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

Optimal. Leaf size=266 \[ -\frac {5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{6144 c^4}+\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2} \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{384 c^3}-\frac {5 b^6 \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{16384 c^{11/2}}+\frac {5 b^4 (b+2 c x) \sqrt {b x+c x^2} \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{16384 c^5}+\frac {9 e \left (b x+c x^2\right )^{7/2} (2 c d-b e)}{112 c^2}+\frac {e \left (b x+c x^2\right )^{7/2} (d+e x)}{8 c} \]

[Out]

-5/6144*b^2*(9*b^2*e^2-32*b*c*d*e+32*c^2*d^2)*(2*c*x+b)*(c*x^2+b*x)^(3/2)/c^4+1/384*(9*b^2*e^2-32*b*c*d*e+32*c
^2*d^2)*(2*c*x+b)*(c*x^2+b*x)^(5/2)/c^3+9/112*e*(-b*e+2*c*d)*(c*x^2+b*x)^(7/2)/c^2+1/8*e*(e*x+d)*(c*x^2+b*x)^(
7/2)/c-5/16384*b^6*(9*b^2*e^2-32*b*c*d*e+32*c^2*d^2)*arctanh(x*c^(1/2)/(c*x^2+b*x)^(1/2))/c^(11/2)+5/16384*b^4
*(9*b^2*e^2-32*b*c*d*e+32*c^2*d^2)*(2*c*x+b)*(c*x^2+b*x)^(1/2)/c^5

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Rubi [A]  time = 0.22, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {742, 640, 612, 620, 206} \[ \frac {5 b^4 (b+2 c x) \sqrt {b x+c x^2} \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{16384 c^5}-\frac {5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{6144 c^4}+\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2} \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{384 c^3}-\frac {5 b^6 \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{16384 c^{11/2}}+\frac {9 e \left (b x+c x^2\right )^{7/2} (2 c d-b e)}{112 c^2}+\frac {e \left (b x+c x^2\right )^{7/2} (d+e x)}{8 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*b^4*(32*c^2*d^2 - 32*b*c*d*e + 9*b^2*e^2)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(16384*c^5) - (5*b^2*(32*c^2*d^2 -
32*b*c*d*e + 9*b^2*e^2)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(6144*c^4) + ((32*c^2*d^2 - 32*b*c*d*e + 9*b^2*e^2)*
(b + 2*c*x)*(b*x + c*x^2)^(5/2))/(384*c^3) + (9*e*(2*c*d - b*e)*(b*x + c*x^2)^(7/2))/(112*c^2) + (e*(d + e*x)*
(b*x + c*x^2)^(7/2))/(8*c) - (5*b^6*(32*c^2*d^2 - 32*b*c*d*e + 9*b^2*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2
]])/(16384*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 620

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

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 (b x+c x^2\right )^{5/2} \, dx &=\frac {e (d+e x) \left (b x+c x^2\right )^{7/2}}{8 c}+\frac {\int \left (\frac {1}{2} d (16 c d-7 b e)+\frac {9}{2} e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{5/2} \, dx}{8 c}\\ &=\frac {9 e (2 c d-b e) \left (b x+c x^2\right )^{7/2}}{112 c^2}+\frac {e (d+e x) \left (b x+c x^2\right )^{7/2}}{8 c}+\frac {\left (c d (16 c d-7 b e)-\frac {9}{2} b e (2 c d-b e)\right ) \int \left (b x+c x^2\right )^{5/2} \, dx}{16 c^2}\\ &=\frac {\left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}+\frac {9 e (2 c d-b e) \left (b x+c x^2\right )^{7/2}}{112 c^2}+\frac {e (d+e x) \left (b x+c x^2\right )^{7/2}}{8 c}-\frac {\left (5 b^2 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right )\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{768 c^3}\\ &=-\frac {5 b^2 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}+\frac {9 e (2 c d-b e) \left (b x+c x^2\right )^{7/2}}{112 c^2}+\frac {e (d+e x) \left (b x+c x^2\right )^{7/2}}{8 c}+\frac {\left (5 b^4 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right )\right ) \int \sqrt {b x+c x^2} \, dx}{4096 c^4}\\ &=\frac {5 b^4 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{16384 c^5}-\frac {5 b^2 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}+\frac {9 e (2 c d-b e) \left (b x+c x^2\right )^{7/2}}{112 c^2}+\frac {e (d+e x) \left (b x+c x^2\right )^{7/2}}{8 c}-\frac {\left (5 b^6 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{32768 c^5}\\ &=\frac {5 b^4 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{16384 c^5}-\frac {5 b^2 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}+\frac {9 e (2 c d-b e) \left (b x+c x^2\right )^{7/2}}{112 c^2}+\frac {e (d+e x) \left (b x+c x^2\right )^{7/2}}{8 c}-\frac {\left (5 b^6 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{16384 c^5}\\ &=\frac {5 b^4 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{16384 c^5}-\frac {5 b^2 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}+\frac {9 e (2 c d-b e) \left (b x+c x^2\right )^{7/2}}{112 c^2}+\frac {e (d+e x) \left (b x+c x^2\right )^{7/2}}{8 c}-\frac {5 b^6 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{16384 c^{11/2}}\\ \end {align*}

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Mathematica [A]  time = 0.56, size = 219, normalized size = 0.82 \[ \frac {(x (b+c x))^{5/2} \left (\frac {\left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right ) \left (\sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \left (15 b^5-10 b^4 c x+8 b^3 c^2 x^2+432 b^2 c^3 x^3+640 b c^4 x^4+256 c^5 x^5\right )-15 b^{11/2} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )\right )}{6144 c^{9/2} (b+c x)^2 \sqrt {\frac {c x}{b}+1}}+\frac {9 e x^{7/2} (b+c x) (2 c d-b e)}{14 c}+e x^{7/2} (b+c x) (d+e x)\right )}{8 c x^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((x*(b + c*x))^(5/2)*((9*e*(2*c*d - b*e)*x^(7/2)*(b + c*x))/(14*c) + e*x^(7/2)*(b + c*x)*(d + e*x) + ((32*c^2*
d^2 - 32*b*c*d*e + 9*b^2*e^2)*(Sqrt[c]*Sqrt[x]*Sqrt[1 + (c*x)/b]*(15*b^5 - 10*b^4*c*x + 8*b^3*c^2*x^2 + 432*b^
2*c^3*x^3 + 640*b*c^4*x^4 + 256*c^5*x^5) - 15*b^(11/2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]]))/(6144*c^(9/2)*(b +
c*x)^2*Sqrt[1 + (c*x)/b])))/(8*c*x^(5/2))

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fricas [A]  time = 1.00, size = 643, normalized size = 2.42 \[ \left [\frac {105 \, {\left (32 \, b^{6} c^{2} d^{2} - 32 \, b^{7} c d e + 9 \, b^{8} e^{2}\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (43008 \, c^{8} e^{2} x^{7} + 3360 \, b^{5} c^{3} d^{2} - 3360 \, b^{6} c^{2} d e + 945 \, b^{7} c e^{2} + 3072 \, {\left (32 \, c^{8} d e + 33 \, b c^{7} e^{2}\right )} x^{6} + 256 \, {\left (224 \, c^{8} d^{2} + 928 \, b c^{7} d e + 243 \, b^{2} c^{6} e^{2}\right )} x^{5} + 128 \, {\left (1120 \, b c^{7} d^{2} + 1184 \, b^{2} c^{6} d e + 3 \, b^{3} c^{5} e^{2}\right )} x^{4} + 48 \, {\left (2016 \, b^{2} c^{6} d^{2} + 32 \, b^{3} c^{5} d e - 9 \, b^{4} c^{4} e^{2}\right )} x^{3} + 56 \, {\left (32 \, b^{3} c^{5} d^{2} - 32 \, b^{4} c^{4} d e + 9 \, b^{5} c^{3} e^{2}\right )} x^{2} - 70 \, {\left (32 \, b^{4} c^{4} d^{2} - 32 \, b^{5} c^{3} d e + 9 \, b^{6} c^{2} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{688128 \, c^{6}}, \frac {105 \, {\left (32 \, b^{6} c^{2} d^{2} - 32 \, b^{7} c d e + 9 \, b^{8} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (43008 \, c^{8} e^{2} x^{7} + 3360 \, b^{5} c^{3} d^{2} - 3360 \, b^{6} c^{2} d e + 945 \, b^{7} c e^{2} + 3072 \, {\left (32 \, c^{8} d e + 33 \, b c^{7} e^{2}\right )} x^{6} + 256 \, {\left (224 \, c^{8} d^{2} + 928 \, b c^{7} d e + 243 \, b^{2} c^{6} e^{2}\right )} x^{5} + 128 \, {\left (1120 \, b c^{7} d^{2} + 1184 \, b^{2} c^{6} d e + 3 \, b^{3} c^{5} e^{2}\right )} x^{4} + 48 \, {\left (2016 \, b^{2} c^{6} d^{2} + 32 \, b^{3} c^{5} d e - 9 \, b^{4} c^{4} e^{2}\right )} x^{3} + 56 \, {\left (32 \, b^{3} c^{5} d^{2} - 32 \, b^{4} c^{4} d e + 9 \, b^{5} c^{3} e^{2}\right )} x^{2} - 70 \, {\left (32 \, b^{4} c^{4} d^{2} - 32 \, b^{5} c^{3} d e + 9 \, b^{6} c^{2} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{344064 \, c^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/688128*(105*(32*b^6*c^2*d^2 - 32*b^7*c*d*e + 9*b^8*e^2)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(c)
) + 2*(43008*c^8*e^2*x^7 + 3360*b^5*c^3*d^2 - 3360*b^6*c^2*d*e + 945*b^7*c*e^2 + 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*e^2)*x^5 + 128*(1120*b*c^7*d^2 + 1184*b^2*c^6*d*e + 3
*b^3*c^5*e^2)*x^4 + 48*(2016*b^2*c^6*d^2 + 32*b^3*c^5*d*e - 9*b^4*c^4*e^2)*x^3 + 56*(32*b^3*c^5*d^2 - 32*b^4*c
^4*d*e + 9*b^5*c^3*e^2)*x^2 - 70*(32*b^4*c^4*d^2 - 32*b^5*c^3*d*e + 9*b^6*c^2*e^2)*x)*sqrt(c*x^2 + b*x))/c^6,
1/344064*(105*(32*b^6*c^2*d^2 - 32*b^7*c*d*e + 9*b^8*e^2)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) +
(43008*c^8*e^2*x^7 + 3360*b^5*c^3*d^2 - 3360*b^6*c^2*d*e + 945*b^7*c*e^2 + 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*e^2)*x^5 + 128*(1120*b*c^7*d^2 + 1184*b^2*c^6*d*e + 3*b^3*c
^5*e^2)*x^4 + 48*(2016*b^2*c^6*d^2 + 32*b^3*c^5*d*e - 9*b^4*c^4*e^2)*x^3 + 56*(32*b^3*c^5*d^2 - 32*b^4*c^4*d*e
+ 9*b^5*c^3*e^2)*x^2 - 70*(32*b^4*c^4*d^2 - 32*b^5*c^3*d*e + 9*b^6*c^2*e^2)*x)*sqrt(c*x^2 + b*x))/c^6]

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giac [A]  time = 0.29, size = 350, normalized size = 1.32 \[ \frac {1}{344064} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (12 \, {\left (14 \, c^{2} x e^{2} + \frac {32 \, c^{9} d e + 33 \, b c^{8} e^{2}}{c^{7}}\right )} x + \frac {224 \, c^{9} d^{2} + 928 \, b c^{8} d e + 243 \, b^{2} c^{7} e^{2}}{c^{7}}\right )} x + \frac {1120 \, b c^{8} d^{2} + 1184 \, b^{2} c^{7} d e + 3 \, b^{3} c^{6} e^{2}}{c^{7}}\right )} x + \frac {3 \, {\left (2016 \, b^{2} c^{7} d^{2} + 32 \, b^{3} c^{6} d e - 9 \, b^{4} c^{5} e^{2}\right )}}{c^{7}}\right )} x + \frac {7 \, {\left (32 \, b^{3} c^{6} d^{2} - 32 \, b^{4} c^{5} d e + 9 \, b^{5} c^{4} e^{2}\right )}}{c^{7}}\right )} x - \frac {35 \, {\left (32 \, b^{4} c^{5} d^{2} - 32 \, b^{5} c^{4} d e + 9 \, b^{6} c^{3} e^{2}\right )}}{c^{7}}\right )} x + \frac {105 \, {\left (32 \, b^{5} c^{4} d^{2} - 32 \, b^{6} c^{3} d e + 9 \, b^{7} c^{2} e^{2}\right )}}{c^{7}}\right )} + \frac {5 \, {\left (32 \, b^{6} c^{2} d^{2} - 32 \, b^{7} c d e + 9 \, b^{8} e^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{32768 \, c^{\frac {11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/344064*sqrt(c*x^2 + b*x)*(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)/c^7)*x + (1120*b*c^8*d^2 + 1184*b^2*c^7*d*e + 3*b^3*c^6*e^2)/c^7)*x + 3*(
2016*b^2*c^7*d^2 + 32*b^3*c^6*d*e - 9*b^4*c^5*e^2)/c^7)*x + 7*(32*b^3*c^6*d^2 - 32*b^4*c^5*d*e + 9*b^5*c^4*e^2
)/c^7)*x - 35*(32*b^4*c^5*d^2 - 32*b^5*c^4*d*e + 9*b^6*c^3*e^2)/c^7)*x + 105*(32*b^5*c^4*d^2 - 32*b^6*c^3*d*e
+ 9*b^7*c^2*e^2)/c^7) + 5/32768*(32*b^6*c^2*d^2 - 32*b^7*c*d*e + 9*b^8*e^2)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2
+ b*x))*sqrt(c) - b))/c^(11/2)

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maple [B]  time = 0.07, size = 553, normalized size = 2.08 \[ -\frac {45 b^{8} e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{32768 c^{\frac {11}{2}}}+\frac {5 b^{7} d e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{1024 c^{\frac {9}{2}}}-\frac {5 b^{6} d^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{1024 c^{\frac {7}{2}}}+\frac {45 \sqrt {c \,x^{2}+b x}\, b^{6} e^{2} x}{8192 c^{4}}-\frac {5 \sqrt {c \,x^{2}+b x}\, b^{5} d e x}{256 c^{3}}+\frac {5 \sqrt {c \,x^{2}+b x}\, b^{4} d^{2} x}{256 c^{2}}+\frac {45 \sqrt {c \,x^{2}+b x}\, b^{7} e^{2}}{16384 c^{5}}-\frac {5 \sqrt {c \,x^{2}+b x}\, b^{6} d e}{512 c^{4}}+\frac {5 \sqrt {c \,x^{2}+b x}\, b^{5} d^{2}}{512 c^{3}}-\frac {15 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{4} e^{2} x}{1024 c^{3}}+\frac {5 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{3} d e x}{96 c^{2}}-\frac {5 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{2} d^{2} x}{96 c}-\frac {15 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{5} e^{2}}{2048 c^{4}}+\frac {5 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{4} d e}{192 c^{3}}-\frac {5 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{3} d^{2}}{192 c^{2}}+\frac {3 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} b^{2} e^{2} x}{64 c^{2}}-\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} b d e x}{6 c}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} d^{2} x}{6}+\frac {3 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} b^{3} e^{2}}{128 c^{3}}-\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} b^{2} d e}{12 c^{2}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} b \,d^{2}}{12 c}+\frac {\left (c \,x^{2}+b x \right )^{\frac {7}{2}} e^{2} x}{8 c}-\frac {9 \left (c \,x^{2}+b x \right )^{\frac {7}{2}} b \,e^{2}}{112 c^{2}}+\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}} d e}{7 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*e^2*x*(c*x^2+b*x)^(7/2)/c-9/112*e^2*b/c^2*(c*x^2+b*x)^(7/2)+3/64*e^2*b^2/c^2*x*(c*x^2+b*x)^(5/2)+3/128*e^2
*b^3/c^3*(c*x^2+b*x)^(5/2)-15/1024*e^2*b^4/c^3*(c*x^2+b*x)^(3/2)*x-15/2048*e^2*b^5/c^4*(c*x^2+b*x)^(3/2)+45/81
92*e^2*b^6/c^4*(c*x^2+b*x)^(1/2)*x+45/16384*e^2*b^7/c^5*(c*x^2+b*x)^(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)^(1/2))+2/7*d*e*(c*x^2+b*x)^(7/2)/c-1/6*d*e*b/c*x*(c*x^2+b*x)^(5/2)-1/12*d*e*b^2/c^2*(
c*x^2+b*x)^(5/2)+5/96*d*e*b^3/c^2*(c*x^2+b*x)^(3/2)*x+5/192*d*e*b^4/c^3*(c*x^2+b*x)^(3/2)-5/256*d*e*b^5/c^3*(c
*x^2+b*x)^(1/2)*x-5/512*d*e*b^6/c^4*(c*x^2+b*x)^(1/2)+5/1024*d*e*b^7/c^(9/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x
)^(1/2))+1/6*d^2*x*(c*x^2+b*x)^(5/2)+1/12*d^2/c*(c*x^2+b*x)^(5/2)*b-5/96*d^2*b^2/c*(c*x^2+b*x)^(3/2)*x-5/192*d
^2*b^3/c^2*(c*x^2+b*x)^(3/2)+5/256*d^2*b^4/c^2*(c*x^2+b*x)^(1/2)*x+5/512*d^2*b^5/c^3*(c*x^2+b*x)^(1/2)-5/1024*
d^2*b^6/c^(7/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2))

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maxima [B]  time = 1.50, size = 549, normalized size = 2.06 \[ \frac {1}{6} \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} d^{2} x + \frac {5 \, \sqrt {c x^{2} + b x} b^{4} d^{2} x}{256 \, c^{2}} - \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2} d^{2} x}{96 \, c} - \frac {5 \, \sqrt {c x^{2} + b x} b^{5} d e x}{256 \, c^{3}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3} d e x}{96 \, c^{2}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} b d e x}{6 \, c} + \frac {45 \, \sqrt {c x^{2} + b x} b^{6} e^{2} x}{8192 \, c^{4}} - \frac {15 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{4} e^{2} x}{1024 \, c^{3}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{2} e^{2} x}{64 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {7}{2}} e^{2} x}{8 \, c} - \frac {5 \, b^{6} d^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{1024 \, c^{\frac {7}{2}}} + \frac {5 \, b^{7} d e \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{1024 \, c^{\frac {9}{2}}} - \frac {45 \, b^{8} e^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{32768 \, c^{\frac {11}{2}}} + \frac {5 \, \sqrt {c x^{2} + b x} b^{5} d^{2}}{512 \, c^{3}} - \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3} d^{2}}{192 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} b d^{2}}{12 \, c} - \frac {5 \, \sqrt {c x^{2} + b x} b^{6} d e}{512 \, c^{4}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{4} d e}{192 \, c^{3}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{2} d e}{12 \, c^{2}} + \frac {2 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} d e}{7 \, c} + \frac {45 \, \sqrt {c x^{2} + b x} b^{7} e^{2}}{16384 \, c^{5}} - \frac {15 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{5} e^{2}}{2048 \, c^{4}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{3} e^{2}}{128 \, c^{3}} - \frac {9 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} b e^{2}}{112 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(c*x^2 + b*x)^(5/2)*d^2*x + 5/256*sqrt(c*x^2 + b*x)*b^4*d^2*x/c^2 - 5/96*(c*x^2 + b*x)^(3/2)*b^2*d^2*x/c -
5/256*sqrt(c*x^2 + b*x)*b^5*d*e*x/c^3 + 5/96*(c*x^2 + b*x)^(3/2)*b^3*d*e*x/c^2 - 1/6*(c*x^2 + b*x)^(5/2)*b*d*
e*x/c + 45/8192*sqrt(c*x^2 + b*x)*b^6*e^2*x/c^4 - 15/1024*(c*x^2 + b*x)^(3/2)*b^4*e^2*x/c^3 + 3/64*(c*x^2 + b*
x)^(5/2)*b^2*e^2*x/c^2 + 1/8*(c*x^2 + b*x)^(7/2)*e^2*x/c - 5/1024*b^6*d^2*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*
sqrt(c))/c^(7/2) + 5/1024*b^7*d*e*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(9/2) - 45/32768*b^8*e^2*log(
2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(11/2) + 5/512*sqrt(c*x^2 + b*x)*b^5*d^2/c^3 - 5/192*(c*x^2 + b*x)^
(3/2)*b^3*d^2/c^2 + 1/12*(c*x^2 + b*x)^(5/2)*b*d^2/c - 5/512*sqrt(c*x^2 + b*x)*b^6*d*e/c^4 + 5/192*(c*x^2 + b*
x)^(3/2)*b^4*d*e/c^3 - 1/12*(c*x^2 + b*x)^(5/2)*b^2*d*e/c^2 + 2/7*(c*x^2 + b*x)^(7/2)*d*e/c + 45/16384*sqrt(c*
x^2 + b*x)*b^7*e^2/c^5 - 15/2048*(c*x^2 + b*x)^(3/2)*b^5*e^2/c^4 + 3/128*(c*x^2 + b*x)^(5/2)*b^3*e^2/c^3 - 9/1
12*(c*x^2 + b*x)^(7/2)*b*e^2/c^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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