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

Optimal. Leaf size=333 \[ -\frac {\sqrt {d+e x} \sqrt {f+g x} (e f-d g) \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )}{64 e^2 g^4}+\frac {(d+e x)^{3/2} \sqrt {f+g x} \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )}{96 e^2 g^3}+\frac {(e f-d g)^2 \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right ) \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )}{64 e^{5/2} g^{9/2}}-\frac {(d+e x)^{5/2} \sqrt {f+g x} (-8 b e g+9 c d g+7 c e f)}{24 e^2 g^2}+\frac {c (d+e x)^{7/2} \sqrt {f+g x}}{4 e^2 g} \]

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Rubi [A]  time = 0.35, antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {951, 80, 50, 63, 217, 206} \begin {gather*} \frac {(d+e x)^{3/2} \sqrt {f+g x} \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )}{96 e^2 g^3}-\frac {\sqrt {d+e x} \sqrt {f+g x} (e f-d g) \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )}{64 e^2 g^4}+\frac {(e f-d g)^2 \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right ) \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )}{64 e^{5/2} g^{9/2}}-\frac {(d+e x)^{5/2} \sqrt {f+g x} (-8 b e g+9 c d g+7 c e f)}{24 e^2 g^2}+\frac {c (d+e x)^{7/2} \sqrt {f+g x}}{4 e^2 g} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((e*f - d*g)*(c*(35*e^2*f^2 + 10*d*e*f*g + 3*d^2*g^2) + 8*e*g*(6*a*e*g - b*(5*e*f + d*g)))*Sqrt[d + e*x]*Sqrt
[f + g*x])/(64*e^2*g^4) + ((c*(35*e^2*f^2 + 10*d*e*f*g + 3*d^2*g^2) + 8*e*g*(6*a*e*g - b*(5*e*f + d*g)))*(d +
e*x)^(3/2)*Sqrt[f + g*x])/(96*e^2*g^3) - ((7*c*e*f + 9*c*d*g - 8*b*e*g)*(d + e*x)^(5/2)*Sqrt[f + g*x])/(24*e^2
*g^2) + (c*(d + e*x)^(7/2)*Sqrt[f + g*x])/(4*e^2*g) + ((e*f - d*g)^2*(c*(35*e^2*f^2 + 10*d*e*f*g + 3*d^2*g^2)
+ 8*e*g*(6*a*e*g - b*(5*e*f + d*g)))*ArcTanh[(Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/(64*e^(5/2)*g^(
9/2))

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

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

Rule 951

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Simp[(c^p*(d + e*x)^(m + 2*p)*(f + g*x)^(n + 1))/(g*e^(2*p)*(m + n + 2*p + 1)), x] + Dist[1/(g*e^(2*p)*(m +
n + 2*p + 1)), Int[(d + e*x)^m*(f + g*x)^n*ExpandToSum[g*(m + n + 2*p + 1)*(e^(2*p)*(a + b*x + c*x^2)^p - c^p*
(d + e*x)^(2*p)) - c^p*(e*f - d*g)*(m + 2*p)*(d + e*x)^(2*p - 1), x], x], 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] && IGtQ[p, 0] && NeQ[m + n + 2*
p + 1, 0] && (IntegerQ[n] ||  !IntegerQ[m])

Rubi steps

\begin {align*} \int \frac {(d+e x)^{3/2} \left (a+b x+c x^2\right )}{\sqrt {f+g x}} \, dx &=\frac {c (d+e x)^{7/2} \sqrt {f+g x}}{4 e^2 g}+\frac {\int \frac {(d+e x)^{3/2} \left (\frac {1}{2} \left (8 a e^2 g-c d (7 e f+d g)\right )-\frac {1}{2} e (7 c e f+9 c d g-8 b e g) x\right )}{\sqrt {f+g x}} \, dx}{4 e^2 g}\\ &=-\frac {(7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt {f+g x}}{24 e^2 g^2}+\frac {c (d+e x)^{7/2} \sqrt {f+g x}}{4 e^2 g}+\frac {\left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) \int \frac {(d+e x)^{3/2}}{\sqrt {f+g x}} \, dx}{48 e^2 g^2}\\ &=\frac {\left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) (d+e x)^{3/2} \sqrt {f+g x}}{96 e^2 g^3}-\frac {(7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt {f+g x}}{24 e^2 g^2}+\frac {c (d+e x)^{7/2} \sqrt {f+g x}}{4 e^2 g}-\frac {\left ((e f-d g) \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {f+g x}} \, dx}{64 e^2 g^3}\\ &=-\frac {(e f-d g) \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) \sqrt {d+e x} \sqrt {f+g x}}{64 e^2 g^4}+\frac {\left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) (d+e x)^{3/2} \sqrt {f+g x}}{96 e^2 g^3}-\frac {(7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt {f+g x}}{24 e^2 g^2}+\frac {c (d+e x)^{7/2} \sqrt {f+g x}}{4 e^2 g}+\frac {\left ((e f-d g)^2 \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x}} \, dx}{128 e^2 g^4}\\ &=-\frac {(e f-d g) \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) \sqrt {d+e x} \sqrt {f+g x}}{64 e^2 g^4}+\frac {\left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) (d+e x)^{3/2} \sqrt {f+g x}}{96 e^2 g^3}-\frac {(7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt {f+g x}}{24 e^2 g^2}+\frac {c (d+e x)^{7/2} \sqrt {f+g x}}{4 e^2 g}+\frac {\left ((e f-d g)^2 \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {f-\frac {d g}{e}+\frac {g x^2}{e}}} \, dx,x,\sqrt {d+e x}\right )}{64 e^3 g^4}\\ &=-\frac {(e f-d g) \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) \sqrt {d+e x} \sqrt {f+g x}}{64 e^2 g^4}+\frac {\left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) (d+e x)^{3/2} \sqrt {f+g x}}{96 e^2 g^3}-\frac {(7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt {f+g x}}{24 e^2 g^2}+\frac {c (d+e x)^{7/2} \sqrt {f+g x}}{4 e^2 g}+\frac {\left ((e f-d g)^2 \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {g x^2}{e}} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{64 e^3 g^4}\\ &=-\frac {(e f-d g) \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) \sqrt {d+e x} \sqrt {f+g x}}{64 e^2 g^4}+\frac {\left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) (d+e x)^{3/2} \sqrt {f+g x}}{96 e^2 g^3}-\frac {(7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt {f+g x}}{24 e^2 g^2}+\frac {c (d+e x)^{7/2} \sqrt {f+g x}}{4 e^2 g}+\frac {(e f-d g)^2 \left (c \left (35 e^2 f^2+10 d e f g+3 d^2 g^2\right )+8 e g (6 a e g-b (5 e f+d g))\right ) \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{64 e^{5/2} g^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 1.54, size = 313, normalized size = 0.94 \begin {gather*} \frac {3 (e f-d g)^{5/2} \sqrt {\frac {e (f+g x)}{e f-d g}} \sinh ^{-1}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e f-d g}}\right ) \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )-e \sqrt {g} \sqrt {d+e x} (f+g x) \left (c \left (9 d^3 g^3+3 d^2 e g^2 (5 f-2 g x)+d e^2 g \left (-145 f^2+92 f g x-72 g^2 x^2\right )+e^3 \left (105 f^3-70 f^2 g x+56 f g^2 x^2-48 g^3 x^3\right )\right )-8 e g \left (6 a e g (5 d g-3 e f+2 e g x)+b \left (3 d^2 g^2+2 d e g (7 g x-11 f)+e^2 \left (15 f^2-10 f g x+8 g^2 x^2\right )\right )\right )\right )}{192 e^3 g^{9/2} \sqrt {f+g x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(e*Sqrt[g]*Sqrt[d + e*x]*(f + g*x)*(c*(9*d^3*g^3 + 3*d^2*e*g^2*(5*f - 2*g*x) + d*e^2*g*(-145*f^2 + 92*f*g*x
- 72*g^2*x^2) + e^3*(105*f^3 - 70*f^2*g*x + 56*f*g^2*x^2 - 48*g^3*x^3)) - 8*e*g*(6*a*e*g*(-3*e*f + 5*d*g + 2*e
*g*x) + b*(3*d^2*g^2 + 2*d*e*g*(-11*f + 7*g*x) + e^2*(15*f^2 - 10*f*g*x + 8*g^2*x^2))))) + 3*(e*f - d*g)^(5/2)
*(c*(35*e^2*f^2 + 10*d*e*f*g + 3*d^2*g^2) + 8*e*g*(6*a*e*g - b*(5*e*f + d*g)))*Sqrt[(e*(f + g*x))/(e*f - d*g)]
*ArcSinh[(Sqrt[g]*Sqrt[d + e*x])/Sqrt[e*f - d*g]])/(192*e^3*g^(9/2)*Sqrt[f + g*x])

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IntegrateAlgebraic [A]  time = 0.73, size = 643, normalized size = 1.93 \begin {gather*} \frac {(e f-d g)^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {g} \sqrt {d+e x}}\right ) \left (48 a e^2 g^2-8 b d e g^2-40 b e^2 f g+3 c d^2 g^2+10 c d e f g+35 c e^2 f^2\right )}{64 e^{5/2} g^{9/2}}-\frac {\sqrt {f+g x} (e f-d g)^2 \left (\frac {144 a e^5 g^2 (f+g x)^3}{(d+e x)^3}-\frac {528 a e^4 g^3 (f+g x)^2}{(d+e x)^2}+\frac {624 a e^3 g^4 (f+g x)}{d+e x}-240 a e^2 g^5-\frac {120 b e^5 f g (f+g x)^3}{(d+e x)^3}-\frac {24 b d e^4 g^2 (f+g x)^3}{(d+e x)^3}+\frac {440 b e^4 f g^2 (f+g x)^2}{(d+e x)^2}+\frac {88 b d e^3 g^3 (f+g x)^2}{(d+e x)^2}-\frac {584 b e^3 f g^3 (f+g x)}{d+e x}-\frac {40 b d e^2 g^4 (f+g x)}{d+e x}-24 b d e g^5+264 b e^2 f g^4+\frac {9 c d^2 e^3 g^2 (f+g x)^3}{(d+e x)^3}-\frac {33 c d^2 e^2 g^3 (f+g x)^2}{(d+e x)^2}-\frac {33 c d^2 e g^4 (f+g x)}{d+e x}+9 c d^2 g^5+\frac {105 c e^5 f^2 (f+g x)^3}{(d+e x)^3}-\frac {385 c e^4 f^2 g (f+g x)^2}{(d+e x)^2}+\frac {30 c d e^4 f g (f+g x)^3}{(d+e x)^3}+\frac {511 c e^3 f^2 g^2 (f+g x)}{d+e x}-\frac {110 c d e^3 f g^2 (f+g x)^2}{(d+e x)^2}+\frac {146 c d e^2 f g^3 (f+g x)}{d+e x}+30 c d e f g^4-279 c e^2 f^2 g^3\right )}{192 e^2 g^4 \sqrt {d+e x} \left (\frac {e (f+g x)}{d+e x}-g\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/192*((e*f - d*g)^2*Sqrt[f + g*x]*(-279*c*e^2*f^2*g^3 + 30*c*d*e*f*g^4 + 264*b*e^2*f*g^4 + 9*c*d^2*g^5 - 24*
b*d*e*g^5 - 240*a*e^2*g^5 + (511*c*e^3*f^2*g^2*(f + g*x))/(d + e*x) + (146*c*d*e^2*f*g^3*(f + g*x))/(d + e*x)
- (584*b*e^3*f*g^3*(f + g*x))/(d + e*x) - (33*c*d^2*e*g^4*(f + g*x))/(d + e*x) - (40*b*d*e^2*g^4*(f + g*x))/(d
+ e*x) + (624*a*e^3*g^4*(f + g*x))/(d + e*x) - (385*c*e^4*f^2*g*(f + g*x)^2)/(d + e*x)^2 - (110*c*d*e^3*f*g^2
*(f + g*x)^2)/(d + e*x)^2 + (440*b*e^4*f*g^2*(f + g*x)^2)/(d + e*x)^2 - (33*c*d^2*e^2*g^3*(f + g*x)^2)/(d + e*
x)^2 + (88*b*d*e^3*g^3*(f + g*x)^2)/(d + e*x)^2 - (528*a*e^4*g^3*(f + g*x)^2)/(d + e*x)^2 + (105*c*e^5*f^2*(f
+ g*x)^3)/(d + e*x)^3 + (30*c*d*e^4*f*g*(f + g*x)^3)/(d + e*x)^3 - (120*b*e^5*f*g*(f + g*x)^3)/(d + e*x)^3 + (
9*c*d^2*e^3*g^2*(f + g*x)^3)/(d + e*x)^3 - (24*b*d*e^4*g^2*(f + g*x)^3)/(d + e*x)^3 + (144*a*e^5*g^2*(f + g*x)
^3)/(d + e*x)^3))/(e^2*g^4*Sqrt[d + e*x]*(-g + (e*(f + g*x))/(d + e*x))^4) + ((e*f - d*g)^2*(35*c*e^2*f^2 + 10
*c*d*e*f*g - 40*b*e^2*f*g + 3*c*d^2*g^2 - 8*b*d*e*g^2 + 48*a*e^2*g^2)*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/(Sqrt[g]
*Sqrt[d + e*x])])/(64*e^(5/2)*g^(9/2))

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fricas [A]  time = 0.66, size = 852, normalized size = 2.56 \begin {gather*} \left [\frac {3 \, {\left (35 \, c e^{4} f^{4} - 20 \, {\left (3 \, c d e^{3} + 2 \, b e^{4}\right )} f^{3} g + 6 \, {\left (3 \, c d^{2} e^{2} + 12 \, b d e^{3} + 8 \, a e^{4}\right )} f^{2} g^{2} + 4 \, {\left (c d^{3} e - 6 \, b d^{2} e^{2} - 24 \, a d e^{3}\right )} f g^{3} + {\left (3 \, c d^{4} - 8 \, b d^{3} e + 48 \, a d^{2} e^{2}\right )} g^{4}\right )} \sqrt {e g} \log \left (8 \, e^{2} g^{2} x^{2} + e^{2} f^{2} + 6 \, d e f g + d^{2} g^{2} + 4 \, {\left (2 \, e g x + e f + d g\right )} \sqrt {e g} \sqrt {e x + d} \sqrt {g x + f} + 8 \, {\left (e^{2} f g + d e g^{2}\right )} x\right ) + 4 \, {\left (48 \, c e^{4} g^{4} x^{3} - 105 \, c e^{4} f^{3} g + 5 \, {\left (29 \, c d e^{3} + 24 \, b e^{4}\right )} f^{2} g^{2} - {\left (15 \, c d^{2} e^{2} + 176 \, b d e^{3} + 144 \, a e^{4}\right )} f g^{3} - 3 \, {\left (3 \, c d^{3} e - 8 \, b d^{2} e^{2} - 80 \, a d e^{3}\right )} g^{4} - 8 \, {\left (7 \, c e^{4} f g^{3} - {\left (9 \, c d e^{3} + 8 \, b e^{4}\right )} g^{4}\right )} x^{2} + 2 \, {\left (35 \, c e^{4} f^{2} g^{2} - 2 \, {\left (23 \, c d e^{3} + 20 \, b e^{4}\right )} f g^{3} + {\left (3 \, c d^{2} e^{2} + 56 \, b d e^{3} + 48 \, a e^{4}\right )} g^{4}\right )} x\right )} \sqrt {e x + d} \sqrt {g x + f}}{768 \, e^{3} g^{5}}, -\frac {3 \, {\left (35 \, c e^{4} f^{4} - 20 \, {\left (3 \, c d e^{3} + 2 \, b e^{4}\right )} f^{3} g + 6 \, {\left (3 \, c d^{2} e^{2} + 12 \, b d e^{3} + 8 \, a e^{4}\right )} f^{2} g^{2} + 4 \, {\left (c d^{3} e - 6 \, b d^{2} e^{2} - 24 \, a d e^{3}\right )} f g^{3} + {\left (3 \, c d^{4} - 8 \, b d^{3} e + 48 \, a d^{2} e^{2}\right )} g^{4}\right )} \sqrt {-e g} \arctan \left (\frac {{\left (2 \, e g x + e f + d g\right )} \sqrt {-e g} \sqrt {e x + d} \sqrt {g x + f}}{2 \, {\left (e^{2} g^{2} x^{2} + d e f g + {\left (e^{2} f g + d e g^{2}\right )} x\right )}}\right ) - 2 \, {\left (48 \, c e^{4} g^{4} x^{3} - 105 \, c e^{4} f^{3} g + 5 \, {\left (29 \, c d e^{3} + 24 \, b e^{4}\right )} f^{2} g^{2} - {\left (15 \, c d^{2} e^{2} + 176 \, b d e^{3} + 144 \, a e^{4}\right )} f g^{3} - 3 \, {\left (3 \, c d^{3} e - 8 \, b d^{2} e^{2} - 80 \, a d e^{3}\right )} g^{4} - 8 \, {\left (7 \, c e^{4} f g^{3} - {\left (9 \, c d e^{3} + 8 \, b e^{4}\right )} g^{4}\right )} x^{2} + 2 \, {\left (35 \, c e^{4} f^{2} g^{2} - 2 \, {\left (23 \, c d e^{3} + 20 \, b e^{4}\right )} f g^{3} + {\left (3 \, c d^{2} e^{2} + 56 \, b d e^{3} + 48 \, a e^{4}\right )} g^{4}\right )} x\right )} \sqrt {e x + d} \sqrt {g x + f}}{384 \, e^{3} g^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/768*(3*(35*c*e^4*f^4 - 20*(3*c*d*e^3 + 2*b*e^4)*f^3*g + 6*(3*c*d^2*e^2 + 12*b*d*e^3 + 8*a*e^4)*f^2*g^2 + 4*
(c*d^3*e - 6*b*d^2*e^2 - 24*a*d*e^3)*f*g^3 + (3*c*d^4 - 8*b*d^3*e + 48*a*d^2*e^2)*g^4)*sqrt(e*g)*log(8*e^2*g^2
*x^2 + e^2*f^2 + 6*d*e*f*g + d^2*g^2 + 4*(2*e*g*x + e*f + d*g)*sqrt(e*g)*sqrt(e*x + d)*sqrt(g*x + f) + 8*(e^2*
f*g + d*e*g^2)*x) + 4*(48*c*e^4*g^4*x^3 - 105*c*e^4*f^3*g + 5*(29*c*d*e^3 + 24*b*e^4)*f^2*g^2 - (15*c*d^2*e^2
+ 176*b*d*e^3 + 144*a*e^4)*f*g^3 - 3*(3*c*d^3*e - 8*b*d^2*e^2 - 80*a*d*e^3)*g^4 - 8*(7*c*e^4*f*g^3 - (9*c*d*e^
3 + 8*b*e^4)*g^4)*x^2 + 2*(35*c*e^4*f^2*g^2 - 2*(23*c*d*e^3 + 20*b*e^4)*f*g^3 + (3*c*d^2*e^2 + 56*b*d*e^3 + 48
*a*e^4)*g^4)*x)*sqrt(e*x + d)*sqrt(g*x + f))/(e^3*g^5), -1/384*(3*(35*c*e^4*f^4 - 20*(3*c*d*e^3 + 2*b*e^4)*f^3
*g + 6*(3*c*d^2*e^2 + 12*b*d*e^3 + 8*a*e^4)*f^2*g^2 + 4*(c*d^3*e - 6*b*d^2*e^2 - 24*a*d*e^3)*f*g^3 + (3*c*d^4
- 8*b*d^3*e + 48*a*d^2*e^2)*g^4)*sqrt(-e*g)*arctan(1/2*(2*e*g*x + e*f + d*g)*sqrt(-e*g)*sqrt(e*x + d)*sqrt(g*x
+ f)/(e^2*g^2*x^2 + d*e*f*g + (e^2*f*g + d*e*g^2)*x)) - 2*(48*c*e^4*g^4*x^3 - 105*c*e^4*f^3*g + 5*(29*c*d*e^3
+ 24*b*e^4)*f^2*g^2 - (15*c*d^2*e^2 + 176*b*d*e^3 + 144*a*e^4)*f*g^3 - 3*(3*c*d^3*e - 8*b*d^2*e^2 - 80*a*d*e^
3)*g^4 - 8*(7*c*e^4*f*g^3 - (9*c*d*e^3 + 8*b*e^4)*g^4)*x^2 + 2*(35*c*e^4*f^2*g^2 - 2*(23*c*d*e^3 + 20*b*e^4)*f
*g^3 + (3*c*d^2*e^2 + 56*b*d*e^3 + 48*a*e^4)*g^4)*x)*sqrt(e*x + d)*sqrt(g*x + f))/(e^3*g^5)]

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giac [A]  time = 0.42, size = 448, normalized size = 1.35 \begin {gather*} \frac {1}{192} \, \sqrt {{\left (x e + d\right )} g e - d g e + f e^{2}} {\left (2 \, {\left (4 \, {\left (x e + d\right )} {\left (\frac {6 \, {\left (x e + d\right )} c e^{\left (-3\right )}}{g} - \frac {{\left (9 \, c d g^{6} e^{6} + 7 \, c f g^{5} e^{7} - 8 \, b g^{6} e^{7}\right )} e^{\left (-9\right )}}{g^{7}}\right )} + \frac {{\left (3 \, c d^{2} g^{6} e^{6} + 10 \, c d f g^{5} e^{7} - 8 \, b d g^{6} e^{7} + 35 \, c f^{2} g^{4} e^{8} - 40 \, b f g^{5} e^{8} + 48 \, a g^{6} e^{8}\right )} e^{\left (-9\right )}}{g^{7}}\right )} {\left (x e + d\right )} + \frac {3 \, {\left (3 \, c d^{3} g^{6} e^{6} + 7 \, c d^{2} f g^{5} e^{7} - 8 \, b d^{2} g^{6} e^{7} + 25 \, c d f^{2} g^{4} e^{8} - 32 \, b d f g^{5} e^{8} + 48 \, a d g^{6} e^{8} - 35 \, c f^{3} g^{3} e^{9} + 40 \, b f^{2} g^{4} e^{9} - 48 \, a f g^{5} e^{9}\right )} e^{\left (-9\right )}}{g^{7}}\right )} \sqrt {x e + d} - \frac {{\left (3 \, c d^{4} g^{4} + 4 \, c d^{3} f g^{3} e - 8 \, b d^{3} g^{4} e + 18 \, c d^{2} f^{2} g^{2} e^{2} - 24 \, b d^{2} f g^{3} e^{2} + 48 \, a d^{2} g^{4} e^{2} - 60 \, c d f^{3} g e^{3} + 72 \, b d f^{2} g^{2} e^{3} - 96 \, a d f g^{3} e^{3} + 35 \, c f^{4} e^{4} - 40 \, b f^{3} g e^{4} + 48 \, a f^{2} g^{2} e^{4}\right )} e^{\left (-\frac {5}{2}\right )} \log \left ({\left | -\sqrt {x e + d} \sqrt {g} e^{\frac {1}{2}} + \sqrt {{\left (x e + d\right )} g e - d g e + f e^{2}} \right |}\right )}{64 \, g^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*sqrt((x*e + d)*g*e - d*g*e + f*e^2)*(2*(4*(x*e + d)*(6*(x*e + d)*c*e^(-3)/g - (9*c*d*g^6*e^6 + 7*c*f*g^5
*e^7 - 8*b*g^6*e^7)*e^(-9)/g^7) + (3*c*d^2*g^6*e^6 + 10*c*d*f*g^5*e^7 - 8*b*d*g^6*e^7 + 35*c*f^2*g^4*e^8 - 40*
b*f*g^5*e^8 + 48*a*g^6*e^8)*e^(-9)/g^7)*(x*e + d) + 3*(3*c*d^3*g^6*e^6 + 7*c*d^2*f*g^5*e^7 - 8*b*d^2*g^6*e^7 +
25*c*d*f^2*g^4*e^8 - 32*b*d*f*g^5*e^8 + 48*a*d*g^6*e^8 - 35*c*f^3*g^3*e^9 + 40*b*f^2*g^4*e^9 - 48*a*f*g^5*e^9
)*e^(-9)/g^7)*sqrt(x*e + d) - 1/64*(3*c*d^4*g^4 + 4*c*d^3*f*g^3*e - 8*b*d^3*g^4*e + 18*c*d^2*f^2*g^2*e^2 - 24*
b*d^2*f*g^3*e^2 + 48*a*d^2*g^4*e^2 - 60*c*d*f^3*g*e^3 + 72*b*d*f^2*g^2*e^3 - 96*a*d*f*g^3*e^3 + 35*c*f^4*e^4 -
40*b*f^3*g*e^4 + 48*a*f^2*g^2*e^4)*e^(-5/2)*log(abs(-sqrt(x*e + d)*sqrt(g)*e^(1/2) + sqrt((x*e + d)*g*e - d*g
*e + f*e^2)))/g^(9/2)

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maple [B]  time = 0.03, size = 1207, normalized size = 3.62

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(e*x+d)^(1/2)*(g*x+f)^(1/2)*(9*ln(1/2*(2*e*g*x+d*g+e*f+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2))/(e*g)^(1/2
))*c*d^4*g^4+105*ln(1/2*(2*e*g*x+d*g+e*f+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2))/(e*g)^(1/2))*c*e^4*f^4-72*ln(1
/2*(2*e*g*x+d*g+e*f+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2))/(e*g)^(1/2))*b*d^2*e^2*f*g^3-30*(e*g)^(1/2)*((e*x+d
)*(g*x+f))^(1/2)*c*d^2*e*f*g^2+224*(e*g)^(1/2)*((e*x+d)*(g*x+f))^(1/2)*x*b*d*e^2*g^3-160*(e*g)^(1/2)*((e*x+d)*
(g*x+f))^(1/2)*x*b*e^3*f*g^2+12*(e*g)^(1/2)*((e*x+d)*(g*x+f))^(1/2)*x*c*d^2*e*g^3+140*(e*g)^(1/2)*((e*x+d)*(g*
x+f))^(1/2)*x*c*e^3*f^2*g-112*x^2*c*e^3*f*g^2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)-352*(e*g)^(1/2)*((e*x+d)*(g*
x+f))^(1/2)*b*d*e^2*f*g^2+290*(e*g)^(1/2)*((e*x+d)*(g*x+f))^(1/2)*c*d*e^2*f^2*g+144*x^2*c*d*e^2*g^3*((e*x+d)*(
g*x+f))^(1/2)*(e*g)^(1/2)+144*ln(1/2*(2*e*g*x+d*g+e*f+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2))/(e*g)^(1/2))*a*d^
2*e^2*g^4-184*(e*g)^(1/2)*((e*x+d)*(g*x+f))^(1/2)*x*c*d*e^2*f*g^2+144*ln(1/2*(2*e*g*x+d*g+e*f+2*((e*x+d)*(g*x+
f))^(1/2)*(e*g)^(1/2))/(e*g)^(1/2))*a*e^4*f^2*g^2-120*ln(1/2*(2*e*g*x+d*g+e*f+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^
(1/2))/(e*g)^(1/2))*b*e^4*f^3*g-210*(e*g)^(1/2)*((e*x+d)*(g*x+f))^(1/2)*c*e^3*f^3-24*ln(1/2*(2*e*g*x+d*g+e*f+2
*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2))/(e*g)^(1/2))*b*d^3*e*g^4-18*(e*g)^(1/2)*((e*x+d)*(g*x+f))^(1/2)*c*d^3*g^
3+216*ln(1/2*(2*e*g*x+d*g+e*f+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2))/(e*g)^(1/2))*b*d*e^3*f^2*g^2+54*ln(1/2*(2
*e*g*x+d*g+e*f+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2))/(e*g)^(1/2))*c*d^2*e^2*f^2*g^2-180*ln(1/2*(2*e*g*x+d*g+e
*f+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2))/(e*g)^(1/2))*c*d*e^3*f^3*g+480*(e*g)^(1/2)*((e*x+d)*(g*x+f))^(1/2)*a
*d*e^2*g^3-288*(e*g)^(1/2)*((e*x+d)*(g*x+f))^(1/2)*a*e^3*f*g^2+240*(e*g)^(1/2)*((e*x+d)*(g*x+f))^(1/2)*b*e^3*f
^2*g-288*ln(1/2*(2*e*g*x+d*g+e*f+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2))/(e*g)^(1/2))*a*d*e^3*f*g^3+48*(e*g)^(1
/2)*((e*x+d)*(g*x+f))^(1/2)*b*d^2*e*g^3+192*(e*g)^(1/2)*((e*x+d)*(g*x+f))^(1/2)*x*a*e^3*g^3+96*x^3*c*e^3*g^3*(
(e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+128*x^2*b*e^3*g^3*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+12*ln(1/2*(2*e*g*x+d*
g+e*f+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2))/(e*g)^(1/2))*c*d^3*e*f*g^3)/e^2/g^4/((e*x+d)*(g*x+f))^(1/2)/(e*g)
^(1/2)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(c*x^2+b*x+a)/(g*x+f)^(1/2),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(d*g-e*f>0)', see `assume?` for
more details)Is d*g-e*f zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^{3/2}\,\left (c\,x^2+b\,x+a\right )}{\sqrt {f+g\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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