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

Optimal. Leaf size=207 \[ -\frac {5 \left (b^2-4 a c\right )^3 (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{9/2}}+\frac {5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2} (2 c d-b e)}{1024 c^4}-\frac {5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{384 c^3}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{24 c^2}+\frac {e \left (a+b x+c x^2\right )^{7/2}}{7 c} \]

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Rubi [A]  time = 0.09, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {640, 612, 621, 206} \begin {gather*} -\frac {5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{384 c^3}+\frac {5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2} (2 c d-b e)}{1024 c^4}-\frac {5 \left (b^2-4 a c\right )^3 (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{9/2}}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{24 c^2}+\frac {e \left (a+b x+c x^2\right )^{7/2}}{7 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(b^2 - 4*a*c)^2*(2*c*d - b*e)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(1024*c^4) - (5*(b^2 - 4*a*c)*(2*c*d - b*e
)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(384*c^3) + ((2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(24*c^2
) + (e*(a + b*x + c*x^2)^(7/2))/(7*c) - (5*(b^2 - 4*a*c)^3*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a
+ b*x + c*x^2])])/(2048*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 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]

Rubi steps

\begin {align*} \int (d+e x) \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac {e \left (a+b x+c x^2\right )^{7/2}}{7 c}+\frac {(2 c d-b e) \int \left (a+b x+c x^2\right )^{5/2} \, dx}{2 c}\\ &=\frac {(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{24 c^2}+\frac {e \left (a+b x+c x^2\right )^{7/2}}{7 c}-\frac {\left (5 \left (b^2-4 a c\right ) (2 c d-b e)\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{48 c^2}\\ &=-\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{24 c^2}+\frac {e \left (a+b x+c x^2\right )^{7/2}}{7 c}+\frac {\left (5 \left (b^2-4 a c\right )^2 (2 c d-b e)\right ) \int \sqrt {a+b x+c x^2} \, dx}{256 c^3}\\ &=\frac {5 \left (b^2-4 a c\right )^2 (2 c d-b e) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^4}-\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{24 c^2}+\frac {e \left (a+b x+c x^2\right )^{7/2}}{7 c}-\frac {\left (5 \left (b^2-4 a c\right )^3 (2 c d-b e)\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2048 c^4}\\ &=\frac {5 \left (b^2-4 a c\right )^2 (2 c d-b e) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^4}-\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{24 c^2}+\frac {e \left (a+b x+c x^2\right )^{7/2}}{7 c}-\frac {\left (5 \left (b^2-4 a c\right )^3 (2 c d-b e)\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 )}{1024 c^4}\\ &=\frac {5 \left (b^2-4 a c\right )^2 (2 c d-b e) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^4}-\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{24 c^2}+\frac {e \left (a+b x+c x^2\right )^{7/2}}{7 c}-\frac {5 \left (b^2-4 a c\right )^3 (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 0.27, size = 180, normalized size = 0.87 \begin {gather*} \frac {(2 c d-b e) \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 {e (a+x (b+c x))^{7/2}}{7 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*(a + x*(b + c*x))^(7/2))/(7*c) + ((2*c*d - b*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))

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IntegrateAlgebraic [B]  time = 1.70, size = 422, normalized size = 2.04 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (3072 a^3 c^3 e-3696 a^2 b^2 c^2 e+7392 a^2 b c^3 d+1824 a^2 b c^3 e x+14784 a^2 c^4 d x+9216 a^2 c^4 e x^2+1120 a b^4 c e-2240 a b^3 c^2 d-672 a b^3 c^2 e x+1344 a b^2 c^3 d x+480 a b^2 c^3 e x^2+17472 a b c^4 d x^2+12608 a b c^4 e x^3+11648 a c^5 d x^3+9216 a c^5 e x^4-105 b^6 e+210 b^5 c d+70 b^5 c e x-140 b^4 c^2 d x-56 b^4 c^2 e x^2+112 b^3 c^3 d x^2+48 b^3 c^3 e x^3+6048 b^2 c^4 d x^3+4736 b^2 c^4 e x^4+8960 b c^5 d x^4+7424 b c^5 e x^5+3584 c^6 d x^5+3072 c^6 e x^6\right )}{21504 c^4}-\frac {5 \left (-64 a^3 b c^3 e+128 a^3 c^4 d+48 a^2 b^3 c^2 e-96 a^2 b^2 c^3 d-12 a b^5 c e+24 a b^4 c^2 d+b^7 e-2 b^6 c d\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{2048 c^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x + c*x^2]*(210*b^5*c*d - 2240*a*b^3*c^2*d + 7392*a^2*b*c^3*d - 105*b^6*e + 1120*a*b^4*c*e - 3696*
a^2*b^2*c^2*e + 3072*a^3*c^3*e - 140*b^4*c^2*d*x + 1344*a*b^2*c^3*d*x + 14784*a^2*c^4*d*x + 70*b^5*c*e*x - 672
*a*b^3*c^2*e*x + 1824*a^2*b*c^3*e*x + 112*b^3*c^3*d*x^2 + 17472*a*b*c^4*d*x^2 - 56*b^4*c^2*e*x^2 + 480*a*b^2*c
^3*e*x^2 + 9216*a^2*c^4*e*x^2 + 6048*b^2*c^4*d*x^3 + 11648*a*c^5*d*x^3 + 48*b^3*c^3*e*x^3 + 12608*a*b*c^4*e*x^
3 + 8960*b*c^5*d*x^4 + 4736*b^2*c^4*e*x^4 + 9216*a*c^5*e*x^4 + 3584*c^6*d*x^5 + 7424*b*c^5*e*x^5 + 3072*c^6*e*
x^6))/(21504*c^4) - (5*(-2*b^6*c*d + 24*a*b^4*c^2*d - 96*a^2*b^2*c^3*d + 128*a^3*c^4*d + b^7*e - 12*a*b^5*c*e
+ 48*a^2*b^3*c^2*e - 64*a^3*b*c^3*e)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + b*x + c*x^2]])/(2048*c^(9/2))

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fricas [B]  time = 0.52, size = 857, normalized size = 4.14 \begin {gather*} \left [\frac {105 \, {\left (2 \, {\left (b^{6} c - 12 \, a b^{4} c^{2} + 48 \, a^{2} b^{2} c^{3} - 64 \, a^{3} c^{4}\right )} d - {\left (b^{7} - 12 \, a b^{5} c + 48 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} e\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (3072 \, c^{7} e x^{6} + 256 \, {\left (14 \, c^{7} d + 29 \, b c^{6} e\right )} x^{5} + 128 \, {\left (70 \, b c^{6} d + {\left (37 \, b^{2} c^{5} + 72 \, a c^{6}\right )} e\right )} x^{4} + 16 \, {\left (14 \, {\left (27 \, b^{2} c^{5} + 52 \, a c^{6}\right )} d + {\left (3 \, b^{3} c^{4} + 788 \, a b c^{5}\right )} e\right )} x^{3} + 8 \, {\left (14 \, {\left (b^{3} c^{4} + 156 \, a b c^{5}\right )} d - {\left (7 \, b^{4} c^{3} - 60 \, a b^{2} c^{4} - 1152 \, a^{2} c^{5}\right )} e\right )} x^{2} + 14 \, {\left (15 \, b^{5} c^{2} - 160 \, a b^{3} c^{3} + 528 \, a^{2} b c^{4}\right )} d - {\left (105 \, b^{6} c - 1120 \, a b^{4} c^{2} + 3696 \, a^{2} b^{2} c^{3} - 3072 \, a^{3} c^{4}\right )} e - 2 \, {\left (14 \, {\left (5 \, b^{4} c^{3} - 48 \, a b^{2} c^{4} - 528 \, a^{2} c^{5}\right )} d - {\left (35 \, b^{5} c^{2} - 336 \, a b^{3} c^{3} + 912 \, a^{2} b c^{4}\right )} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{86016 \, c^{5}}, \frac {105 \, {\left (2 \, {\left (b^{6} c - 12 \, a b^{4} c^{2} + 48 \, a^{2} b^{2} c^{3} - 64 \, a^{3} c^{4}\right )} d - {\left (b^{7} - 12 \, a b^{5} c + 48 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (3072 \, c^{7} e x^{6} + 256 \, {\left (14 \, c^{7} d + 29 \, b c^{6} e\right )} x^{5} + 128 \, {\left (70 \, b c^{6} d + {\left (37 \, b^{2} c^{5} + 72 \, a c^{6}\right )} e\right )} x^{4} + 16 \, {\left (14 \, {\left (27 \, b^{2} c^{5} + 52 \, a c^{6}\right )} d + {\left (3 \, b^{3} c^{4} + 788 \, a b c^{5}\right )} e\right )} x^{3} + 8 \, {\left (14 \, {\left (b^{3} c^{4} + 156 \, a b c^{5}\right )} d - {\left (7 \, b^{4} c^{3} - 60 \, a b^{2} c^{4} - 1152 \, a^{2} c^{5}\right )} e\right )} x^{2} + 14 \, {\left (15 \, b^{5} c^{2} - 160 \, a b^{3} c^{3} + 528 \, a^{2} b c^{4}\right )} d - {\left (105 \, b^{6} c - 1120 \, a b^{4} c^{2} + 3696 \, a^{2} b^{2} c^{3} - 3072 \, a^{3} c^{4}\right )} e - 2 \, {\left (14 \, {\left (5 \, b^{4} c^{3} - 48 \, a b^{2} c^{4} - 528 \, a^{2} c^{5}\right )} d - {\left (35 \, b^{5} c^{2} - 336 \, a b^{3} c^{3} + 912 \, a^{2} b c^{4}\right )} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{43008 \, c^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/86016*(105*(2*(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 -
64*a^3*b*c^3)*e)*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*(3072*c^7*e*x^6 + 256*(14*c^7*d + 29*b*c^6*e)*x^5 + 128*(70*b*c^6*d + (37*b^2*c^5 + 72*a*c^6)*e)*x^4 + 1
6*(14*(27*b^2*c^5 + 52*a*c^6)*d + (3*b^3*c^4 + 788*a*b*c^5)*e)*x^3 + 8*(14*(b^3*c^4 + 156*a*b*c^5)*d - (7*b^4*
c^3 - 60*a*b^2*c^4 - 1152*a^2*c^5)*e)*x^2 + 14*(15*b^5*c^2 - 160*a*b^3*c^3 + 528*a^2*b*c^4)*d - (105*b^6*c - 1
120*a*b^4*c^2 + 3696*a^2*b^2*c^3 - 3072*a^3*c^4)*e - 2*(14*(5*b^4*c^3 - 48*a*b^2*c^4 - 528*a^2*c^5)*d - (35*b^
5*c^2 - 336*a*b^3*c^3 + 912*a^2*b*c^4)*e)*x)*sqrt(c*x^2 + b*x + a))/c^5, 1/43008*(105*(2*(b^6*c - 12*a*b^4*c^2
+ 48*a^2*b^2*c^3 - 64*a^3*c^4)*d - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*e)*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*(3072*c^7*e*x^6 + 256*(14*c^7*d + 29*b
*c^6*e)*x^5 + 128*(70*b*c^6*d + (37*b^2*c^5 + 72*a*c^6)*e)*x^4 + 16*(14*(27*b^2*c^5 + 52*a*c^6)*d + (3*b^3*c^4
+ 788*a*b*c^5)*e)*x^3 + 8*(14*(b^3*c^4 + 156*a*b*c^5)*d - (7*b^4*c^3 - 60*a*b^2*c^4 - 1152*a^2*c^5)*e)*x^2 +
14*(15*b^5*c^2 - 160*a*b^3*c^3 + 528*a^2*b*c^4)*d - (105*b^6*c - 1120*a*b^4*c^2 + 3696*a^2*b^2*c^3 - 3072*a^3*
c^4)*e - 2*(14*(5*b^4*c^3 - 48*a*b^2*c^4 - 528*a^2*c^5)*d - (35*b^5*c^2 - 336*a*b^3*c^3 + 912*a^2*b*c^4)*e)*x)
*sqrt(c*x^2 + b*x + a))/c^5]

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giac [B]  time = 0.25, size = 445, normalized size = 2.15 \begin {gather*} \frac {1}{21504} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (12 \, c^{2} x e + \frac {14 \, c^{8} d + 29 \, b c^{7} e}{c^{6}}\right )} x + \frac {70 \, b c^{7} d + 37 \, b^{2} c^{6} e + 72 \, a c^{7} e}{c^{6}}\right )} x + \frac {378 \, b^{2} c^{6} d + 728 \, a c^{7} d + 3 \, b^{3} c^{5} e + 788 \, a b c^{6} e}{c^{6}}\right )} x + \frac {14 \, b^{3} c^{5} d + 2184 \, a b c^{6} d - 7 \, b^{4} c^{4} e + 60 \, a b^{2} c^{5} e + 1152 \, a^{2} c^{6} e}{c^{6}}\right )} x - \frac {70 \, b^{4} c^{4} d - 672 \, a b^{2} c^{5} d - 7392 \, a^{2} c^{6} d - 35 \, b^{5} c^{3} e + 336 \, a b^{3} c^{4} e - 912 \, a^{2} b c^{5} e}{c^{6}}\right )} x + \frac {210 \, b^{5} c^{3} d - 2240 \, a b^{3} c^{4} d + 7392 \, a^{2} b c^{5} d - 105 \, b^{6} c^{2} e + 1120 \, a b^{4} c^{3} e - 3696 \, a^{2} b^{2} c^{4} e + 3072 \, a^{3} c^{5} e}{c^{6}}\right )} + \frac {5 \, {\left (2 \, b^{6} c d - 24 \, a b^{4} c^{2} d + 96 \, a^{2} b^{2} c^{3} d - 128 \, a^{3} c^{4} d - b^{7} e + 12 \, a b^{5} c e - 48 \, a^{2} b^{3} c^{2} e + 64 \, a^{3} b c^{3} e\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{2048 \, c^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21504*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(12*c^2*x*e + (14*c^8*d + 29*b*c^7*e)/c^6)*x + (70*b*c^7*d + 37*b
^2*c^6*e + 72*a*c^7*e)/c^6)*x + (378*b^2*c^6*d + 728*a*c^7*d + 3*b^3*c^5*e + 788*a*b*c^6*e)/c^6)*x + (14*b^3*c
^5*d + 2184*a*b*c^6*d - 7*b^4*c^4*e + 60*a*b^2*c^5*e + 1152*a^2*c^6*e)/c^6)*x - (70*b^4*c^4*d - 672*a*b^2*c^5*
d - 7392*a^2*c^6*d - 35*b^5*c^3*e + 336*a*b^3*c^4*e - 912*a^2*b*c^5*e)/c^6)*x + (210*b^5*c^3*d - 2240*a*b^3*c^
4*d + 7392*a^2*b*c^5*d - 105*b^6*c^2*e + 1120*a*b^4*c^3*e - 3696*a^2*b^2*c^4*e + 3072*a^3*c^5*e)/c^6) + 5/2048
*(2*b^6*c*d - 24*a*b^4*c^2*d + 96*a^2*b^2*c^3*d - 128*a^3*c^4*d - b^7*e + 12*a*b^5*c*e - 48*a^2*b^3*c^2*e + 64
*a^3*b*c^3*e)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(9/2)

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maple [B]  time = 0.05, size = 807, normalized size = 3.90 \begin {gather*} -\frac {5 a^{3} b e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{32 c^{\frac {3}{2}}}+\frac {5 a^{3} d \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 \sqrt {c}}+\frac {15 a^{2} b^{3} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{128 c^{\frac {5}{2}}}-\frac {15 a^{2} b^{2} d \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{64 c^{\frac {3}{2}}}-\frac {15 a \,b^{5} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{512 c^{\frac {7}{2}}}+\frac {15 a \,b^{4} d \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {5}{2}}}+\frac {5 b^{7} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2048 c^{\frac {9}{2}}}-\frac {5 b^{6} d \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{1024 c^{\frac {7}{2}}}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, a^{2} b e x}{32 c}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, a^{2} d x}{16}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, a \,b^{3} e x}{64 c^{2}}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} d x}{32 c}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, b^{5} e x}{512 c^{3}}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, b^{4} d x}{256 c^{2}}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{2} e}{64 c^{2}}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, a^{2} b d}{32 c}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, a \,b^{4} e}{128 c^{3}}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, a \,b^{3} d}{64 c^{2}}-\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a b e x}{48 c}+\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a d x}{24}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, b^{6} e}{1024 c^{4}}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, b^{5} d}{512 c^{3}}+\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{3} e x}{192 c^{2}}-\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{2} d x}{96 c}-\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \,b^{2} e}{96 c^{2}}+\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a b d}{48 c}+\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{4} e}{384 c^{3}}-\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{3} d}{192 c^{2}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b e x}{12 c}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} d x}{6}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b^{2} e}{24 c^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b d}{12 c}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} e}{7 c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*e*b^2/c^2*(c*x^2+b*x+a)^(5/2)+5/384*e*b^4/c^3*(c*x^2+b*x+a)^(3/2)+5/2048*e*b^7/c^(9/2)*ln((c*x+1/2*b)/c^
(1/2)+(c*x^2+b*x+a)^(1/2))+5/16*d*(c*x^2+b*x+a)^(1/2)*x*a^2-5/1024*e*b^6/c^4*(c*x^2+b*x+a)^(1/2)-5/192*d/c^2*(
c*x^2+b*x+a)^(3/2)*b^3+5/24*d*(c*x^2+b*x+a)^(3/2)*x*a+1/12*d/c*(c*x^2+b*x+a)^(5/2)*b+5/16*d/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/c^(7/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^6+5/512*d
/c^3*(c*x^2+b*x+a)^(1/2)*b^5-5/48*e*b/c*(c*x^2+b*x+a)^(3/2)*x*a-5/32*e*b/c*(c*x^2+b*x+a)^(1/2)*x*a^2-5/32*d/c*
(c*x^2+b*x+a)^(1/2)*x*a*b^2+5/64*e*b^3/c^2*(c*x^2+b*x+a)^(1/2)*x*a-5/96*e*b^2/c^2*(c*x^2+b*x+a)^(3/2)*a+15/256
*d/c^(5/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^4*a-1/12*e*b/c*x*(c*x^2+b*x+a)^(5/2)+5/192*e*b^3/c^2*
(c*x^2+b*x+a)^(3/2)*x+1/6*d*x*(c*x^2+b*x+a)^(5/2)+15/128*e*b^3/c^(5/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1
/2))*a^2-15/512*e*b^5/c^(7/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-15/64*d/c^(3/2)*ln((c*x+1/2*b)/c^(
1/2)+(c*x^2+b*x+a)^(1/2))*b^2*a^2-5/32*e*b/c^(3/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3+5/256*d/c^2
*(c*x^2+b*x+a)^(1/2)*x*b^4+5/32*d/c*(c*x^2+b*x+a)^(1/2)*b*a^2-5/64*d/c^2*(c*x^2+b*x+a)^(1/2)*b^3*a+5/48*d/c*(c
*x^2+b*x+a)^(3/2)*b*a-5/96*d/c*(c*x^2+b*x+a)^(3/2)*x*b^2-5/512*e*b^5/c^3*(c*x^2+b*x+a)^(1/2)*x-5/64*e*b^2/c^2*
(c*x^2+b*x+a)^(1/2)*a^2+5/128*e*b^4/c^3*(c*x^2+b*x+a)^(1/2)*a+1/7*e*(c*x^2+b*x+a)^(7/2)/c

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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)*(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 )\,{\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)*(a + b*x + c*x^2)^(5/2),x)

[Out]

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

[Out]

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

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