3.380 \(\int (1+4 x-7 x^2)^3 (2+5 x+x^2) (3+2 x+5 x^2)^{3/2} \, dx\)

Optimal. Leaf size=231 \[ \frac {2173004363 \left (5 x^2+2 x+3\right )^{5/2} x^2}{173250000}+\frac {837379699 \left (5 x^2+2 x+3\right )^{5/2} x}{72187500}-\frac {6133820867 \left (5 x^2+2 x+3\right )^{5/2}}{1203125000}-\frac {22840599 (5 x+1) \left (5 x^2+2 x+3\right )^{3/2}}{62500000}-\frac {479652579 (5 x+1) \sqrt {5 x^2+2 x+3}}{312500000}-\frac {343}{60} \left (5 x^2+2 x+3\right )^{5/2} x^7-\frac {61103 \left (5 x^2+2 x+3\right )^{5/2} x^6}{3300}+\frac {1031177 \left (5 x^2+2 x+3\right )^{5/2} x^5}{20625}-\frac {796559 \left (5 x^2+2 x+3\right )^{5/2} x^4}{123750}-\frac {190236913 \left (5 x^2+2 x+3\right )^{5/2} x^3}{4950000}-\frac {3357568053 \sinh ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{156250000 \sqrt {5}} \]

[Out]

-22840599/62500000*(1+5*x)*(5*x^2+2*x+3)^(3/2)-6133820867/1203125000*(5*x^2+2*x+3)^(5/2)[phone]/72187500*x*
(5*x^2+2*x+3)^(5/2)[phone]/173250000*x^2*(5*x^2+2*x+3)^(5/2)-190236913/4950000*x^3*(5*x^2+2*x+3)^(5/2)-796
559/123750*x^4*(5*x^2+2*x+3)^(5/2)+1031177/20625*x^5*(5*x^2+2*x+3)^(5/2)-61103/3300*x^6*(5*x^2+2*x+3)^(5/2)-34
3/60*x^7*(5*x^2+2*x+3)^(5/2)-3357568053/781250000*arcsinh(1/14*(1+5*x)*14^(1/2))*5^(1/2)-479652579/312500000*(
1+5*x)*(5*x^2+2*x+3)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.36, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1661, 640, 612, 619, 215} \[ -\frac {343}{60} \left (5 x^2+2 x+3\right )^{5/2} x^7-\frac {61103 \left (5 x^2+2 x+3\right )^{5/2} x^6}{3300}+\frac {1031177 \left (5 x^2+2 x+3\right )^{5/2} x^5}{20625}-\frac {796559 \left (5 x^2+2 x+3\right )^{5/2} x^4}{123750}-\frac {190236913 \left (5 x^2+2 x+3\right )^{5/2} x^3}{4950000}+\frac {2173004363 \left (5 x^2+2 x+3\right )^{5/2} x^2}{173250000}+\frac {837379699 \left (5 x^2+2 x+3\right )^{5/2} x}{72187500}-\frac {6133820867 \left (5 x^2+2 x+3\right )^{5/2}}{1203125000}-\frac {22840599 (5 x+1) \left (5 x^2+2 x+3\right )^{3/2}}{62500000}-\frac {479652579 (5 x+1) \sqrt {5 x^2+2 x+3}}{312500000}-\frac {3357568053 \sinh ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )}{156250000 \sqrt {5}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + 4*x - 7*x^2)^3*(2 + 5*x + x^2)*(3 + 2*x + 5*x^2)^(3/2),x]

[Out]

(-479652579*(1 + 5*x)*Sqrt[3 + 2*x + 5*x^2])/312500000 - (22840599*(1 + 5*x)*(3 + 2*x + 5*x^2)^(3/2))/62500000
- (6133820867*(3 + 2*x + 5*x^2)^(5/2))/1203125000 + (837379699*x*(3 + 2*x + 5*x^2)^(5/2))/72187500 + (2173004
363*x^2*(3 + 2*x + 5*x^2)^(5/2))/173250000 - (190236913*x^3*(3 + 2*x + 5*x^2)^(5/2))/4950000 - (796559*x^4*(3
+ 2*x + 5*x^2)^(5/2))/123750 + (1031177*x^5*(3 + 2*x + 5*x^2)^(5/2))/20625 - (61103*x^6*(3 + 2*x + 5*x^2)^(5/2
))/3300 - (343*x^7*(3 + 2*x + 5*x^2)^(5/2))/60 - (3357568053*ArcSinh[(1 + 5*x)/Sqrt[14]])/(156250000*Sqrt[5])

Rule 215

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

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 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/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 1661

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expo
n[Pq, x]]}, Simp[(e*x^(q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*(q + 2*p + 1)), x] + Dist[1/(c*(q + 2*p + 1)), Int
[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + p)*x^(q - 1) - c*e*(q +
2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rubi steps

\begin {align*} \int \left (1+4 x-7 x^2\right )^3 \left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2} \, dx &=-\frac {343}{60} x^7 \left (3+2 x+5 x^2\right )^{5/2}+\frac {1}{60} \int \left (3+2 x+5 x^2\right )^{3/2} \left (120+1740 x+6900 x^2-3660 x^3-52260 x^4+7620 x^5+131103 x^6-61103 x^7\right ) \, dx\\ &=-\frac {61103 x^6 \left (3+2 x+5 x^2\right )^{5/2}}{3300}-\frac {343}{60} x^7 \left (3+2 x+5 x^2\right )^{5/2}+\frac {\int \left (3+2 x+5 x^2\right )^{3/2} \left (6600+95700 x+379500 x^2-201300 x^3-2874300 x^4+1518954 x^5+8249416 x^6\right ) \, dx}{3300}\\ &=\frac {1031177 x^5 \left (3+2 x+5 x^2\right )^{5/2}}{20625}-\frac {61103 x^6 \left (3+2 x+5 x^2\right )^{5/2}}{3300}-\frac {343}{60} x^7 \left (3+2 x+5 x^2\right )^{5/2}+\frac {\int \left (3+2 x+5 x^2\right )^{3/2} \left (330000+4785000 x+18975000 x^2-10065000 x^3-267456240 x^4-47793540 x^5\right ) \, dx}{165000}\\ &=-\frac {796559 x^4 \left (3+2 x+5 x^2\right )^{5/2}}{123750}+\frac {1031177 x^5 \left (3+2 x+5 x^2\right )^{5/2}}{20625}-\frac {61103 x^6 \left (3+2 x+5 x^2\right )^{5/2}}{3300}-\frac {343}{60} x^7 \left (3+2 x+5 x^2\right )^{5/2}+\frac {\int \left (3+2 x+5 x^2\right )^{3/2} \left (14850000+215325000 x+853875000 x^2+120597480 x^3-11414214780 x^4\right ) \, dx}{7425000}\\ &=-\frac {190236913 x^3 \left (3+2 x+5 x^2\right )^{5/2}}{4950000}-\frac {796559 x^4 \left (3+2 x+5 x^2\right )^{5/2}}{123750}+\frac {1031177 x^5 \left (3+2 x+5 x^2\right )^{5/2}}{20625}-\frac {61103 x^6 \left (3+2 x+5 x^2\right )^{5/2}}{3300}-\frac {343}{60} x^7 \left (3+2 x+5 x^2\right )^{5/2}+\frac {\int \left (3+2 x+5 x^2\right )^{3/2} \left (594000000+8613000000 x+136882933020 x^2+130380261780 x^3\right ) \, dx}{297000000}\\ &=\frac {2173004363 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{173250000}-\frac {190236913 x^3 \left (3+2 x+5 x^2\right )^{5/2}}{4950000}-\frac {796559 x^4 \left (3+2 x+5 x^2\right )^{5/2}}{123750}+\frac {1031177 x^5 \left (3+2 x+5 x^2\right )^{5/2}}{20625}-\frac {61103 x^6 \left (3+2 x+5 x^2\right )^{5/2}}{3300}-\frac {343}{60} x^7 \left (3+2 x+5 x^2\right )^{5/2}+\frac {\int \left (3+2 x+5 x^2\right )^{3/2} \left (20790000000-480826570680 x+3617480299680 x^2\right ) \, dx}{10395000000}\\ &=\frac {837379699 x \left (3+2 x+5 x^2\right )^{5/2}}{72187500}+\frac {2173004363 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{173250000}-\frac {190236913 x^3 \left (3+2 x+5 x^2\right )^{5/2}}{4950000}-\frac {796559 x^4 \left (3+2 x+5 x^2\right )^{5/2}}{123750}+\frac {1031177 x^5 \left (3+2 x+5 x^2\right )^{5/2}}{20625}-\frac {61103 x^6 \left (3+2 x+5 x^2\right )^{5/2}}{3300}-\frac {343}{60} x^7 \left (3+2 x+5 x^2\right )^{5/2}+\frac {\int (-10228740899040-39747159218160 x) \left (3+2 x+5 x^2\right )^{3/2} \, dx}{311850000000}\\ &=-\frac {6133820867 \left (3+2 x+5 x^2\right )^{5/2}}{1203125000}+\frac {837379699 x \left (3+2 x+5 x^2\right )^{5/2}}{72187500}+\frac {2173004363 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{173250000}-\frac {190236913 x^3 \left (3+2 x+5 x^2\right )^{5/2}}{4950000}-\frac {796559 x^4 \left (3+2 x+5 x^2\right )^{5/2}}{123750}+\frac {1031177 x^5 \left (3+2 x+5 x^2\right )^{5/2}}{20625}-\frac {61103 x^6 \left (3+2 x+5 x^2\right )^{5/2}}{3300}-\frac {343}{60} x^7 \left (3+2 x+5 x^2\right )^{5/2}-\frac {22840599 \int \left (3+2 x+5 x^2\right )^{3/2} \, dx}{3125000}\\ &=-\frac {22840599 (1+5 x) \left (3+2 x+5 x^2\right )^{3/2}}{62500000}-\frac {6133820867 \left (3+2 x+5 x^2\right )^{5/2}}{1203125000}+\frac {837379699 x \left (3+2 x+5 x^2\right )^{5/2}}{72187500}+\frac {2173004363 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{173250000}-\frac {190236913 x^3 \left (3+2 x+5 x^2\right )^{5/2}}{4950000}-\frac {796559 x^4 \left (3+2 x+5 x^2\right )^{5/2}}{123750}+\frac {1031177 x^5 \left (3+2 x+5 x^2\right )^{5/2}}{20625}-\frac {61103 x^6 \left (3+2 x+5 x^2\right )^{5/2}}{3300}-\frac {343}{60} x^7 \left (3+2 x+5 x^2\right )^{5/2}-\frac {479652579 \int \sqrt {3+2 x+5 x^2} \, dx}{31250000}\\ &=-\frac {479652579 (1+5 x) \sqrt {3+2 x+5 x^2}}{312500000}-\frac {22840599 (1+5 x) \left (3+2 x+5 x^2\right )^{3/2}}{62500000}-\frac {6133820867 \left (3+2 x+5 x^2\right )^{5/2}}{1203125000}+\frac {837379699 x \left (3+2 x+5 x^2\right )^{5/2}}{72187500}+\frac {2173004363 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{173250000}-\frac {190236913 x^3 \left (3+2 x+5 x^2\right )^{5/2}}{4950000}-\frac {796559 x^4 \left (3+2 x+5 x^2\right )^{5/2}}{123750}+\frac {1031177 x^5 \left (3+2 x+5 x^2\right )^{5/2}}{20625}-\frac {61103 x^6 \left (3+2 x+5 x^2\right )^{5/2}}{3300}-\frac {343}{60} x^7 \left (3+2 x+5 x^2\right )^{5/2}-\frac {3357568053 \int \frac {1}{\sqrt {3+2 x+5 x^2}} \, dx}{156250000}\\ &=-\frac {479652579 (1+5 x) \sqrt {3+2 x+5 x^2}}{312500000}-\frac {22840599 (1+5 x) \left (3+2 x+5 x^2\right )^{3/2}}{62500000}-\frac {6133820867 \left (3+2 x+5 x^2\right )^{5/2}}{1203125000}+\frac {837379699 x \left (3+2 x+5 x^2\right )^{5/2}}{72187500}+\frac {2173004363 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{173250000}-\frac {190236913 x^3 \left (3+2 x+5 x^2\right )^{5/2}}{4950000}-\frac {796559 x^4 \left (3+2 x+5 x^2\right )^{5/2}}{123750}+\frac {1031177 x^5 \left (3+2 x+5 x^2\right )^{5/2}}{20625}-\frac {61103 x^6 \left (3+2 x+5 x^2\right )^{5/2}}{3300}-\frac {343}{60} x^7 \left (3+2 x+5 x^2\right )^{5/2}-\frac {\left (479652579 \sqrt {\frac {7}{10}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{56}}} \, dx,x,2+10 x\right )}{312500000}\\ &=-\frac {479652579 (1+5 x) \sqrt {3+2 x+5 x^2}}{312500000}-\frac {22840599 (1+5 x) \left (3+2 x+5 x^2\right )^{3/2}}{62500000}-\frac {6133820867 \left (3+2 x+5 x^2\right )^{5/2}}{1203125000}+\frac {837379699 x \left (3+2 x+5 x^2\right )^{5/2}}{72187500}+\frac {2173004363 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{173250000}-\frac {190236913 x^3 \left (3+2 x+5 x^2\right )^{5/2}}{4950000}-\frac {796559 x^4 \left (3+2 x+5 x^2\right )^{5/2}}{123750}+\frac {1031177 x^5 \left (3+2 x+5 x^2\right )^{5/2}}{20625}-\frac {61103 x^6 \left (3+2 x+5 x^2\right )^{5/2}}{3300}-\frac {343}{60} x^7 \left (3+2 x+5 x^2\right )^{5/2}-\frac {3357568053 \sinh ^{-1}\left (\frac {1+5 x}{\sqrt {14}}\right )}{156250000 \sqrt {5}}\\ \end {align*}

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Mathematica [A]  time = 0.42, size = 95, normalized size = 0.41 \[ \frac {-4653589321458 \sqrt {5} \sinh ^{-1}\left (\frac {5 x+1}{\sqrt {14}}\right )-5 \sqrt {5 x^2+2 x+3} \left (30950390625000 x^{11}[phone] x^{10}-148393743750000 x^9-30505457500000 x^8-72918247281250 x^7+52106830406250 x^6+85130334087500 x^5-2573089891000 x^4-19041688239675 x^3-15865844408685 x^2-6352777129950 x+10506617068392\right )}{1082812500000} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + 4*x - 7*x^2)^3*(2 + 5*x + x^2)*(3 + 2*x + 5*x^2)^(3/2),x]

[Out]

(-5*Sqrt[3 + 2*x + 5*x^2]*(10506617068392 - 6352777129950*x - 15865844408685*x^2 - 19041688239675*x^3 - 257308
9891000*x^4 + 85130334087500*x^5 + 52106830406250*x^6 - 72918247281250*x^7 - 30505457500000*x^8 - 148393743750
000*x^9 + 125007421875000*x^10 + 30950390625000*x^11) - 4653589321458*Sqrt[5]*ArcSinh[(1 + 5*x)/Sqrt[14]])/108
2812500000

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fricas [A]  time = 0.88, size = 107, normalized size = 0.46 \[ -\frac {1}{216562500000} \, {\left (30950390625000 \, x^{11} + 125007421875000 \, x^{10} - 148393743750000 \, x^{9} - 30505457500000 \, x^{8} - 72918247281250 \, x^{7} + 52106830406250 \, x^{6} + 85130334087500 \, x^{5} - 2573089891000 \, x^{4} - 19041688239675 \, x^{3} - 15865844408685 \, x^{2} - 6352777129950 \, x + 10506617068392\right )} \sqrt {5 \, x^{2} + 2 \, x + 3} + \frac {3357568053}{1562500000} \, \sqrt {5} \log \left (\sqrt {5} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (5 \, x + 1\right )} - 25 \, x^{2} - 10 \, x - 8\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-7*x^2+4*x+1)^3*(x^2+5*x+2)*(5*x^2+2*x+3)^(3/2),x, algorithm="fricas")

[Out]

-1/216562500000*(30950390625000*x^11 + 125007421875000*x^10 - 148393743750000*x^9 - 30505457500000*x^8 - 72918
247281250*x^7 + 52106830406250*x^6 + 85130334087500*x^5 - 2573089891000*x^4 - 19041688239675*x^3 - 15865844408
685*x^2 - 6352777129950*x + 10506617068392)*sqrt(5*x^2 + 2*x + 3) + 3357568053/1562500000*sqrt(5)*log(sqrt(5)*
sqrt(5*x^2 + 2*x + 3)*(5*x + 1) - 25*x^2 - 10*x - 8)

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giac [A]  time = 0.23, size = 102, normalized size = 0.44 \[ -\frac {1}{216562500000} \, {\left (5 \, {\left ({\left (5 \, {\left (10 \, {\left (25 \, {\left (5 \, {\left (7 \, {\left (20 \, {\left (105 \, {\left (875 \, {\left (77 \, x + 311\right )} x - 323034\right )} x - 6972676\right )} x - 333340559\right )} x + 1667418573\right )} x + 13620853454\right )} x - 10292359564\right )} x - 761667529587\right )} x - 3173168881737\right )} x - 1270555425990\right )} x + 10506617068392\right )} \sqrt {5 \, x^{2} + 2 \, x + 3} + \frac {3357568053}{781250000} \, \sqrt {5} \log \left (-\sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-7*x^2+4*x+1)^3*(x^2+5*x+2)*(5*x^2+2*x+3)^(3/2),x, algorithm="giac")

[Out]

-1/216562500000*(5*((5*(10*(25*(5*(7*(20*(105*(875*(77*x + 311)*x - 323034)*x - 6972676)*x - 333340559)*x + 16
67418573)*x + 13620853454)*x - 10292359564)*x - 761667529587)*x - 3173168881737)*x - 1270555425990)*x + 105066
17068392)*sqrt(5*x^2 + 2*x + 3) + 3357568053/781250000*sqrt(5)*log(-sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3)
) - 1)

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maple [A]  time = 0.04, size = 185, normalized size = 0.80 \[ -\frac {343 \left (5 x^{2}+2 x +3\right )^{\frac {5}{2}} x^{7}}{60}-\frac {61103 \left (5 x^{2}+2 x +3\right )^{\frac {5}{2}} x^{6}}{3300}+\frac {1031177 \left (5 x^{2}+2 x +3\right )^{\frac {5}{2}} x^{5}}{20625}-\frac {796559 \left (5 x^{2}+2 x +3\right )^{\frac {5}{2}} x^{4}}{123750}-\frac {190236913 \left (5 x^{2}+2 x +3\right )^{\frac {5}{2}} x^{3}}{4950000}+\frac {2173004363 \left (5 x^{2}+2 x +3\right )^{\frac {5}{2}} x^{2}}{173250000}+\frac {837379699 \left (5 x^{2}+2 x +3\right )^{\frac {5}{2}} x}{72187500}-\frac {3357568053 \sqrt {5}\, \arcsinh \left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{781250000}-\frac {6133820867 \left (5 x^{2}+2 x +3\right )^{\frac {5}{2}}}{1203125000}-\frac {479652579 \left (10 x +2\right ) \sqrt {5 x^{2}+2 x +3}}{625000000}-\frac {22840599 \left (10 x +2\right ) \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}}}{125000000} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-7*x^2+4*x+1)^3*(x^2+5*x+2)*(5*x^2+2*x+3)^(3/2),x)

[Out]

-6133820867/1203125000*(5*x^2+2*x+3)^(5/2)-343/60*x^7*(5*x^2+2*x+3)^(5/2)-61103/3300*x^6*(5*x^2+2*x+3)^(5/2)+1
031177/20625*x^5*(5*x^2+2*x+3)^(5/2)-796559/123750*x^4*(5*x^2+2*x+3)^(5/2)-190236913/4950000*x^3*(5*x^2+2*x+3)
^(5/2)[phone]/173250000*x^2*(5*x^2+2*x+3)^(5/2)-3357568053/781250000*5^(1/2)*arcsinh(5/14*14^(1/2)*(x+1/5)
)-479652579/625000000*(10*x+2)*(5*x^2+2*x+3)^(1/2)[phone]/72187500*x*(5*x^2+2*x+3)^(5/2)-22840599/125000000
*(10*x+2)*(5*x^2+2*x+3)^(3/2)

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maxima [A]  time = 1.01, size = 206, normalized size = 0.89 \[ -\frac {343}{60} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {5}{2}} x^{7} - \frac {61103}{3300} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {5}{2}} x^{6} + \frac {1031177}{20625} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {5}{2}} x^{5} - \frac {796559}{123750} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {5}{2}} x^{4} - \frac {190236913}{4950000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {5}{2}} x^{3} + \frac {2173004363}{173250000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {5}{2}} x^{2} + \frac {837379699}{72187500} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {5}{2}} x - \frac {6133820867}{1203125000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {5}{2}} - \frac {22840599}{12500000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x - \frac {22840599}{62500000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} - \frac {479652579}{62500000} \, \sqrt {5 \, x^{2} + 2 \, x + 3} x - \frac {3357568053}{781250000} \, \sqrt {5} \operatorname {arsinh}\left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) - \frac {479652579}{312500000} \, \sqrt {5 \, x^{2} + 2 \, x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-7*x^2+4*x+1)^3*(x^2+5*x+2)*(5*x^2+2*x+3)^(3/2),x, algorithm="maxima")

[Out]

-343/60*(5*x^2 + 2*x + 3)^(5/2)*x^7 - 61103/3300*(5*x^2 + 2*x + 3)^(5/2)*x^6 + 1031177/20625*(5*x^2 + 2*x + 3)
^(5/2)*x^5 - 796559/123750*(5*x^2 + 2*x + 3)^(5/2)*x^4 - 190236913/4950000*(5*x^2 + 2*x + 3)^(5/2)*x^3 + 21730
04363/173250000*(5*x^2 + 2*x + 3)^(5/2)*x^2 + 837379699/72187500*(5*x^2 + 2*x + 3)^(5/2)*x - 6133820867/120312
5000*(5*x^2 + 2*x + 3)^(5/2) - 22840599/12500000*(5*x^2 + 2*x + 3)^(3/2)*x - 22840599/62500000*(5*x^2 + 2*x +
3)^(3/2) - 479652579/62500000*sqrt(5*x^2 + 2*x + 3)*x - 3357568053/781250000*sqrt(5)*arcsinh(1/14*sqrt(14)*(5*
x + 1)) - 479652579/312500000*sqrt(5*x^2 + 2*x + 3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (x^2+5\,x+2\right )\,{\left (5\,x^2+2\,x+3\right )}^{3/2}\,{\left (-7\,x^2+4\,x+1\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + x^2 + 2)*(2*x + 5*x^2 + 3)^(3/2)*(4*x - 7*x^2 + 1)^3,x)

[Out]

int((5*x + x^2 + 2)*(2*x + 5*x^2 + 3)^(3/2)*(4*x - 7*x^2 + 1)^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- 91 x \sqrt {5 x^{2} + 2 x + 3}\right )\, dx - \int \left (- 413 x^{2} \sqrt {5 x^{2} + 2 x + 3}\right )\, dx - \int \left (- 192 x^{3} \sqrt {5 x^{2} + 2 x + 3}\right )\, dx - \int 2160 x^{4} \sqrt {5 x^{2} + 2 x + 3}\, dx - \int 1666 x^{5} \sqrt {5 x^{2} + 2 x + 3}\, dx - \int \left (- 2094 x^{6} \sqrt {5 x^{2} + 2 x + 3}\right )\, dx - \int \left (- 1384 x^{7} \sqrt {5 x^{2} + 2 x + 3}\right )\, dx - \int \left (- 7042 x^{8} \sqrt {5 x^{2} + 2 x + 3}\right )\, dx - \int 6321 x^{9} \sqrt {5 x^{2} + 2 x + 3}\, dx - \int 1715 x^{10} \sqrt {5 x^{2} + 2 x + 3}\, dx - \int \left (- 6 \sqrt {5 x^{2} + 2 x + 3}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-7*x**2+4*x+1)**3*(x**2+5*x+2)*(5*x**2+2*x+3)**(3/2),x)

[Out]

-Integral(-91*x*sqrt(5*x**2 + 2*x + 3), x) - Integral(-413*x**2*sqrt(5*x**2 + 2*x + 3), x) - Integral(-192*x**
3*sqrt(5*x**2 + 2*x + 3), x) - Integral(2160*x**4*sqrt(5*x**2 + 2*x + 3), x) - Integral(1666*x**5*sqrt(5*x**2
+ 2*x + 3), x) - Integral(-2094*x**6*sqrt(5*x**2 + 2*x + 3), x) - Integral(-1384*x**7*sqrt(5*x**2 + 2*x + 3),
x) - Integral(-7042*x**8*sqrt(5*x**2 + 2*x + 3), x) - Integral(6321*x**9*sqrt(5*x**2 + 2*x + 3), x) - Integral
(1715*x**10*sqrt(5*x**2 + 2*x + 3), x) - Integral(-6*sqrt(5*x**2 + 2*x + 3), x)

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