**(5/2)/5 - (f + g*x)**(7/2)/7)/g**4 - 6*c*d*e**2*(-f**5/sqrt(f + g*x) - 5*f**4*s
qrt(f + g*x) + 10*f**3*(f + g*x)**(3/2)/3 - 2*f**2*(f + g*x)**(5/2) + 5*f*(f + g*x)**(7/2)/7 - (f + g*x)**(9/2
)/9)/g**4 - 2*c*e**3*f*(-f**5/sqrt(f + g*x) - 5*f**4*sqrt(f + g*x) + 10*f**3*(f + g*x)**(3/2)/3 - 2*f**2*(f +
g*x)**(5/2) + 5*f*(f + g*x)**(7/2)/7 - (f + g*x)**(9/2)/9)/g**5 - 2*c*e**3*(f**6/sqrt(f + g*x) + 6*f**5*sqrt(f
+ g*x) - 5*f**4*(f + g*x)**(3/2) + 4*f**3*(f + g*x)**(5/2) - 15*f**2*(f + g*x)**(7/2)/7 + 2*f*(f + g*x)**(9/2
)/3 - (f + g*x)**(11/2)/11)/g**5)/g, Ne(g, 0)), ((a*d**3*x + c*e**3*x**6/6 + x**5*(b*e**3 + 3*c*d*e**2)/5 + x*
*4*(a*e**3 + 3*b*d*e**2 + 3*c*d**2*e)/4 + x**3*(3*a*d*e**2 + 3*b*d**2*e + c*d**3)/3 + x**2*(3*a*d**2*e + b*d**
3)/2)/sqrt(f), True))

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