3.3.74 \(\int (d+e x)^2 (b x+c x^2)^{3/2} \, dx\)

Optimal. Leaf size=214 \[ -\frac {b^2 (b+2 c x) \sqrt {b x+c x^2} \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right )}{512 c^4}+\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2} \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right )}{192 c^3}+\frac {b^4 \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{512 c^{9/2}}+\frac {7 e \left (b x+c x^2\right )^{5/2} (2 c d-b e)}{60 c^2}+\frac {e \left (b x+c x^2\right )^{5/2} (d+e x)}{6 c} \]

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Rubi [A]  time = 0.18, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \begin {gather*} -\frac {b^2 (b+2 c x) \sqrt {b x+c x^2} \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right )}{512 c^4}+\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2} \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right )}{192 c^3}+\frac {b^4 \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{512 c^{9/2}}+\frac {7 e \left (b x+c x^2\right )^{5/2} (2 c d-b e)}{60 c^2}+\frac {e \left (b x+c x^2\right )^{5/2} (d+e x)}{6 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^2*(24*c^2*d^2 - 24*b*c*d*e + 7*b^2*e^2)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(512*c^4) + ((24*c^2*d^2 - 24*b*c*d
*e + 7*b^2*e^2)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(192*c^3) + (7*e*(2*c*d - b*e)*(b*x + c*x^2)^(5/2))/(60*c^2)
+ (e*(d + e*x)*(b*x + c*x^2)^(5/2))/(6*c) + (b^4*(24*c^2*d^2 - 24*b*c*d*e + 7*b^2*e^2)*ArcTanh[(Sqrt[c]*x)/Sqr
t[b*x + c*x^2]])/(512*c^(9/2))

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

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Mathematica [A]  time = 0.43, size = 197, normalized size = 0.92 \begin {gather*} \frac {(x (b+c x))^{3/2} \left (\frac {\left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right ) \left (3 b^{7/2} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )+\sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \left (-3 b^3+2 b^2 c x+24 b c^2 x^2+16 c^3 x^3\right )\right )}{256 c^{7/2} (b+c x) \sqrt {\frac {c x}{b}+1}}+\frac {7 e x^{5/2} (b+c x) (2 c d-b e)}{10 c}+e x^{5/2} (b+c x) (d+e x)\right )}{6 c x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((x*(b + c*x))^(3/2)*((7*e*(2*c*d - b*e)*x^(5/2)*(b + c*x))/(10*c) + e*x^(5/2)*(b + c*x)*(d + e*x) + ((24*c^2*
d^2 - 24*b*c*d*e + 7*b^2*e^2)*(Sqrt[c]*Sqrt[x]*Sqrt[1 + (c*x)/b]*(-3*b^3 + 2*b^2*c*x + 24*b*c^2*x^2 + 16*c^3*x
^3) + 3*b^(7/2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]]))/(256*c^(7/2)*(b + c*x)*Sqrt[1 + (c*x)/b])))/(6*c*x^(3/2))

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IntegrateAlgebraic [A]  time = 0.89, size = 254, normalized size = 1.19 \begin {gather*} \frac {\left (-7 b^6 e^2+24 b^5 c d e-24 b^4 c^2 d^2\right ) \log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right )}{1024 c^{9/2}}+\frac {\sqrt {b x+c x^2} \left (-105 b^5 e^2+360 b^4 c d e+70 b^4 c e^2 x-360 b^3 c^2 d^2-240 b^3 c^2 d e x-56 b^3 c^2 e^2 x^2+240 b^2 c^3 d^2 x+192 b^2 c^3 d e x^2+48 b^2 c^3 e^2 x^3+2880 b c^4 d^2 x^2+4224 b c^4 d e x^3+1664 b c^4 e^2 x^4+1920 c^5 d^2 x^3+3072 c^5 d e x^4+1280 c^5 e^2 x^5\right )}{7680 c^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b*x + c*x^2]*(-360*b^3*c^2*d^2 + 360*b^4*c*d*e - 105*b^5*e^2 + 240*b^2*c^3*d^2*x - 240*b^3*c^2*d*e*x + 7
0*b^4*c*e^2*x + 2880*b*c^4*d^2*x^2 + 192*b^2*c^3*d*e*x^2 - 56*b^3*c^2*e^2*x^2 + 1920*c^5*d^2*x^3 + 4224*b*c^4*
d*e*x^3 + 48*b^2*c^3*e^2*x^3 + 3072*c^5*d*e*x^4 + 1664*b*c^4*e^2*x^4 + 1280*c^5*e^2*x^5))/(7680*c^4) + ((-24*b
^4*c^2*d^2 + 24*b^5*c*d*e - 7*b^6*e^2)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[b*x + c*x^2]])/(1024*c^(9/2))

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fricas [A]  time = 0.43, size = 490, normalized size = 2.29 \begin {gather*} \left [\frac {15 \, {\left (24 \, b^{4} c^{2} d^{2} - 24 \, b^{5} c d e + 7 \, b^{6} e^{2}\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (1280 \, c^{6} e^{2} x^{5} - 360 \, b^{3} c^{3} d^{2} + 360 \, b^{4} c^{2} d e - 105 \, b^{5} c e^{2} + 128 \, {\left (24 \, c^{6} d e + 13 \, b c^{5} e^{2}\right )} x^{4} + 48 \, {\left (40 \, c^{6} d^{2} + 88 \, b c^{5} d e + b^{2} c^{4} e^{2}\right )} x^{3} + 8 \, {\left (360 \, b c^{5} d^{2} + 24 \, b^{2} c^{4} d e - 7 \, b^{3} c^{3} e^{2}\right )} x^{2} + 10 \, {\left (24 \, b^{2} c^{4} d^{2} - 24 \, b^{3} c^{3} d e + 7 \, b^{4} c^{2} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{15360 \, c^{5}}, -\frac {15 \, {\left (24 \, b^{4} c^{2} d^{2} - 24 \, b^{5} c d e + 7 \, b^{6} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (1280 \, c^{6} e^{2} x^{5} - 360 \, b^{3} c^{3} d^{2} + 360 \, b^{4} c^{2} d e - 105 \, b^{5} c e^{2} + 128 \, {\left (24 \, c^{6} d e + 13 \, b c^{5} e^{2}\right )} x^{4} + 48 \, {\left (40 \, c^{6} d^{2} + 88 \, b c^{5} d e + b^{2} c^{4} e^{2}\right )} x^{3} + 8 \, {\left (360 \, b c^{5} d^{2} + 24 \, b^{2} c^{4} d e - 7 \, b^{3} c^{3} e^{2}\right )} x^{2} + 10 \, {\left (24 \, b^{2} c^{4} d^{2} - 24 \, b^{3} c^{3} d e + 7 \, b^{4} c^{2} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{7680 \, c^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/15360*(15*(24*b^4*c^2*d^2 - 24*b^5*c*d*e + 7*b^6*e^2)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))
+ 2*(1280*c^6*e^2*x^5 - 360*b^3*c^3*d^2 + 360*b^4*c^2*d*e - 105*b^5*c*e^2 + 128*(24*c^6*d*e + 13*b*c^5*e^2)*x^
4 + 48*(40*c^6*d^2 + 88*b*c^5*d*e + b^2*c^4*e^2)*x^3 + 8*(360*b*c^5*d^2 + 24*b^2*c^4*d*e - 7*b^3*c^3*e^2)*x^2
+ 10*(24*b^2*c^4*d^2 - 24*b^3*c^3*d*e + 7*b^4*c^2*e^2)*x)*sqrt(c*x^2 + b*x))/c^5, -1/7680*(15*(24*b^4*c^2*d^2
- 24*b^5*c*d*e + 7*b^6*e^2)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - (1280*c^6*e^2*x^5 - 360*b^3*c^
3*d^2 + 360*b^4*c^2*d*e - 105*b^5*c*e^2 + 128*(24*c^6*d*e + 13*b*c^5*e^2)*x^4 + 48*(40*c^6*d^2 + 88*b*c^5*d*e
+ b^2*c^4*e^2)*x^3 + 8*(360*b*c^5*d^2 + 24*b^2*c^4*d*e - 7*b^3*c^3*e^2)*x^2 + 10*(24*b^2*c^4*d^2 - 24*b^3*c^3*
d*e + 7*b^4*c^2*e^2)*x)*sqrt(c*x^2 + b*x))/c^5]

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giac [A]  time = 0.28, size = 262, normalized size = 1.22 \begin {gather*} \frac {1}{7680} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, c x e^{2} + \frac {24 \, c^{6} d e + 13 \, b c^{5} e^{2}}{c^{5}}\right )} x + \frac {3 \, {\left (40 \, c^{6} d^{2} + 88 \, b c^{5} d e + b^{2} c^{4} e^{2}\right )}}{c^{5}}\right )} x + \frac {360 \, b c^{5} d^{2} + 24 \, b^{2} c^{4} d e - 7 \, b^{3} c^{3} e^{2}}{c^{5}}\right )} x + \frac {5 \, {\left (24 \, b^{2} c^{4} d^{2} - 24 \, b^{3} c^{3} d e + 7 \, b^{4} c^{2} e^{2}\right )}}{c^{5}}\right )} x - \frac {15 \, {\left (24 \, b^{3} c^{3} d^{2} - 24 \, b^{4} c^{2} d e + 7 \, b^{5} c e^{2}\right )}}{c^{5}}\right )} - \frac {{\left (24 \, b^{4} c^{2} d^{2} - 24 \, b^{5} c d e + 7 \, b^{6} e^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{1024 \, c^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7680*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(10*c*x*e^2 + (24*c^6*d*e + 13*b*c^5*e^2)/c^5)*x + 3*(40*c^6*d^2 + 88*b*c
^5*d*e + b^2*c^4*e^2)/c^5)*x + (360*b*c^5*d^2 + 24*b^2*c^4*d*e - 7*b^3*c^3*e^2)/c^5)*x + 5*(24*b^2*c^4*d^2 - 2
4*b^3*c^3*d*e + 7*b^4*c^2*e^2)/c^5)*x - 15*(24*b^3*c^3*d^2 - 24*b^4*c^2*d*e + 7*b^5*c*e^2)/c^5) - 1/1024*(24*b
^4*c^2*d^2 - 24*b^5*c*d*e + 7*b^6*e^2)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(9/2)

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maple [B]  time = 0.05, size = 420, normalized size = 1.96 \begin {gather*} \frac {7 b^{6} e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{1024 c^{\frac {9}{2}}}-\frac {3 b^{5} d e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{128 c^{\frac {7}{2}}}+\frac {3 b^{4} d^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{128 c^{\frac {5}{2}}}-\frac {7 \sqrt {c \,x^{2}+b x}\, b^{4} e^{2} x}{256 c^{3}}+\frac {3 \sqrt {c \,x^{2}+b x}\, b^{3} d e x}{32 c^{2}}-\frac {3 \sqrt {c \,x^{2}+b x}\, b^{2} d^{2} x}{32 c}-\frac {7 \sqrt {c \,x^{2}+b x}\, b^{5} e^{2}}{512 c^{4}}+\frac {3 \sqrt {c \,x^{2}+b x}\, b^{4} d e}{64 c^{3}}-\frac {3 \sqrt {c \,x^{2}+b x}\, b^{3} d^{2}}{64 c^{2}}+\frac {7 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{2} e^{2} x}{96 c^{2}}-\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} b d e x}{4 c}+\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} d^{2} x}{4}+\frac {7 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{3} e^{2}}{192 c^{3}}-\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{2} d e}{8 c^{2}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} b \,d^{2}}{8 c}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} e^{2} x}{6 c}-\frac {7 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} b \,e^{2}}{60 c^{2}}+\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} d e}{5 c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*e^2*x*(c*x^2+b*x)^(5/2)/c-7/60*e^2*b/c^2*(c*x^2+b*x)^(5/2)+7/96*e^2*b^2/c^2*x*(c*x^2+b*x)^(3/2)+7/192*e^2*
b^3/c^3*(c*x^2+b*x)^(3/2)-7/256*e^2*b^4/c^3*(c*x^2+b*x)^(1/2)*x-7/512*e^2*b^5/c^4*(c*x^2+b*x)^(1/2)+7/1024*e^2
*b^6/c^(9/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2))+2/5*d*e*(c*x^2+b*x)^(5/2)/c-1/4*d*e*b/c*x*(c*x^2+b*x)^(
3/2)-1/8*d*e*b^2/c^2*(c*x^2+b*x)^(3/2)+3/32*d*e*b^3/c^2*(c*x^2+b*x)^(1/2)*x+3/64*d*e*b^4/c^3*(c*x^2+b*x)^(1/2)
-3/128*d*e*b^5/c^(7/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2))+1/4*d^2*x*(c*x^2+b*x)^(3/2)+1/8*d^2/c*(c*x^2+
b*x)^(3/2)*b-3/32*d^2*b^2/c*(c*x^2+b*x)^(1/2)*x-3/64*d^2*b^3/c^2*(c*x^2+b*x)^(1/2)+3/128*d^2*b^4/c^(5/2)*ln((c
*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2))

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maxima [B]  time = 1.57, size = 416, normalized size = 1.94 \begin {gather*} \frac {1}{4} \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} d^{2} x - \frac {3 \, \sqrt {c x^{2} + b x} b^{2} d^{2} x}{32 \, c} + \frac {3 \, \sqrt {c x^{2} + b x} b^{3} d e x}{32 \, c^{2}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} b d e x}{4 \, c} - \frac {7 \, \sqrt {c x^{2} + b x} b^{4} e^{2} x}{256 \, c^{3}} + \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2} e^{2} x}{96 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} e^{2} x}{6 \, c} + \frac {3 \, b^{4} d^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {5}{2}}} - \frac {3 \, b^{5} d e \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {7}{2}}} + \frac {7 \, b^{6} e^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{1024 \, c^{\frac {9}{2}}} - \frac {3 \, \sqrt {c x^{2} + b x} b^{3} d^{2}}{64 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} b d^{2}}{8 \, c} + \frac {3 \, \sqrt {c x^{2} + b x} b^{4} d e}{64 \, c^{3}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2} d e}{8 \, c^{2}} + \frac {2 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} d e}{5 \, c} - \frac {7 \, \sqrt {c x^{2} + b x} b^{5} e^{2}}{512 \, c^{4}} + \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3} e^{2}}{192 \, c^{3}} - \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b e^{2}}{60 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(c*x^2 + b*x)^(3/2)*d^2*x - 3/32*sqrt(c*x^2 + b*x)*b^2*d^2*x/c + 3/32*sqrt(c*x^2 + b*x)*b^3*d*e*x/c^2 - 1/
4*(c*x^2 + b*x)^(3/2)*b*d*e*x/c - 7/256*sqrt(c*x^2 + b*x)*b^4*e^2*x/c^3 + 7/96*(c*x^2 + b*x)^(3/2)*b^2*e^2*x/c
^2 + 1/6*(c*x^2 + b*x)^(5/2)*e^2*x/c + 3/128*b^4*d^2*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(5/2) - 3/
128*b^5*d*e*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(7/2) + 7/1024*b^6*e^2*log(2*c*x + b + 2*sqrt(c*x^2
+ b*x)*sqrt(c))/c^(9/2) - 3/64*sqrt(c*x^2 + b*x)*b^3*d^2/c^2 + 1/8*(c*x^2 + b*x)^(3/2)*b*d^2/c + 3/64*sqrt(c*
x^2 + b*x)*b^4*d*e/c^3 - 1/8*(c*x^2 + b*x)^(3/2)*b^2*d*e/c^2 + 2/5*(c*x^2 + b*x)^(5/2)*d*e/c - 7/512*sqrt(c*x^
2 + b*x)*b^5*e^2/c^4 + 7/192*(c*x^2 + b*x)^(3/2)*b^3*e^2/c^3 - 7/60*(c*x^2 + b*x)^(5/2)*b*e^2/c^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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