3.24.53 \(\int \frac {(a+b x+c x^2)^{3/2}}{(d+e x)^7} \, dx\) [2353]

Optimal. Leaf size=409 \[ -\frac {\left (b^2-4 a c\right ) \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{512 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^2}+\frac {\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{192 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^4}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{6 \left (c d^2-b d e+a e^2\right ) (d+e x)^6}-\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^5}+\frac {\left (b^2-4 a c\right )^2 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{1024 \left (c d^2-b d e+a e^2\right )^{9/2}} \]

[Out]

1/192*(24*c^2*d^2+7*b^2*e^2-4*c*e*(a*e+6*b*d))*(b*d-2*a*e+(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(3/2)/(a*e^2-b*d*e+c*d
^2)^3/(e*x+d)^4-1/6*e*(c*x^2+b*x+a)^(5/2)/(a*e^2-b*d*e+c*d^2)/(e*x+d)^6-7/60*e*(-b*e+2*c*d)*(c*x^2+b*x+a)^(5/2
)/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^5+1/1024*(-4*a*c+b^2)^2*(24*c^2*d^2+7*b^2*e^2-4*c*e*(a*e+6*b*d))*arctanh(1/2*(
b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/(a*e^2-b*d*e+c*d^2)^(9/2)-1/512*(-4*a
*c+b^2)*(24*c^2*d^2+7*b^2*e^2-4*c*e*(a*e+6*b*d))*(b*d-2*a*e+(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(1/2)/(a*e^2-b*d*e+c
*d^2)^4/(e*x+d)^2

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Rubi [A]
time = 0.39, antiderivative size = 409, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {758, 820, 734, 738, 212} \begin {gather*} \frac {\left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{192 (d+e x)^4 \left (a e^2-b d e+c d^2\right )^3}-\frac {\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{512 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^4}+\frac {\left (b^2-4 a c\right )^2 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{1024 \left (a e^2-b d e+c d^2\right )^{9/2}}-\frac {7 e \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{60 (d+e x)^5 \left (a e^2-b d e+c d^2\right )^2}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2-b d e+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^(3/2)/(d + e*x)^7,x]

[Out]

-1/512*((b^2 - 4*a*c)*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*(b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a +
b*x + c*x^2])/((c*d^2 - b*d*e + a*e^2)^4*(d + e*x)^2) + ((24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*(b*d -
2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(192*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^4) - (e*(a + b*x +
c*x^2)^(5/2))/(6*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^6) - (7*e*(2*c*d - b*e)*(a + b*x + c*x^2)^(5/2))/(60*(c*d^
2 - b*d*e + a*e^2)^2*(d + e*x)^5) + ((b^2 - 4*a*c)^2*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*ArcTanh[(b
*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(1024*(c*d^2 - b*d*e + a
*e^2)^(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 758

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(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)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, x]*(a + b*x + c*x^2)^p, x], x]
/; FreeQ[{a, b, c, d, e, m, p}, 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] && NeQ[m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ
[p]) || ILtQ[Simplify[m + 2*p + 3], 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]

Rubi steps

\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx &=-\frac {e \left (a+b x+c x^2\right )^{5/2}}{6 \left (c d^2-b d e+a e^2\right ) (d+e x)^6}-\frac {\int \frac {\left (\frac {1}{2} (-12 c d+7 b e)+c e x\right ) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx}{6 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {e \left (a+b x+c x^2\right )^{5/2}}{6 \left (c d^2-b d e+a e^2\right ) (d+e x)^6}-\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^5}+\frac {\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx}{24 \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac {\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{192 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^4}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{6 \left (c d^2-b d e+a e^2\right ) (d+e x)^6}-\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^5}-\frac {\left (\left (b^2-4 a c\right ) \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right )\right ) \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3} \, dx}{128 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac {\left (b^2-4 a c\right ) \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{512 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^2}+\frac {\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{192 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^4}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{6 \left (c d^2-b d e+a e^2\right ) (d+e x)^6}-\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^5}+\frac {\left (\left (b^2-4 a c\right )^2 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{1024 \left (c d^2-b d e+a e^2\right )^4}\\ &=-\frac {\left (b^2-4 a c\right ) \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{512 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^2}+\frac {\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{192 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^4}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{6 \left (c d^2-b d e+a e^2\right ) (d+e x)^6}-\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^5}-\frac {\left (\left (b^2-4 a c\right )^2 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right )\right ) \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac {-b d+2 a e-(2 c d-b e) x}{\sqrt {a+b x+c x^2}}\right )}{512 \left (c d^2-b d e+a e^2\right )^4}\\ &=-\frac {\left (b^2-4 a c\right ) \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{512 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^2}+\frac {\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{192 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^4}-\frac {e \left (a+b x+c x^2\right )^{5/2}}{6 \left (c d^2-b d e+a e^2\right ) (d+e x)^6}-\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^5}+\frac {\left (b^2-4 a c\right )^2 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{1024 \left (c d^2-b d e+a e^2\right )^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 11.50, size = 349, normalized size = 0.85 \begin {gather*} -\frac {e (a+x (b+c x))^{5/2}}{6 \left (c d^2+e (-b d+a e)\right ) (d+e x)^6}-\frac {7 e (2 c d-b e) (a+x (b+c x))^{5/2}}{60 \left (c d^2+e (-b d+a e)\right )^2 (d+e x)^5}-\frac {\left (12 c^2 d^2+\frac {7 b^2 e^2}{2}-2 c e (6 b d+a e)\right ) \left (\frac {2 (-b d+2 a e-2 c d x+b e x) (a+x (b+c x))^{3/2}}{(d+e x)^4}+3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+x (b+c x)} (-2 a e+2 c d x+b (d-e x))}{4 \left (c d^2+e (-b d+a e)\right ) (d+e x)^2}+\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{8 \left (c d^2+e (-b d+a e)\right )^{3/2}}\right )\right )}{192 \left (c d^2+e (-b d+a e)\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x)^7,x]

[Out]

-1/6*(e*(a + x*(b + c*x))^(5/2))/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^6) - (7*e*(2*c*d - b*e)*(a + x*(b + c*x
))^(5/2))/(60*(c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x)^5) - ((12*c^2*d^2 + (7*b^2*e^2)/2 - 2*c*e*(6*b*d + a*e))*
((2*(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)*(a + x*(b + c*x))^(3/2))/(d + e*x)^4 + 3*(b^2 - 4*a*c)*((Sqrt[a + x*(b
+ c*x)]*(-2*a*e + 2*c*d*x + b*(d - e*x)))/(4*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + ((b^2 - 4*a*c)*ArcTanh[
(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(8*(c*d^2 + e*(-
(b*d) + a*e))^(3/2)))))/(192*(c*d^2 + e*(-(b*d) + a*e))^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(10396\) vs. \(2(383)=766\).
time = 0.83, size = 10397, normalized size = 25.42

method	result	size
default	\(\text {Expression too large to display}\)	\(10397\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(3/2)/(e*x+d)^7,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((c*x^2+b*x+a)^(3/2)/(e*x+d)^7,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^2-%e*b*d+%e^2*a>0)', see `
assume?` for

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Fricas [F(-1)] Timed out
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((c*x^2+b*x+a)^(3/2)/(e*x+d)^7,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{7}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**(3/2)/(e*x+d)**7,x)

[Out]

Integral((a + b*x + c*x**2)**(3/2)/(d + e*x)**7, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 14083 vs. \(2 (399) = 798\).
time = 29.96, size = 14083, normalized size = 34.43 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(3/2)/(e*x+d)^7,x, algorithm="giac")

[Out]

1/512*(24*b^4*c^2*d^2 - 192*a*b^2*c^3*d^2 + 384*a^2*c^4*d^2 - 24*b^5*c*d*e + 192*a*b^3*c^2*d*e - 384*a^2*b*c^3
*d*e + 7*b^6*e^2 - 60*a*b^4*c*e^2 + 144*a^2*b^2*c^2*e^2 - 64*a^3*c^3*e^2)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b
*x + a))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/((c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 + 4*a*c^3*
d^6*e^2 - 4*b^3*c*d^5*e^3 - 12*a*b*c^2*d^5*e^3 + b^4*d^4*e^4 + 12*a*b^2*c*d^4*e^4 + 6*a^2*c^2*d^4*e^4 - 4*a*b^
3*d^3*e^5 - 12*a^2*b*c*d^3*e^5 + 6*a^2*b^2*d^2*e^6 + 4*a^3*c*d^2*e^6 - 4*a^3*b*d*e^7 + a^4*e^8)*sqrt(-c*d^2 +
b*d*e - a*e^2)) + 1/7680*(24576*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*c^8*d^10*e + 8192*(sqrt(c)*x - sqrt(c*x^
2 + b*x + a))^6*c^(17/2)*d^11 + 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*c^(15/2)*d^9*e^2 + 40960*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))^6*b*c^(15/2)*d^10*e + 24576*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b*c^8*d^11 + 2048
0*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*c^7*d^8*e^3 - 24576*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*c^8*d^10*e
+ 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^2*c^(15/2)*d^11 - 46080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8
*b*c^(13/2)*d^8*e^3 - 101376*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^2*c^(13/2)*d^9*e^2 - 4096*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^6*a*c^(15/2)*d^9*e^2 - 46080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^3*c^(13/2)*d^10*e -
61440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b*c^(15/2)*d^10*e + 20480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*
b^3*c^7*d^11 - 81920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b*c^6*d^7*e^4 - 119808*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^7*b^2*c^6*d^8*e^3 + 110592*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*c^7*d^8*e^3 - 119808*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^5*b^3*c^6*d^9*e^2 + 110592*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b*c^7*d^9*e^2 - 43520
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^4*c^6*d^10*e - 61440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^2*c^7*
d^10*e + 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^4*c^(13/2)*d^11 - 122880*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^8*b^2*c^(11/2)*d^7*e^4 + 122880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*c^(13/2)*d^7*e^4 - 55296*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))^6*b^3*c^(11/2)*d^8*e^3 + 405504*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b*c^(13/2)
*d^8*e^3 - 51840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^4*c^(11/2)*d^9*e^2 + 276480*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^4*a*b^2*c^(13/2)*d^9*e^2 + 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*c^(15/2)*d^9*e^2 - 18432
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^5*c^(11/2)*d^10*e - 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^3
*c^(13/2)*d^10*e + 1536*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^5*c^6*d^11 + 122880*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^9*b^2*c^5*d^6*e^5 + 81920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*c^6*d^6*e^5 - 12288*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))^7*b^3*c^5*d^7*e^4 + 49152*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b*c^6*d^7*e^4 + 41472*(s
qrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^4*c^5*d^8*e^3 + 414720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^2*c^6*d
^8*e^3 - 110592*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*c^7*d^8*e^3 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^3*b^5*c^5*d^9*e^2 + 266240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^3*c^6*d^9*e^2 + 61440*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))^3*a^2*b*c^7*d^9*e^2 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^6*c^5*d^10*e - 7680*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))*a*b^4*c^6*d^10*e + 128*b^6*c^(11/2)*d^11 + 337920*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^8*b^3*c^(9/2)*d^6*e^5 - 61440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b*c^(11/2)*d^6*e^5 + 100800*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^6*b^4*c^(9/2)*d^7*e^4 - 450048*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b^2*c^(1
1/2)*d^7*e^4 - 549888*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*c^(13/2)*d^7*e^4 + 60480*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^4*b^5*c^(9/2)*d^8*e^3 + 69120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^3*c^(11/2)*d^8*e^3 - 41
4720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b*c^(13/2)*d^8*e^3 + 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2
*b^6*c^(9/2)*d^9*e^2 + 126720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^4*c^(11/2)*d^9*e^2 + 46080*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^2*a^2*b^2*c^(13/2)*d^9*e^2 - 320*b^7*c^(9/2)*d^10*e - 768*a*b^5*c^(11/2)*d^10*e - 819
20*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^3*c^4*d^5*e^6 - 245760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b*c^
5*d^5*e^6 + 336960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^4*c^4*d^6*e^5 + 93696*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^7*a*b^2*c^5*d^6*e^5 - 605184*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^2*c^6*d^6*e^5 + 87360*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^5*b^5*c^4*d^7*e^4 - 600576*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^3*c^5*d^7*e^4 - 1
207296*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b*c^6*d^7*e^4 + 33920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b
^6*c^4*d^8*e^3 - 130560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^4*c^5*d^8*e^3 - 537600*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^3*a^2*b^2*c^6*d^8*e^3 - 20480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*c^7*d^8*e^3 + 1152*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))*b^7*c^4*d^9*e^2...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (d+e\,x\right )}^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^(3/2)/(d + e*x)^7,x)

[Out]

int((a + b*x + c*x^2)^(3/2)/(d + e*x)^7, x)

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