3.12.71 \(\int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^6} \, dx\) [1171]

Optimal. Leaf size=449 \[ \frac {\left (32 A c^3 d^3-16 b c^2 d^2 (B d+3 A e)+6 b^2 c d e (2 B d+5 A e)-b^3 e^2 (3 B d+7 A e)\right ) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{128 d^4 (c d-b e)^4 (d+e x)^2}+\frac {(B d-A e) \left (b x+c x^2\right )^{3/2}}{5 d (c d-b e) (d+e x)^5}-\frac {(7 A e (2 c d-b e)-B d (4 c d+3 b e)) \left (b x+c x^2\right )^{3/2}}{40 d^2 (c d-b e)^2 (d+e x)^4}+\frac {\left (B d \left (8 c^2 d^2+42 b c d e-15 b^2 e^2\right )-A e \left (108 c^2 d^2-108 b c d e+35 b^2 e^2\right )\right ) \left (b x+c x^2\right )^{3/2}}{240 d^3 (c d-b e)^3 (d+e x)^3}-\frac {b^2 \left (32 A c^3 d^3-16 b c^2 d^2 (B d+3 A e)+6 b^2 c d e (2 B d+5 A e)-b^3 e^2 (3 B d+7 A e)\right ) \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{256 d^{9/2} (c d-b e)^{9/2}} \]

[Out]

1/5*(-A*e+B*d)*(c*x^2+b*x)^(3/2)/d/(-b*e+c*d)/(e*x+d)^5-1/40*(7*A*e*(-b*e+2*c*d)-B*d*(3*b*e+4*c*d))*(c*x^2+b*x
)^(3/2)/d^2/(-b*e+c*d)^2/(e*x+d)^4+1/240*(B*d*(-15*b^2*e^2+42*b*c*d*e+8*c^2*d^2)-A*e*(35*b^2*e^2-108*b*c*d*e+1
08*c^2*d^2))*(c*x^2+b*x)^(3/2)/d^3/(-b*e+c*d)^3/(e*x+d)^3-1/256*b^2*(32*A*c^3*d^3-16*b*c^2*d^2*(3*A*e+B*d)+6*b
^2*c*d*e*(5*A*e+2*B*d)-b^3*e^2*(7*A*e+3*B*d))*arctanh(1/2*(b*d+(-b*e+2*c*d)*x)/d^(1/2)/(-b*e+c*d)^(1/2)/(c*x^2
+b*x)^(1/2))/d^(9/2)/(-b*e+c*d)^(9/2)+1/128*(32*A*c^3*d^3-16*b*c^2*d^2*(3*A*e+B*d)+6*b^2*c*d*e*(5*A*e+2*B*d)-b
^3*e^2*(7*A*e+3*B*d))*(b*d+(-b*e+2*c*d)*x)*(c*x^2+b*x)^(1/2)/d^4/(-b*e+c*d)^4/(e*x+d)^2

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Rubi [A]
time = 0.52, antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {848, 820, 734, 738, 212} \begin {gather*} \frac {\left (b x+c x^2\right )^{3/2} \left (B d \left (-15 b^2 e^2+42 b c d e+8 c^2 d^2\right )-A e \left (35 b^2 e^2-108 b c d e+108 c^2 d^2\right )\right )}{240 d^3 (d+e x)^3 (c d-b e)^3}-\frac {b^2 \left (b^3 \left (-e^2\right ) (7 A e+3 B d)+6 b^2 c d e (5 A e+2 B d)-16 b c^2 d^2 (3 A e+B d)+32 A c^3 d^3\right ) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{256 d^{9/2} (c d-b e)^{9/2}}+\frac {\sqrt {b x+c x^2} (x (2 c d-b e)+b d) \left (b^3 \left (-e^2\right ) (7 A e+3 B d)+6 b^2 c d e (5 A e+2 B d)-16 b c^2 d^2 (3 A e+B d)+32 A c^3 d^3\right )}{128 d^4 (d+e x)^2 (c d-b e)^4}-\frac {\left (b x+c x^2\right )^{3/2} (7 A e (2 c d-b e)-B d (3 b e+4 c d))}{40 d^2 (d+e x)^4 (c d-b e)^2}+\frac {\left (b x+c x^2\right )^{3/2} (B d-A e)}{5 d (d+e x)^5 (c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*Sqrt[b*x + c*x^2])/(d + e*x)^6,x]

[Out]

((32*A*c^3*d^3 - 16*b*c^2*d^2*(B*d + 3*A*e) + 6*b^2*c*d*e*(2*B*d + 5*A*e) - b^3*e^2*(3*B*d + 7*A*e))*(b*d + (2
*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(128*d^4*(c*d - b*e)^4*(d + e*x)^2) + ((B*d - A*e)*(b*x + c*x^2)^(3/2))/(5*d
*(c*d - b*e)*(d + e*x)^5) - ((7*A*e*(2*c*d - b*e) - B*d*(4*c*d + 3*b*e))*(b*x + c*x^2)^(3/2))/(40*d^2*(c*d - b
*e)^2*(d + e*x)^4) + ((B*d*(8*c^2*d^2 + 42*b*c*d*e - 15*b^2*e^2) - A*e*(108*c^2*d^2 - 108*b*c*d*e + 35*b^2*e^2
))*(b*x + c*x^2)^(3/2))/(240*d^3*(c*d - b*e)^3*(d + e*x)^3) - (b^2*(32*A*c^3*d^3 - 16*b*c^2*d^2*(B*d + 3*A*e)
+ 6*b^2*c*d*e*(2*B*d + 5*A*e) - b^3*e^2*(3*B*d + 7*A*e))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d -
b*e]*Sqrt[b*x + c*x^2])])/(256*d^(9/2)*(c*d - b*e)^(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 734

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))
*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[p*((b^2
- 4*a*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; Free
Q[{a, b, c, d, e}, 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] && EqQ[m
+ 2*p + 2, 0] && GtQ[p, 0]

Rule 738

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

Rule 820

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Dist[
(b*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x]
, x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[S
implify[m + 2*p + 3], 0]

Rule 848

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rubi steps

\begin {align*} \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^6} \, dx &=\frac {(B d-A e) \left (b x+c x^2\right )^{3/2}}{5 d (c d-b e) (d+e x)^5}-\frac {\int \frac {\left (\frac {1}{2} (-10 A c d+b (3 B d+7 A e))-2 c (B d-A e) x\right ) \sqrt {b x+c x^2}}{(d+e x)^5} \, dx}{5 d (c d-b e)}\\ &=\frac {(B d-A e) \left (b x+c x^2\right )^{3/2}}{5 d (c d-b e) (d+e x)^5}-\frac {(7 A e (2 c d-b e)-B d (4 c d+3 b e)) \left (b x+c x^2\right )^{3/2}}{40 d^2 (c d-b e)^2 (d+e x)^4}+\frac {\int \frac {\left (\frac {1}{4} \left (80 A c^2 d^2+5 b^2 e (3 B d+7 A e)-2 b c d (18 B d+47 A e)\right )-\frac {1}{2} c (7 A e (2 c d-b e)-B d (4 c d+3 b e)) x\right ) \sqrt {b x+c x^2}}{(d+e x)^4} \, dx}{20 d^2 (c d-b e)^2}\\ &=\frac {(B d-A e) \left (b x+c x^2\right )^{3/2}}{5 d (c d-b e) (d+e x)^5}-\frac {(7 A e (2 c d-b e)-B d (4 c d+3 b e)) \left (b x+c x^2\right )^{3/2}}{40 d^2 (c d-b e)^2 (d+e x)^4}+\frac {\left (B d \left (8 c^2 d^2+42 b c d e-15 b^2 e^2\right )-A e \left (108 c^2 d^2-108 b c d e+35 b^2 e^2\right )\right ) \left (b x+c x^2\right )^{3/2}}{240 d^3 (c d-b e)^3 (d+e x)^3}+\frac {\left (32 A c^3 d^3-16 b c^2 d^2 (B d+3 A e)+6 b^2 c d e (2 B d+5 A e)-b^3 e^2 (3 B d+7 A e)\right ) \int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx}{32 d^3 (c d-b e)^3}\\ &=\frac {\left (32 A c^3 d^3-16 b c^2 d^2 (B d+3 A e)+6 b^2 c d e (2 B d+5 A e)-b^3 e^2 (3 B d+7 A e)\right ) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{128 d^4 (c d-b e)^4 (d+e x)^2}+\frac {(B d-A e) \left (b x+c x^2\right )^{3/2}}{5 d (c d-b e) (d+e x)^5}-\frac {(7 A e (2 c d-b e)-B d (4 c d+3 b e)) \left (b x+c x^2\right )^{3/2}}{40 d^2 (c d-b e)^2 (d+e x)^4}+\frac {\left (B d \left (8 c^2 d^2+42 b c d e-15 b^2 e^2\right )-A e \left (108 c^2 d^2-108 b c d e+35 b^2 e^2\right )\right ) \left (b x+c x^2\right )^{3/2}}{240 d^3 (c d-b e)^3 (d+e x)^3}-\frac {\left (b^2 \left (32 A c^3 d^3-16 b c^2 d^2 (B d+3 A e)+6 b^2 c d e (2 B d+5 A e)-b^3 e^2 (3 B d+7 A e)\right )\right ) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{256 d^4 (c d-b e)^4}\\ &=\frac {\left (32 A c^3 d^3-16 b c^2 d^2 (B d+3 A e)+6 b^2 c d e (2 B d+5 A e)-b^3 e^2 (3 B d+7 A e)\right ) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{128 d^4 (c d-b e)^4 (d+e x)^2}+\frac {(B d-A e) \left (b x+c x^2\right )^{3/2}}{5 d (c d-b e) (d+e x)^5}-\frac {(7 A e (2 c d-b e)-B d (4 c d+3 b e)) \left (b x+c x^2\right )^{3/2}}{40 d^2 (c d-b e)^2 (d+e x)^4}+\frac {\left (B d \left (8 c^2 d^2+42 b c d e-15 b^2 e^2\right )-A e \left (108 c^2 d^2-108 b c d e+35 b^2 e^2\right )\right ) \left (b x+c x^2\right )^{3/2}}{240 d^3 (c d-b e)^3 (d+e x)^3}+\frac {\left (b^2 \left (32 A c^3 d^3-16 b c^2 d^2 (B d+3 A e)+6 b^2 c d e (2 B d+5 A e)-b^3 e^2 (3 B d+7 A e)\right )\right ) \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac {-b d-(2 c d-b e) x}{\sqrt {b x+c x^2}}\right )}{128 d^4 (c d-b e)^4}\\ &=\frac {\left (32 A c^3 d^3-16 b c^2 d^2 (B d+3 A e)+6 b^2 c d e (2 B d+5 A e)-b^3 e^2 (3 B d+7 A e)\right ) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{128 d^4 (c d-b e)^4 (d+e x)^2}+\frac {(B d-A e) \left (b x+c x^2\right )^{3/2}}{5 d (c d-b e) (d+e x)^5}-\frac {(7 A e (2 c d-b e)-B d (4 c d+3 b e)) \left (b x+c x^2\right )^{3/2}}{40 d^2 (c d-b e)^2 (d+e x)^4}+\frac {\left (B d \left (8 c^2 d^2+42 b c d e-15 b^2 e^2\right )-A e \left (108 c^2 d^2-108 b c d e+35 b^2 e^2\right )\right ) \left (b x+c x^2\right )^{3/2}}{240 d^3 (c d-b e)^3 (d+e x)^3}-\frac {b^2 \left (32 A c^3 d^3-16 b c^2 d^2 (B d+3 A e)+6 b^2 c d e (2 B d+5 A e)-b^3 e^2 (3 B d+7 A e)\right ) \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{256 d^{9/2} (c d-b e)^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 11.81, size = 387, normalized size = 0.86 \begin {gather*} -\frac {\sqrt {x (b+c x)} \left (384 (B d-A e) x^{3/2} (b+c x)+\frac {48 (7 A e (-2 c d+b e)+B d (4 c d+3 b e)) x^{3/2} (b+c x) (d+e x)}{d (c d-b e)}+\frac {8 \left (A e \left (-108 c^2 d^2+108 b c d e-35 b^2 e^2\right )+B d \left (8 c^2 d^2+42 b c d e-15 b^2 e^2\right )\right ) x^{3/2} (b+c x) (d+e x)^2}{d^2 (c d-b e)^2}+\frac {15 \left (-32 A c^3 d^3+16 b c^2 d^2 (B d+3 A e)-6 b^2 c d e (2 B d+5 A e)+b^3 e^2 (3 B d+7 A e)\right ) (d+e x)^3 \left (\sqrt {d} \sqrt {c d-b e} \sqrt {x} \sqrt {b+c x} (-b d-2 c d x+b e x)+b^2 (d+e x)^2 \tanh ^{-1}\left (\frac {\sqrt {c d-b e} \sqrt {x}}{\sqrt {d} \sqrt {b+c x}}\right )\right )}{d^{7/2} (c d-b e)^{7/2} \sqrt {b+c x}}\right )}{1920 d (-c d+b e) \sqrt {x} (d+e x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*Sqrt[b*x + c*x^2])/(d + e*x)^6,x]

[Out]

-1/1920*(Sqrt[x*(b + c*x)]*(384*(B*d - A*e)*x^(3/2)*(b + c*x) + (48*(7*A*e*(-2*c*d + b*e) + B*d*(4*c*d + 3*b*e
))*x^(3/2)*(b + c*x)*(d + e*x))/(d*(c*d - b*e)) + (8*(A*e*(-108*c^2*d^2 + 108*b*c*d*e - 35*b^2*e^2) + B*d*(8*c
^2*d^2 + 42*b*c*d*e - 15*b^2*e^2))*x^(3/2)*(b + c*x)*(d + e*x)^2)/(d^2*(c*d - b*e)^2) + (15*(-32*A*c^3*d^3 + 1
6*b*c^2*d^2*(B*d + 3*A*e) - 6*b^2*c*d*e*(2*B*d + 5*A*e) + b^3*e^2*(3*B*d + 7*A*e))*(d + e*x)^3*(Sqrt[d]*Sqrt[c
*d - b*e]*Sqrt[x]*Sqrt[b + c*x]*(-(b*d) - 2*c*d*x + b*e*x) + b^2*(d + e*x)^2*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])
/(Sqrt[d]*Sqrt[b + c*x])]))/(d^(7/2)*(c*d - b*e)^(7/2)*Sqrt[b + c*x])))/(d*(-(c*d) + b*e)*Sqrt[x]*(d + e*x)^5)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(5574\) vs. \(2(419)=838\).
time = 0.61, size = 5575, normalized size = 12.42

method	result	size
default	\(\text {Expression too large to display}\)	\(5575\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^(1/2)/(e*x+d)^6,x,method=_RETURNVERBOSE)

[Out]

result too large to display

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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((B*x+A)*(c*x^2+b*x)^(1/2)/(e*x+d)^6,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(c*d-%e*b>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1712 vs. \(2 (441) = 882\).
time = 3.77, size = 3437, normalized size = 7.65 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(1/2)/(e*x+d)^6,x, algorithm="fricas")

[Out]

[1/3840*(15*(7*A*b^5*x^5*e^8 + 16*(B*b^3*c^2 - 2*A*b^2*c^3)*d^8 + (35*A*b^5*d*x^4 + 3*(B*b^5 - 10*A*b^4*c)*d*x
^5)*e^7 + (70*A*b^5*d^2*x^3 - 12*(B*b^4*c - 4*A*b^3*c^2)*d^2*x^5 + 15*(B*b^5 - 10*A*b^4*c)*d^2*x^4)*e^6 + 2*(3
5*A*b^5*d^3*x^2 + 8*(B*b^3*c^2 - 2*A*b^2*c^3)*d^3*x^5 - 30*(B*b^4*c - 4*A*b^3*c^2)*d^3*x^4 + 15*(B*b^5 - 10*A*
b^4*c)*d^3*x^3)*e^5 + 5*(7*A*b^5*d^4*x + 16*(B*b^3*c^2 - 2*A*b^2*c^3)*d^4*x^4 - 24*(B*b^4*c - 4*A*b^3*c^2)*d^4
*x^3 + 6*(B*b^5 - 10*A*b^4*c)*d^4*x^2)*e^4 + (7*A*b^5*d^5 + 160*(B*b^3*c^2 - 2*A*b^2*c^3)*d^5*x^3 - 120*(B*b^4
*c - 4*A*b^3*c^2)*d^5*x^2 + 15*(B*b^5 - 10*A*b^4*c)*d^5*x)*e^3 + (160*(B*b^3*c^2 - 2*A*b^2*c^3)*d^6*x^2 - 60*(
B*b^4*c - 4*A*b^3*c^2)*d^6*x + 3*(B*b^5 - 10*A*b^4*c)*d^6)*e^2 + 4*(20*(B*b^3*c^2 - 2*A*b^2*c^3)*d^7*x - 3*(B*
b^4*c - 4*A*b^3*c^2)*d^7)*e)*sqrt(c*d^2 - b*d*e)*log((2*c*d*x - b*x*e + b*d + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2
+ b*x))/(x*e + d)) + 2*(640*B*c^5*d^9*x^2 - 105*A*b^5*d*x^4*e^8 + 160*(B*b*c^4 + 6*A*c^5)*d^9*x - 240*(B*b^2*
c^3 - 2*A*b*c^4)*d^9 - 5*(98*A*b^5*d^2*x^3 + (9*B*b^5 - 97*A*b^4*c)*d^2*x^4)*e^7 - (896*A*b^5*d^3*x^2 - (195*B
*b^4*c - 856*A*b^3*c^2)*d^3*x^4 + 42*(5*B*b^5 - 54*A*b^4*c)*d^3*x^3)*e^6 - 2*(395*A*b^5*d^4*x + (147*B*b^3*c^2
- 334*A*b^2*c^3)*d^4*x^4 - 3*(152*B*b^4*c - 669*A*b^3*c^2)*d^4*x^3 + 3*(64*B*b^5 - 693*A*b^4*c)*d^4*x^2)*e^5
+ (105*A*b^5*d^5 + 32*(11*B*b^2*c^3 - 9*A*b*c^4)*d^5*x^4 - 2*(763*B*b^3*c^2 - 1622*A*b^2*c^3)*d^5*x^3 + 66*(27
*B*b^4*c - 113*A*b^3*c^2)*d^5*x^2 + 10*(21*B*b^5 + 374*A*b^4*c)*d^5*x)*e^4 - (16*(17*B*b*c^4 - 6*A*c^5)*d^6*x^
4 - 24*(79*B*b^2*c^3 - 62*A*b*c^4)*d^6*x^3 + 2*(1691*B*b^3*c^2 - 3178*A*b^2*c^3)*d^6*x^2 + 10*(108*B*b^4*c + 6
97*A*b^3*c^2)*d^6*x - 15*(3*B*b^5 - 37*A*b^4*c)*d^6)*e^3 + (64*B*c^5*d^7*x^4 - 48*(29*B*b*c^4 - 10*A*c^5)*d^7*
x^3 + 48*(88*B*b^2*c^3 - 65*A*b*c^4)*d^7*x^2 + 10*(211*B*b^3*c^2 + 642*A*b^2*c^3)*d^7*x - 45*(5*B*b^4*c - 26*A
*b^3*c^2)*d^7)*e^2 + 20*(16*B*c^5*d^8*x^3 - 48*(3*B*b*c^4 - A*c^5)*d^8*x^2 - 14*(5*B*b^2*c^3 + 12*A*b*c^4)*d^8
*x + 3*(7*B*b^3*c^2 - 20*A*b^2*c^3)*d^8)*e)*sqrt(c*x^2 + b*x))/(c^5*d^15 - b^5*d^5*x^5*e^10 + 5*(b^4*c*d^6*x^5
- b^5*d^6*x^4)*e^9 - 5*(2*b^3*c^2*d^7*x^5 - 5*b^4*c*d^7*x^4 + 2*b^5*d^7*x^3)*e^8 + 10*(b^2*c^3*d^8*x^5 - 5*b^
3*c^2*d^8*x^4 + 5*b^4*c*d^8*x^3 - b^5*d^8*x^2)*e^7 - 5*(b*c^4*d^9*x^5 - 10*b^2*c^3*d^9*x^4 + 20*b^3*c^2*d^9*x^
3 - 10*b^4*c*d^9*x^2 + b^5*d^9*x)*e^6 + (c^5*d^10*x^5 - 25*b*c^4*d^10*x^4 + 100*b^2*c^3*d^10*x^3 - 100*b^3*c^2
*d^10*x^2 + 25*b^4*c*d^10*x - b^5*d^10)*e^5 + 5*(c^5*d^11*x^4 - 10*b*c^4*d^11*x^3 + 20*b^2*c^3*d^11*x^2 - 10*b
^3*c^2*d^11*x + b^4*c*d^11)*e^4 + 10*(c^5*d^12*x^3 - 5*b*c^4*d^12*x^2 + 5*b^2*c^3*d^12*x - b^3*c^2*d^12)*e^3 +
5*(2*c^5*d^13*x^2 - 5*b*c^4*d^13*x + 2*b^2*c^3*d^13)*e^2 + 5*(c^5*d^14*x - b*c^4*d^14)*e), 1/1920*(15*(7*A*b^
5*x^5*e^8 + 16*(B*b^3*c^2 - 2*A*b^2*c^3)*d^8 + (35*A*b^5*d*x^4 + 3*(B*b^5 - 10*A*b^4*c)*d*x^5)*e^7 + (70*A*b^5
*d^2*x^3 - 12*(B*b^4*c - 4*A*b^3*c^2)*d^2*x^5 + 15*(B*b^5 - 10*A*b^4*c)*d^2*x^4)*e^6 + 2*(35*A*b^5*d^3*x^2 + 8
*(B*b^3*c^2 - 2*A*b^2*c^3)*d^3*x^5 - 30*(B*b^4*c - 4*A*b^3*c^2)*d^3*x^4 + 15*(B*b^5 - 10*A*b^4*c)*d^3*x^3)*e^5
+ 5*(7*A*b^5*d^4*x + 16*(B*b^3*c^2 - 2*A*b^2*c^3)*d^4*x^4 - 24*(B*b^4*c - 4*A*b^3*c^2)*d^4*x^3 + 6*(B*b^5 - 1
0*A*b^4*c)*d^4*x^2)*e^4 + (7*A*b^5*d^5 + 160*(B*b^3*c^2 - 2*A*b^2*c^3)*d^5*x^3 - 120*(B*b^4*c - 4*A*b^3*c^2)*d
^5*x^2 + 15*(B*b^5 - 10*A*b^4*c)*d^5*x)*e^3 + (160*(B*b^3*c^2 - 2*A*b^2*c^3)*d^6*x^2 - 60*(B*b^4*c - 4*A*b^3*c
^2)*d^6*x + 3*(B*b^5 - 10*A*b^4*c)*d^6)*e^2 + 4*(20*(B*b^3*c^2 - 2*A*b^2*c^3)*d^7*x - 3*(B*b^4*c - 4*A*b^3*c^2
)*d^7)*e)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/(c*d*x - b*x*e)) + (640*B*c^5*d^
9*x^2 - 105*A*b^5*d*x^4*e^8 + 160*(B*b*c^4 + 6*A*c^5)*d^9*x - 240*(B*b^2*c^3 - 2*A*b*c^4)*d^9 - 5*(98*A*b^5*d^
2*x^3 + (9*B*b^5 - 97*A*b^4*c)*d^2*x^4)*e^7 - (896*A*b^5*d^3*x^2 - (195*B*b^4*c - 856*A*b^3*c^2)*d^3*x^4 + 42*
(5*B*b^5 - 54*A*b^4*c)*d^3*x^3)*e^6 - 2*(395*A*b^5*d^4*x + (147*B*b^3*c^2 - 334*A*b^2*c^3)*d^4*x^4 - 3*(152*B*
b^4*c - 669*A*b^3*c^2)*d^4*x^3 + 3*(64*B*b^5 - 693*A*b^4*c)*d^4*x^2)*e^5 + (105*A*b^5*d^5 + 32*(11*B*b^2*c^3 -
9*A*b*c^4)*d^5*x^4 - 2*(763*B*b^3*c^2 - 1622*A*b^2*c^3)*d^5*x^3 + 66*(27*B*b^4*c - 113*A*b^3*c^2)*d^5*x^2 + 1
0*(21*B*b^5 + 374*A*b^4*c)*d^5*x)*e^4 - (16*(17*B*b*c^4 - 6*A*c^5)*d^6*x^4 - 24*(79*B*b^2*c^3 - 62*A*b*c^4)*d^
6*x^3 + 2*(1691*B*b^3*c^2 - 3178*A*b^2*c^3)*d^6*x^2 + 10*(108*B*b^4*c + 697*A*b^3*c^2)*d^6*x - 15*(3*B*b^5 - 3
7*A*b^4*c)*d^6)*e^3 + (64*B*c^5*d^7*x^4 - 48*(29*B*b*c^4 - 10*A*c^5)*d^7*x^3 + 48*(88*B*b^2*c^3 - 65*A*b*c^4)*
d^7*x^2 + 10*(211*B*b^3*c^2 + 642*A*b^2*c^3)*d^7*x - 45*(5*B*b^4*c - 26*A*b^3*c^2)*d^7)*e^2 + 20*(16*B*c^5*d^8
*x^3 - 48*(3*B*b*c^4 - A*c^5)*d^8*x^2 - 14*(5*B*b^2*c^3 + 12*A*b*c^4)*d^8*x + 3*(7*B*b^3*c^2 - 20*A*b^2*c^3)*d
^8)*e)*sqrt(c*x^2 + b*x))/(c^5*d^15 - b^5*d^5*x^5*e^10 + 5*(b^4*c*d^6*x^5 - b^5*d^6*x^4)*e^9 - 5*(2*b^3*c^2*d^
7*x^5 - 5*b^4*c*d^7*x^4 + 2*b^5*d^7*x^3)*e^8 + 10*(b^2*c^3*d^8*x^5 - 5*b^3*c^2*d^8*x^4 + 5*b^4*c*d^8*x^3 - b^5
*d^8*x^2)*e^7 - 5*(b*c^4*d^9*x^5 - 10*b^2*c^3*d...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x \left (b + c x\right )} \left (A + B x\right )}{\left (d + e x\right )^{6}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**(1/2)/(e*x+d)**6,x)

[Out]

Integral(sqrt(x*(b + c*x))*(A + B*x)/(d + e*x)**6, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 4102 vs. \(2 (441) = 882\).
time = 2.25, size = 4102, normalized size = 9.14 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(1/2)/(e*x+d)^6,x, algorithm="giac")

[Out]

1/128*(16*B*b^3*c^2*d^3 - 32*A*b^2*c^3*d^3 - 12*B*b^4*c*d^2*e + 48*A*b^3*c^2*d^2*e + 3*B*b^5*d*e^2 - 30*A*b^4*
c*d*e^2 + 7*A*b^5*e^3)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))/((c^4*d^8
- 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 - 4*b^3*c*d^5*e^3 + b^4*d^4*e^4)*sqrt(-c*d^2 + b*d*e)) + 1/1920*(5120*(sq
rt(c)*x - sqrt(c*x^2 + b*x))^6*B*c^(13/2)*d^9*e + 2048*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*c^7*d^10 + 5120*(sq
rt(c)*x - sqrt(c*x^2 + b*x))^7*B*c^6*d^8*e^2 + 3584*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b*c^6*d^9*e + 3072*(sq
rt(c)*x - sqrt(c*x^2 + b*x))^5*A*c^7*d^9*e + 5120*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b*c^(13/2)*d^10 - 8960*(
sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b*c^(11/2)*d^8*e^2 + 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*c^(13/2)*d^8*
e^2 - 8960*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^2*c^(11/2)*d^9*e + 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b
*c^(13/2)*d^9*e + 5120*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^2*c^6*d^10 - 20480*(sqrt(c)*x - sqrt(c*x^2 + b*x)
)^7*B*b*c^5*d^7*e^3 - 24832*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^2*c^5*d^8*e^2 + 9216*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^5*A*b*c^6*d^8*e^2 - 14080*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^3*c^5*d^9*e + 7680*(sqrt(c)*x - sqrt(
c*x^2 + b*x))^3*A*b^2*c^6*d^9*e + 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^3*c^(11/2)*d^10 - 15360*(sqrt(c)*
x - sqrt(c*x^2 + b*x))^6*B*b^2*c^(9/2)*d^7*e^3 - 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b*c^(11/2)*d^7*e^3
- 12800*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^3*c^(9/2)*d^8*e^2 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^2
*c^(11/2)*d^8*e^2 - 8000*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^4*c^(9/2)*d^9*e + 3840*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^2*A*b^3*c^(11/2)*d^9*e + 640*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^4*c^5*d^10 + 30720*(sqrt(c)*x - sqrt(
c*x^2 + b*x))^7*B*b^2*c^4*d^6*e^4 + 13760*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^3*c^4*d^7*e^3 - 50048*(sqrt(c)
*x - sqrt(c*x^2 + b*x))^5*A*b^2*c^5*d^7*e^3 + 3200*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^4*c^4*d^8*e^2 - 11520
*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^3*c^5*d^8*e^2 - 2080*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^5*c^4*d^9*e +
960*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^4*c^5*d^9*e + 64*B*b^5*c^(9/2)*d^10 + 36320*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^6*B*b^3*c^(7/2)*d^6*e^4 + 70720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^2*c^(9/2)*d^6*e^4 + 15520*(sqrt(c)
*x - sqrt(c*x^2 + b*x))^4*B*b^4*c^(7/2)*d^7*e^3 - 17600*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^3*c^(9/2)*d^7*e^
3 + 4720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^5*c^(7/2)*d^8*e^2 - 7200*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^
4*c^(9/2)*d^8*e^2 - 208*B*b^6*c^(7/2)*d^9*e + 96*A*b^5*c^(9/2)*d^9*e - 28000*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7
*B*b^3*c^3*d^5*e^5 + 15040*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^2*c^4*d^5*e^5 + 2000*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^5*B*b^4*c^3*d^6*e^4 + 129280*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^3*c^4*d^6*e^4 + 1280*(sqrt(c)*x - s
qrt(c*x^2 + b*x))^3*B*b^5*c^3*d^7*e^3 + 14080*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^4*c^4*d^7*e^3 + 1440*(sqrt
(c)*x - sqrt(c*x^2 + b*x))*B*b^6*c^3*d^8*e^2 - 1920*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^5*c^4*d^8*e^2 - 2160*(
sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b^3*c^(5/2)*d^4*e^6 + 4320*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b^2*c^(7/2)*
d^4*e^6 - 39560*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^4*c^(5/2)*d^5*e^5 - 52000*(sqrt(c)*x - sqrt(c*x^2 + b*x)
)^6*A*b^3*c^(7/2)*d^5*e^5 - 14360*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^5*c^(5/2)*d^6*e^4 + 81920*(sqrt(c)*x -
sqrt(c*x^2 + b*x))^4*A*b^4*c^(7/2)*d^6*e^4 - 3120*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^6*c^(5/2)*d^7*e^3 + 1
3760*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^5*c^(7/2)*d^7*e^3 + 144*B*b^7*c^(5/2)*d^8*e^2 - 192*A*b^6*c^(7/2)*d
^8*e^2 - 240*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b^3*c^2*d^3*e^7 + 480*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*b^2
*c^3*d^3*e^7 + 9640*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^4*c^2*d^4*e^6 - 20320*(sqrt(c)*x - sqrt(c*x^2 + b*x)
)^7*A*b^3*c^3*d^4*e^6 - 17284*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^5*c^2*d^5*e^5 - 120680*(sqrt(c)*x - sqrt(c
*x^2 + b*x))^5*A*b^4*c^3*d^5*e^5 - 6920*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^6*c^2*d^6*e^4 + 14080*(sqrt(c)*x
- sqrt(c*x^2 + b*x))^3*A*b^5*c^3*d^6*e^4 - 1260*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^7*c^2*d^7*e^3 + 4280*(sqr
t(c)*x - sqrt(c*x^2 + b*x))*A*b^6*c^3*d^7*e^3 + 1620*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b^4*c^(3/2)*d^3*e^7 -
6480*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b^3*c^(5/2)*d^3*e^7 + 15090*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^5*
c^(3/2)*d^4*e^6 + 7260*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^4*c^(5/2)*d^4*e^6 + 330*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^4*B*b^6*c^(3/2)*d^5*e^5 - 85780*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^5*c^(5/2)*d^5*e^5 - 570*(sqrt(c)*
x - sqrt(c*x^2 + b*x))^2*B*b^7*c^(3/2)*d^6*e^4 - 6340*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^6*c^(5/2)*d^6*e^4
- 150*B*b^8*c^(3/2)*d^7*e^3 + 476*A*b^7*c^(5/2)*d^7*e^3 + 180*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b^4*c*d^2*e^
8 - 720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*b^3*c^2*d^2*e^8 - 570*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^5*c*d^
3*e^7 + 10740*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {c\,x^2+b\,x}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x + c*x^2)^(1/2)*(A + B*x))/(d + e*x)^6,x)

[Out]

int(((b*x + c*x^2)^(1/2)*(A + B*x))/(d + e*x)^6, x)

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