3.21.25 \(\int (d+e x)^2 (a+b x+c x^2)^{5/2} \, dx\)

Optimal. Leaf size=323 \[ -\frac {5 \left (b^2-4 a c\right )^3 \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{11/2}}+\frac {5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{16384 c^5}-\frac {5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{6144 c^4}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{384 c^3}+\frac {9 e \left (a+b x+c x^2\right )^{7/2} (2 c d-b e)}{112 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c} \]

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Rubi [A]  time = 0.44, antiderivative size = 323, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {742, 640, 612, 621, 206} \begin {gather*} \frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{384 c^3}-\frac {5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{6144 c^4}+\frac {5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2} \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{16384 c^5}-\frac {5 \left (b^2-4 a c\right )^3 \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{11/2}}+\frac {9 e \left (a+b x+c x^2\right )^{7/2} (2 c d-b e)}{112 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2*(a + b*x + c*x^2)^(5/2),x]

[Out]

(5*(b^2 - 4*a*c)^2*(32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(16384*c^
5) - (5*(b^2 - 4*a*c)*(32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(614
4*c^4) + ((32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(384*c^3) + (9*e
*(2*c*d - b*e)*(a + b*x + c*x^2)^(7/2))/(112*c^2) + (e*(d + e*x)*(a + b*x + c*x^2)^(7/2))/(8*c) - (5*(b^2 - 4*
a*c)^3*(32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/
(32768*c^(11/2))

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 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 621

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

Rule 640

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

Rule 742

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))/(c*(m + 2*p + 1)), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2
*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*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]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,
p, x]

Rubi steps

\begin {align*} \int (d+e x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac {e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}+\frac {\int \left (\frac {1}{2} \left (16 c d^2-2 e \left (\frac {7 b d}{2}+a e\right )\right )+\frac {9}{2} e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2} \, dx}{8 c}\\ &=\frac {9 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}+\frac {\left (-\frac {9}{2} b e (2 c d-b e)+c \left (16 c d^2-2 e \left (\frac {7 b d}{2}+a e\right )\right )\right ) \int \left (a+b x+c x^2\right )^{5/2} \, dx}{16 c^2}\\ &=\frac {\left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}+\frac {9 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}-\frac {\left (5 \left (b^2-4 a c\right ) \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{768 c^3}\\ &=-\frac {5 \left (b^2-4 a c\right ) \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}+\frac {9 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}+\frac {\left (5 \left (b^2-4 a c\right )^2 \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{4096 c^4}\\ &=\frac {5 \left (b^2-4 a c\right )^2 \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^5}-\frac {5 \left (b^2-4 a c\right ) \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}+\frac {9 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}-\frac {\left (5 \left (b^2-4 a c\right )^3 \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{32768 c^5}\\ &=\frac {5 \left (b^2-4 a c\right )^2 \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^5}-\frac {5 \left (b^2-4 a c\right ) \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}+\frac {9 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}-\frac {\left (5 \left (b^2-4 a c\right )^3 \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{16384 c^5}\\ &=\frac {5 \left (b^2-4 a c\right )^2 \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^5}-\frac {5 \left (b^2-4 a c\right ) \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}+\frac {9 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}-\frac {5 \left (b^2-4 a c\right )^3 \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{11/2}}\\ \end {align*}

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Mathematica [A]  time = 0.59, size = 238, normalized size = 0.74 \begin {gather*} \frac {-\frac {\left (2 c e (a e+8 b d)-\frac {9 b^2 e^2}{2}-16 c^2 d^2\right ) \left (256 c^{5/2} (b+2 c x) (a+x (b+c x))^{5/2}-5 \left (b^2-4 a c\right ) \left (16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left (b^2-4 a c\right ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )\right )\right )}{6144 c^{9/2}}+\frac {9 e (a+x (b+c x))^{7/2} (2 c d-b e)}{14 c}+e (d+e x) (a+x (b+c x))^{7/2}}{8 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^2*(a + b*x + c*x^2)^(5/2),x]

[Out]

((9*e*(2*c*d - b*e)*(a + x*(b + c*x))^(7/2))/(14*c) + e*(d + e*x)*(a + x*(b + c*x))^(7/2) - ((-16*c^2*d^2 - (9
*b^2*e^2)/2 + 2*c*e*(8*b*d + a*e))*(256*c^(5/2)*(b + 2*c*x)*(a + x*(b + c*x))^(5/2) - 5*(b^2 - 4*a*c)*(16*c^(3
/2)*(b + 2*c*x)*(a + x*(b + c*x))^(3/2) - 3*(b^2 - 4*a*c)*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] - (b^2
- 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))))/(6144*c^(9/2)))/(8*c)

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IntegrateAlgebraic [B]  time = 3.66, size = 801, normalized size = 2.48 \begin {gather*} \frac {\sqrt {c x^2+b x+a} \left (945 e^2 b^7-3360 c d e b^6-630 c e^2 x b^6+3360 c^2 d^2 b^5-10500 a c e^2 b^5+504 c^2 e^2 x^2 b^5+2240 c^2 d e x b^5-432 c^3 e^2 x^3 b^4-1792 c^3 d e x^2 b^4+35840 a c^2 d e b^4-2240 c^3 d^2 x b^4+6328 a c^2 e^2 x b^4+384 c^4 e^2 x^4 b^3+1536 c^4 d e x^3 b^3-35840 a c^3 d^2 b^3+37744 a^2 c^2 e^2 b^3+1792 c^4 d^2 x^2 b^3-4544 a c^3 e^2 x^2 b^3-21504 a c^3 d e x b^3+62208 c^5 e^2 x^5 b^2+151552 c^5 d e x^4 b^2+96768 c^5 d^2 x^3 b^2+3456 a c^4 e^2 x^3 b^2+15360 a c^4 d e x^2 b^2-118272 a^2 c^3 d e b^2+21504 a c^4 d^2 x b^2-19104 a^2 c^3 e^2 x b^2+101376 c^6 e^2 x^6 b+237568 c^6 d e x^5 b+143360 c^6 d^2 x^4 b+157184 a c^5 e^2 x^4 b+403456 a c^5 d e x^3 b+118272 a^2 c^4 d^2 b-42432 a^3 c^3 e^2 b+279552 a c^5 d^2 x^2 b+11136 a^2 c^4 e^2 x^2 b+58368 a^2 c^4 d e x b+43008 c^7 e^2 x^7+98304 c^7 d e x^6+57344 c^7 d^2 x^5+121856 a c^6 e^2 x^5+294912 a c^6 d e x^4+186368 a c^6 d^2 x^3+105728 a^2 c^5 e^2 x^3+294912 a^2 c^5 d e x^2+98304 a^3 c^4 d e+236544 a^2 c^5 d^2 x+13440 a^3 c^4 e^2 x\right )}{344064 c^5}+\frac {5 \left (9 e^2 b^8-32 c d e b^7+32 c^2 d^2 b^6-112 a c e^2 b^6+384 a c^2 d e b^5-384 a c^3 d^2 b^4+480 a^2 c^2 e^2 b^4-1536 a^2 c^3 d e b^3+1536 a^2 c^4 d^2 b^2-768 a^3 c^3 e^2 b^2+2048 a^3 c^4 d e b-2048 a^3 c^5 d^2+256 a^4 c^4 e^2\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {c x^2+b x+a}\right )}{32768 c^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)^2*(a + b*x + c*x^2)^(5/2),x]

[Out]

(Sqrt[a + b*x + c*x^2]*(3360*b^5*c^2*d^2 - 35840*a*b^3*c^3*d^2 + 118272*a^2*b*c^4*d^2 - 3360*b^6*c*d*e + 35840
*a*b^4*c^2*d*e - 118272*a^2*b^2*c^3*d*e + 98304*a^3*c^4*d*e + 945*b^7*e^2 - 10500*a*b^5*c*e^2 + 37744*a^2*b^3*
c^2*e^2 - 42432*a^3*b*c^3*e^2 - 2240*b^4*c^3*d^2*x + 21504*a*b^2*c^4*d^2*x + 236544*a^2*c^5*d^2*x + 2240*b^5*c
^2*d*e*x - 21504*a*b^3*c^3*d*e*x + 58368*a^2*b*c^4*d*e*x - 630*b^6*c*e^2*x + 6328*a*b^4*c^2*e^2*x - 19104*a^2*
b^2*c^3*e^2*x + 13440*a^3*c^4*e^2*x + 1792*b^3*c^4*d^2*x^2 + 279552*a*b*c^5*d^2*x^2 - 1792*b^4*c^3*d*e*x^2 + 1
5360*a*b^2*c^4*d*e*x^2 + 294912*a^2*c^5*d*e*x^2 + 504*b^5*c^2*e^2*x^2 - 4544*a*b^3*c^3*e^2*x^2 + 11136*a^2*b*c
^4*e^2*x^2 + 96768*b^2*c^5*d^2*x^3 + 186368*a*c^6*d^2*x^3 + 1536*b^3*c^4*d*e*x^3 + 403456*a*b*c^5*d*e*x^3 - 43
2*b^4*c^3*e^2*x^3 + 3456*a*b^2*c^4*e^2*x^3 + 105728*a^2*c^5*e^2*x^3 + 143360*b*c^6*d^2*x^4 + 151552*b^2*c^5*d*
e*x^4 + 294912*a*c^6*d*e*x^4 + 384*b^3*c^4*e^2*x^4 + 157184*a*b*c^5*e^2*x^4 + 57344*c^7*d^2*x^5 + 237568*b*c^6
*d*e*x^5 + 62208*b^2*c^5*e^2*x^5 + 121856*a*c^6*e^2*x^5 + 98304*c^7*d*e*x^6 + 101376*b*c^6*e^2*x^6 + 43008*c^7
*e^2*x^7))/(344064*c^5) + (5*(32*b^6*c^2*d^2 - 384*a*b^4*c^3*d^2 + 1536*a^2*b^2*c^4*d^2 - 2048*a^3*c^5*d^2 - 3
2*b^7*c*d*e + 384*a*b^5*c^2*d*e - 1536*a^2*b^3*c^3*d*e + 2048*a^3*b*c^4*d*e + 9*b^8*e^2 - 112*a*b^6*c*e^2 + 48
0*a^2*b^4*c^2*e^2 - 768*a^3*b^2*c^3*e^2 + 256*a^4*c^4*e^2)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + b*x + c*x^2]])/(
32768*c^(11/2))

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fricas [B]  time = 0.64, size = 1425, normalized size = 4.41

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[1/1376256*(105*(32*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^2 - 32*(b^7*c - 12*a*b^5*c^2 + 48
*a^2*b^3*c^3 - 64*a^3*b*c^4)*d*e + (9*b^8 - 112*a*b^6*c + 480*a^2*b^4*c^2 - 768*a^3*b^2*c^3 + 256*a^4*c^4)*e^2
)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 + 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) + 4*(43008*c^8
*e^2*x^7 + 3072*(32*c^8*d*e + 33*b*c^7*e^2)*x^6 + 256*(224*c^8*d^2 + 928*b*c^7*d*e + (243*b^2*c^6 + 476*a*c^7)
*e^2)*x^5 + 128*(1120*b*c^7*d^2 + 32*(37*b^2*c^6 + 72*a*c^7)*d*e + (3*b^3*c^5 + 1228*a*b*c^6)*e^2)*x^4 + 16*(2
24*(27*b^2*c^6 + 52*a*c^7)*d^2 + 32*(3*b^3*c^5 + 788*a*b*c^6)*d*e - (27*b^4*c^4 - 216*a*b^2*c^5 - 6608*a^2*c^6
)*e^2)*x^3 + 224*(15*b^5*c^3 - 160*a*b^3*c^4 + 528*a^2*b*c^5)*d^2 - 32*(105*b^6*c^2 - 1120*a*b^4*c^3 + 3696*a^
2*b^2*c^4 - 3072*a^3*c^5)*d*e + (945*b^7*c - 10500*a*b^5*c^2 + 37744*a^2*b^3*c^3 - 42432*a^3*b*c^4)*e^2 + 8*(2
24*(b^3*c^5 + 156*a*b*c^6)*d^2 - 32*(7*b^4*c^4 - 60*a*b^2*c^5 - 1152*a^2*c^6)*d*e + (63*b^5*c^3 - 568*a*b^3*c^
4 + 1392*a^2*b*c^5)*e^2)*x^2 - 2*(224*(5*b^4*c^4 - 48*a*b^2*c^5 - 528*a^2*c^6)*d^2 - 32*(35*b^5*c^3 - 336*a*b^
3*c^4 + 912*a^2*b*c^5)*d*e + (315*b^6*c^2 - 3164*a*b^4*c^3 + 9552*a^2*b^2*c^4 - 6720*a^3*c^5)*e^2)*x)*sqrt(c*x
^2 + b*x + a))/c^6, 1/688128*(105*(32*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^2 - 32*(b^7*c -
12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d*e + (9*b^8 - 112*a*b^6*c + 480*a^2*b^4*c^2 - 768*a^3*b^2*c^3
+ 256*a^4*c^4)*e^2)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) +
2*(43008*c^8*e^2*x^7 + 3072*(32*c^8*d*e + 33*b*c^7*e^2)*x^6 + 256*(224*c^8*d^2 + 928*b*c^7*d*e + (243*b^2*c^6
+ 476*a*c^7)*e^2)*x^5 + 128*(1120*b*c^7*d^2 + 32*(37*b^2*c^6 + 72*a*c^7)*d*e + (3*b^3*c^5 + 1228*a*b*c^6)*e^2)
*x^4 + 16*(224*(27*b^2*c^6 + 52*a*c^7)*d^2 + 32*(3*b^3*c^5 + 788*a*b*c^6)*d*e - (27*b^4*c^4 - 216*a*b^2*c^5 -
6608*a^2*c^6)*e^2)*x^3 + 224*(15*b^5*c^3 - 160*a*b^3*c^4 + 528*a^2*b*c^5)*d^2 - 32*(105*b^6*c^2 - 1120*a*b^4*c
^3 + 3696*a^2*b^2*c^4 - 3072*a^3*c^5)*d*e + (945*b^7*c - 10500*a*b^5*c^2 + 37744*a^2*b^3*c^3 - 42432*a^3*b*c^4
)*e^2 + 8*(224*(b^3*c^5 + 156*a*b*c^6)*d^2 - 32*(7*b^4*c^4 - 60*a*b^2*c^5 - 1152*a^2*c^6)*d*e + (63*b^5*c^3 -
568*a*b^3*c^4 + 1392*a^2*b*c^5)*e^2)*x^2 - 2*(224*(5*b^4*c^4 - 48*a*b^2*c^5 - 528*a^2*c^6)*d^2 - 32*(35*b^5*c^
3 - 336*a*b^3*c^4 + 912*a^2*b*c^5)*d*e + (315*b^6*c^2 - 3164*a*b^4*c^3 + 9552*a^2*b^2*c^4 - 6720*a^3*c^5)*e^2)
*x)*sqrt(c*x^2 + b*x + a))/c^6]

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giac [B]  time = 0.31, size = 767, normalized size = 2.37 \begin {gather*} \frac {1}{344064} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (12 \, {\left (14 \, c^{2} x e^{2} + \frac {32 \, c^{9} d e + 33 \, b c^{8} e^{2}}{c^{7}}\right )} x + \frac {224 \, c^{9} d^{2} + 928 \, b c^{8} d e + 243 \, b^{2} c^{7} e^{2} + 476 \, a c^{8} e^{2}}{c^{7}}\right )} x + \frac {1120 \, b c^{8} d^{2} + 1184 \, b^{2} c^{7} d e + 2304 \, a c^{8} d e + 3 \, b^{3} c^{6} e^{2} + 1228 \, a b c^{7} e^{2}}{c^{7}}\right )} x + \frac {6048 \, b^{2} c^{7} d^{2} + 11648 \, a c^{8} d^{2} + 96 \, b^{3} c^{6} d e + 25216 \, a b c^{7} d e - 27 \, b^{4} c^{5} e^{2} + 216 \, a b^{2} c^{6} e^{2} + 6608 \, a^{2} c^{7} e^{2}}{c^{7}}\right )} x + \frac {224 \, b^{3} c^{6} d^{2} + 34944 \, a b c^{7} d^{2} - 224 \, b^{4} c^{5} d e + 1920 \, a b^{2} c^{6} d e + 36864 \, a^{2} c^{7} d e + 63 \, b^{5} c^{4} e^{2} - 568 \, a b^{3} c^{5} e^{2} + 1392 \, a^{2} b c^{6} e^{2}}{c^{7}}\right )} x - \frac {1120 \, b^{4} c^{5} d^{2} - 10752 \, a b^{2} c^{6} d^{2} - 118272 \, a^{2} c^{7} d^{2} - 1120 \, b^{5} c^{4} d e + 10752 \, a b^{3} c^{5} d e - 29184 \, a^{2} b c^{6} d e + 315 \, b^{6} c^{3} e^{2} - 3164 \, a b^{4} c^{4} e^{2} + 9552 \, a^{2} b^{2} c^{5} e^{2} - 6720 \, a^{3} c^{6} e^{2}}{c^{7}}\right )} x + \frac {3360 \, b^{5} c^{4} d^{2} - 35840 \, a b^{3} c^{5} d^{2} + 118272 \, a^{2} b c^{6} d^{2} - 3360 \, b^{6} c^{3} d e + 35840 \, a b^{4} c^{4} d e - 118272 \, a^{2} b^{2} c^{5} d e + 98304 \, a^{3} c^{6} d e + 945 \, b^{7} c^{2} e^{2} - 10500 \, a b^{5} c^{3} e^{2} + 37744 \, a^{2} b^{3} c^{4} e^{2} - 42432 \, a^{3} b c^{5} e^{2}}{c^{7}}\right )} + \frac {5 \, {\left (32 \, b^{6} c^{2} d^{2} - 384 \, a b^{4} c^{3} d^{2} + 1536 \, a^{2} b^{2} c^{4} d^{2} - 2048 \, a^{3} c^{5} d^{2} - 32 \, b^{7} c d e + 384 \, a b^{5} c^{2} d e - 1536 \, a^{2} b^{3} c^{3} d e + 2048 \, a^{3} b c^{4} d e + 9 \, b^{8} e^{2} - 112 \, a b^{6} c e^{2} + 480 \, a^{2} b^{4} c^{2} e^{2} - 768 \, a^{3} b^{2} c^{3} e^{2} + 256 \, a^{4} c^{4} e^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{32768 \, c^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

1/344064*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(12*(14*c^2*x*e^2 + (32*c^9*d*e + 33*b*c^8*e^2)/c^7)*x + (224*c^
9*d^2 + 928*b*c^8*d*e + 243*b^2*c^7*e^2 + 476*a*c^8*e^2)/c^7)*x + (1120*b*c^8*d^2 + 1184*b^2*c^7*d*e + 2304*a*
c^8*d*e + 3*b^3*c^6*e^2 + 1228*a*b*c^7*e^2)/c^7)*x + (6048*b^2*c^7*d^2 + 11648*a*c^8*d^2 + 96*b^3*c^6*d*e + 25
216*a*b*c^7*d*e - 27*b^4*c^5*e^2 + 216*a*b^2*c^6*e^2 + 6608*a^2*c^7*e^2)/c^7)*x + (224*b^3*c^6*d^2 + 34944*a*b
*c^7*d^2 - 224*b^4*c^5*d*e + 1920*a*b^2*c^6*d*e + 36864*a^2*c^7*d*e + 63*b^5*c^4*e^2 - 568*a*b^3*c^5*e^2 + 139
2*a^2*b*c^6*e^2)/c^7)*x - (1120*b^4*c^5*d^2 - 10752*a*b^2*c^6*d^2 - 118272*a^2*c^7*d^2 - 1120*b^5*c^4*d*e + 10
752*a*b^3*c^5*d*e - 29184*a^2*b*c^6*d*e + 315*b^6*c^3*e^2 - 3164*a*b^4*c^4*e^2 + 9552*a^2*b^2*c^5*e^2 - 6720*a
^3*c^6*e^2)/c^7)*x + (3360*b^5*c^4*d^2 - 35840*a*b^3*c^5*d^2 + 118272*a^2*b*c^6*d^2 - 3360*b^6*c^3*d*e + 35840
*a*b^4*c^4*d*e - 118272*a^2*b^2*c^5*d*e + 98304*a^3*c^6*d*e + 945*b^7*c^2*e^2 - 10500*a*b^5*c^3*e^2 + 37744*a^
2*b^3*c^4*e^2 - 42432*a^3*b*c^5*e^2)/c^7) + 5/32768*(32*b^6*c^2*d^2 - 384*a*b^4*c^3*d^2 + 1536*a^2*b^2*c^4*d^2
- 2048*a^3*c^5*d^2 - 32*b^7*c*d*e + 384*a*b^5*c^2*d*e - 1536*a^2*b^3*c^3*d*e + 2048*a^3*b*c^4*d*e + 9*b^8*e^2
- 112*a*b^6*c*e^2 + 480*a^2*b^4*c^2*e^2 - 768*a^3*b^2*c^3*e^2 + 256*a^4*c^4*e^2)*log(abs(-2*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))*sqrt(c) - b))/c^(11/2)

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maple [B]  time = 0.06, size = 1517, normalized size = 4.70

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2*(c*x^2+b*x+a)^(5/2),x)

[Out]

55/512*e^2*b^2/c^2*(c*x^2+b*x+a)^(1/2)*x*a^2-95/2048*e^2*b^4/c^3*(c*x^2+b*x+a)^(1/2)*x*a+25/384*e^2*b^2/c^2*(c
*x^2+b*x+a)^(3/2)*x*a-5/48*d*e*b^2/c^2*(c*x^2+b*x+a)^(3/2)*a-1/6*d*e*b/c*x*(c*x^2+b*x+a)^(5/2)+5/96*d*e*b^3/c^
2*(c*x^2+b*x+a)^(3/2)*x-5/16*d*e*b/c^(3/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3+15/64*d*e*b^3/c^(5/
2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2-15/256*d*e*b^5/c^(7/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)
^(1/2))*a-5/32*d^2/c*(c*x^2+b*x+a)^(1/2)*x*a*b^2-5/256*d*e*b^5/c^3*(c*x^2+b*x+a)^(1/2)*x-5/32*d*e*b^2/c^2*(c*x
^2+b*x+a)^(1/2)*a^2+5/64*d*e*b^4/c^3*(c*x^2+b*x+a)^(1/2)*a+5/32*d*e*b^3/c^2*(c*x^2+b*x+a)^(1/2)*x*a-5/128*e^2*
a^3/c*(c*x^2+b*x+a)^(1/2)*x-5/256*e^2*a^3/c^2*(c*x^2+b*x+a)^(1/2)*b-5/192*e^2*a^2/c*(c*x^2+b*x+a)^(3/2)*x-5/38
4*e^2*a^2/c^2*(c*x^2+b*x+a)^(3/2)*b-1/48*e^2*a/c*x*(c*x^2+b*x+a)^(5/2)-1/96*e^2*a/c^2*(c*x^2+b*x+a)^(5/2)*b+55
/1024*e^2*b^3/c^3*(c*x^2+b*x+a)^(1/2)*a^2+15/256*d^2/c^(5/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^4*a
-5/96*d^2/c*(c*x^2+b*x+a)^(3/2)*x*b^2+5/48*d^2/c*(c*x^2+b*x+a)^(3/2)*b*a+5/256*d^2/c^2*(c*x^2+b*x+a)^(1/2)*x*b
^4+5/32*d^2/c*(c*x^2+b*x+a)^(1/2)*b*a^2-5/64*d^2/c^2*(c*x^2+b*x+a)^(1/2)*b^3*a-15/64*d^2/c^(3/2)*ln((c*x+1/2*b
)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^2*a^2+2/7*d*e*(c*x^2+b*x+a)^(7/2)/c-9/112*e^2*b/c^2*(c*x^2+b*x+a)^(7/2)+1/12*
d^2/c*(c*x^2+b*x+a)^(5/2)*b+5/24*d^2*(c*x^2+b*x+a)^(3/2)*x*a-5/192*d^2/c^2*(c*x^2+b*x+a)^(3/2)*b^3+5/16*d^2*(c
*x^2+b*x+a)^(1/2)*x*a^2+5/512*d^2/c^3*(c*x^2+b*x+a)^(1/2)*b^5+5/16*d^2/c^(1/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b
*x+a)^(1/2))*a^3-5/1024*d^2/c^(7/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^6+45/16384*e^2*b^7/c^5*(c*x^
2+b*x+a)^(1/2)+1/8*e^2*x*(c*x^2+b*x+a)^(7/2)/c-15/2048*e^2*b^5/c^4*(c*x^2+b*x+a)^(3/2)-5/128*e^2*a^4/c^(3/2)*l
n((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-45/32768*e^2*b^8/c^(11/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2
))+3/128*e^2*b^3/c^3*(c*x^2+b*x+a)^(5/2)-5/24*d*e*b/c*(c*x^2+b*x+a)^(3/2)*x*a-5/16*d*e*b/c*(c*x^2+b*x+a)^(1/2)
*x*a^2-95/4096*e^2*b^5/c^4*(c*x^2+b*x+a)^(1/2)*a+3/64*e^2*b^2/c^2*x*(c*x^2+b*x+a)^(5/2)+5/1024*d*e*b^7/c^(9/2)
*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-1/12*d*e*b^2/c^2*(c*x^2+b*x+a)^(5/2)+45/8192*e^2*b^6/c^4*(c*x^2+b
*x+a)^(1/2)*x+25/768*e^2*b^3/c^3*(c*x^2+b*x+a)^(3/2)*a+15/128*e^2*b^2/c^(5/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*
x+a)^(1/2))*a^3-75/1024*e^2*b^4/c^(7/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2+35/2048*e^2*b^6/c^(9/2
)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-15/1024*e^2*b^4/c^3*(c*x^2+b*x+a)^(3/2)*x+5/192*d*e*b^4/c^3*(c
*x^2+b*x+a)^(3/2)-5/512*d*e*b^6/c^4*(c*x^2+b*x+a)^(1/2)+1/6*d^2*x*(c*x^2+b*x+a)^(5/2)

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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)^2*(c*x^2+b*x+a)^(5/2),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(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive, negative or zero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (d+e\,x\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^2*(a + b*x + c*x^2)^(5/2),x)

[Out]

int((d + e*x)^2*(a + b*x + c*x^2)^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d + e x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**2*(c*x**2+b*x+a)**(5/2),x)

[Out]

Integral((d + e*x)**2*(a + b*x + c*x**2)**(5/2), x)

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