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

Optimal. Leaf size=345 \[ \frac {\left (b x+c x^2\right )^{5/2} \left (10 c e x (14 A c e-9 b B e+4 B c d)+14 A c e (24 c d-7 b e)+B \left (63 b^2 e^2-196 b c d e+48 c^2 d^2\right )\right )}{840 c^3}-\frac {b^2 (b+2 c x) \sqrt {b x+c x^2} \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right )}{1024 c^5}+\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2} \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right )}{384 c^4}+\frac {b^4 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right )}{1024 c^{11/2}}+\frac {B \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c} \]

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Rubi [A]  time = 0.34, antiderivative size = 345, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {832, 779, 612, 620, 206} \begin {gather*} -\frac {b^2 (b+2 c x) \sqrt {b x+c x^2} \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right )}{1024 c^5}+\frac {\left (b x+c x^2\right )^{5/2} \left (10 c e x (14 A c e-9 b B e+4 B c d)+14 A c e (24 c d-7 b e)+B \left (63 b^2 e^2-196 b c d e+48 c^2 d^2\right )\right )}{840 c^3}+\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2} \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right )}{384 c^4}+\frac {b^4 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (14 b^2 c e (A e+2 B d)-24 b c^2 d (2 A e+B d)+48 A c^3 d^2-9 b^3 B e^2\right )}{1024 c^{11/2}}+\frac {B \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^2*(48*A*c^3*d^2 - 9*b^3*B*e^2 + 14*b^2*c*e*(2*B*d + A*e) - 24*b*c^2*d*(B*d + 2*A*e))*(b + 2*c*x)*Sqrt[b*x
+ c*x^2])/(1024*c^5) + ((48*A*c^3*d^2 - 9*b^3*B*e^2 + 14*b^2*c*e*(2*B*d + A*e) - 24*b*c^2*d*(B*d + 2*A*e))*(b
+ 2*c*x)*(b*x + c*x^2)^(3/2))/(384*c^4) + (B*(d + e*x)^2*(b*x + c*x^2)^(5/2))/(7*c) + ((14*A*c*e*(24*c*d - 7*b
*e) + B*(48*c^2*d^2 - 196*b*c*d*e + 63*b^2*e^2) + 10*c*e*(4*B*c*d - 9*b*B*e + 14*A*c*e)*x)*(b*x + c*x^2)^(5/2)
)/(840*c^3) + (b^4*(48*A*c^3*d^2 - 9*b^3*B*e^2 + 14*b^2*c*e*(2*B*d + A*e) - 24*b*c^2*d*(B*d + 2*A*e))*ArcTanh[
(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(1024*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 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m
- 1)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p
+ 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 0])

Rubi steps

\begin {align*} \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^{3/2} \, dx &=\frac {B (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {\int (d+e x) \left (-\frac {1}{2} (5 b B-14 A c) d+\frac {1}{2} (4 B c d-9 b B e+14 A c e) x\right ) \left (b x+c x^2\right )^{3/2} \, dx}{7 c}\\ &=\frac {B (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {\left (14 A c e (24 c d-7 b e)+B \left (48 c^2 d^2-196 b c d e+63 b^2 e^2\right )+10 c e (4 B c d-9 b B e+14 A c e) x\right ) \left (b x+c x^2\right )^{5/2}}{840 c^3}+\frac {\left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{48 c^3}\\ &=\frac {\left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^4}+\frac {B (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {\left (14 A c e (24 c d-7 b e)+B \left (48 c^2 d^2-196 b c d e+63 b^2 e^2\right )+10 c e (4 B c d-9 b B e+14 A c e) x\right ) \left (b x+c x^2\right )^{5/2}}{840 c^3}-\frac {\left (b^2 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right )\right ) \int \sqrt {b x+c x^2} \, dx}{256 c^4}\\ &=-\frac {b^2 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^5}+\frac {\left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^4}+\frac {B (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {\left (14 A c e (24 c d-7 b e)+B \left (48 c^2 d^2-196 b c d e+63 b^2 e^2\right )+10 c e (4 B c d-9 b B e+14 A c e) x\right ) \left (b x+c x^2\right )^{5/2}}{840 c^3}+\frac {\left (b^4 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{2048 c^5}\\ &=-\frac {b^2 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^5}+\frac {\left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^4}+\frac {B (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {\left (14 A c e (24 c d-7 b e)+B \left (48 c^2 d^2-196 b c d e+63 b^2 e^2\right )+10 c e (4 B c d-9 b B e+14 A c e) x\right ) \left (b x+c x^2\right )^{5/2}}{840 c^3}+\frac {\left (b^4 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{1024 c^5}\\ &=-\frac {b^2 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^5}+\frac {\left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{384 c^4}+\frac {B (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {\left (14 A c e (24 c d-7 b e)+B \left (48 c^2 d^2-196 b c d e+63 b^2 e^2\right )+10 c e (4 B c d-9 b B e+14 A c e) x\right ) \left (b x+c x^2\right )^{5/2}}{840 c^3}+\frac {b^4 \left (48 A c^3 d^2-9 b^3 B e^2+14 b^2 c e (2 B d+A e)-24 b c^2 d (B d+2 A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{1024 c^{11/2}}\\ \end {align*}

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Mathematica [A]  time = 1.60, size = 415, normalized size = 1.20 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (14 A c \left (\frac {5}{4} \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right ) \left (3 b^{9/2} \sqrt {c} \sqrt {x} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )+b c 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 )+320 b c^4 e x^3 (b+c x)^2 \sqrt {\frac {c x}{b}+1} (d+e x)-224 b c^3 e x^3 (b+c x)^2 \sqrt {\frac {c x}{b}+1} (b e-2 c d)\right )+B \left (\frac {7}{4} \left (9 b^2 e^2-28 b c d e+24 c^2 d^2\right ) \left (b c x \sqrt {\frac {c x}{b}+1} \left (15 b^4-10 b^3 c x+8 b^2 c^2 x^2+176 b c^3 x^3+128 c^4 x^4\right )-15 b^{11/2} \sqrt {c} \sqrt {x} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )\right )+3840 b c^5 e x^4 (b+c x)^2 \sqrt {\frac {c x}{b}+1} (d+e x)+320 b c^4 e x^4 (b+c x)^2 \sqrt {\frac {c x}{b}+1} (16 c d-9 b e)\right )\right )}{26880 b c^6 x \sqrt {\frac {c x}{b}+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x*(b + c*x)]*(14*A*c*(-224*b*c^3*e*(-2*c*d + b*e)*x^3*(b + c*x)^2*Sqrt[1 + (c*x)/b] + 320*b*c^4*e*x^3*(b
+ c*x)^2*Sqrt[1 + (c*x)/b]*(d + e*x) + (5*(24*c^2*d^2 - 24*b*c*d*e + 7*b^2*e^2)*(b*c*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^(9/2)*Sqrt[c]*Sqrt[x]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]]))/
4) + B*(320*b*c^4*e*(16*c*d - 9*b*e)*x^4*(b + c*x)^2*Sqrt[1 + (c*x)/b] + 3840*b*c^5*e*x^4*(b + c*x)^2*Sqrt[1 +
(c*x)/b]*(d + e*x) + (7*(24*c^2*d^2 - 28*b*c*d*e + 9*b^2*e^2)*(b*c*x*Sqrt[1 + (c*x)/b]*(15*b^4 - 10*b^3*c*x +
8*b^2*c^2*x^2 + 176*b*c^3*x^3 + 128*c^4*x^4) - 15*b^(11/2)*Sqrt[c]*Sqrt[x]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]]
))/4)))/(26880*b*c^6*x*Sqrt[1 + (c*x)/b])

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IntegrateAlgebraic [A]  time = 2.26, size = 537, normalized size = 1.56 \begin {gather*} \frac {\log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right ) \left (-14 A b^6 c e^2+48 A b^5 c^2 d e-48 A b^4 c^3 d^2+9 b^7 B e^2-28 b^6 B c d e+24 b^5 B c^2 d^2\right )}{2048 c^{11/2}}+\frac {\sqrt {b x+c x^2} \left (-1470 A b^5 c e^2+5040 A b^4 c^2 d e+980 A b^4 c^2 e^2 x-5040 A b^3 c^3 d^2-3360 A b^3 c^3 d e x-784 A b^3 c^3 e^2 x^2+3360 A b^2 c^4 d^2 x+2688 A b^2 c^4 d e x^2+672 A b^2 c^4 e^2 x^3+40320 A b c^5 d^2 x^2+59136 A b c^5 d e x^3+23296 A b c^5 e^2 x^4+26880 A c^6 d^2 x^3+43008 A c^6 d e x^4+17920 A c^6 e^2 x^5+945 b^6 B e^2-2940 b^5 B c d e-630 b^5 B c e^2 x+2520 b^4 B c^2 d^2+1960 b^4 B c^2 d e x+504 b^4 B c^2 e^2 x^2-1680 b^3 B c^3 d^2 x-1568 b^3 B c^3 d e x^2-432 b^3 B c^3 e^2 x^3+1344 b^2 B c^4 d^2 x^2+1344 b^2 B c^4 d e x^3+384 b^2 B c^4 e^2 x^4+29568 b B c^5 d^2 x^3+46592 b B c^5 d e x^4+19200 b B c^5 e^2 x^5+21504 B c^6 d^2 x^4+35840 B c^6 d e x^5+15360 B c^6 e^2 x^6\right )}{107520 c^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b*x + c*x^2]*(2520*b^4*B*c^2*d^2 - 5040*A*b^3*c^3*d^2 - 2940*b^5*B*c*d*e + 5040*A*b^4*c^2*d*e + 945*b^6*
B*e^2 - 1470*A*b^5*c*e^2 - 1680*b^3*B*c^3*d^2*x + 3360*A*b^2*c^4*d^2*x + 1960*b^4*B*c^2*d*e*x - 3360*A*b^3*c^3
*d*e*x - 630*b^5*B*c*e^2*x + 980*A*b^4*c^2*e^2*x + 1344*b^2*B*c^4*d^2*x^2 + 40320*A*b*c^5*d^2*x^2 - 1568*b^3*B
*c^3*d*e*x^2 + 2688*A*b^2*c^4*d*e*x^2 + 504*b^4*B*c^2*e^2*x^2 - 784*A*b^3*c^3*e^2*x^2 + 29568*b*B*c^5*d^2*x^3
+ 26880*A*c^6*d^2*x^3 + 1344*b^2*B*c^4*d*e*x^3 + 59136*A*b*c^5*d*e*x^3 - 432*b^3*B*c^3*e^2*x^3 + 672*A*b^2*c^4
*e^2*x^3 + 21504*B*c^6*d^2*x^4 + 46592*b*B*c^5*d*e*x^4 + 43008*A*c^6*d*e*x^4 + 384*b^2*B*c^4*e^2*x^4 + 23296*A
*b*c^5*e^2*x^4 + 35840*B*c^6*d*e*x^5 + 19200*b*B*c^5*e^2*x^5 + 17920*A*c^6*e^2*x^5 + 15360*B*c^6*e^2*x^6))/(10
7520*c^5) + ((24*b^5*B*c^2*d^2 - 48*A*b^4*c^3*d^2 - 28*b^6*B*c*d*e + 48*A*b^5*c^2*d*e + 9*b^7*B*e^2 - 14*A*b^6
*c*e^2)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[b*x + c*x^2]])/(2048*c^(11/2))

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fricas [A]  time = 0.46, size = 987, normalized size = 2.86

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/215040*(105*(24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^2 - 4*(7*B*b^6*c - 12*A*b^5*c^2)*d*e + (9*B*b^7 - 14*A*b^6*c)*
e^2)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 2*(15360*B*c^7*e^2*x^6 + 1280*(28*B*c^7*d*e + (15*
B*b*c^6 + 14*A*c^7)*e^2)*x^5 + 128*(168*B*c^7*d^2 + 28*(13*B*b*c^6 + 12*A*c^7)*d*e + (3*B*b^2*c^5 + 182*A*b*c^
6)*e^2)*x^4 + 48*(56*(11*B*b*c^6 + 10*A*c^7)*d^2 + 28*(B*b^2*c^5 + 44*A*b*c^6)*d*e - (9*B*b^3*c^4 - 14*A*b^2*c
^5)*e^2)*x^3 + 2520*(B*b^4*c^3 - 2*A*b^3*c^4)*d^2 - 420*(7*B*b^5*c^2 - 12*A*b^4*c^3)*d*e + 105*(9*B*b^6*c - 14
*A*b^5*c^2)*e^2 + 56*(24*(B*b^2*c^5 + 30*A*b*c^6)*d^2 - 4*(7*B*b^3*c^4 - 12*A*b^2*c^5)*d*e + (9*B*b^4*c^3 - 14
*A*b^3*c^4)*e^2)*x^2 - 70*(24*(B*b^3*c^4 - 2*A*b^2*c^5)*d^2 - 4*(7*B*b^4*c^3 - 12*A*b^3*c^4)*d*e + (9*B*b^5*c^
2 - 14*A*b^4*c^3)*e^2)*x)*sqrt(c*x^2 + b*x))/c^6, 1/107520*(105*(24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^2 - 4*(7*B*b^6
*c - 12*A*b^5*c^2)*d*e + (9*B*b^7 - 14*A*b^6*c)*e^2)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (1536
0*B*c^7*e^2*x^6 + 1280*(28*B*c^7*d*e + (15*B*b*c^6 + 14*A*c^7)*e^2)*x^5 + 128*(168*B*c^7*d^2 + 28*(13*B*b*c^6
+ 12*A*c^7)*d*e + (3*B*b^2*c^5 + 182*A*b*c^6)*e^2)*x^4 + 48*(56*(11*B*b*c^6 + 10*A*c^7)*d^2 + 28*(B*b^2*c^5 +
44*A*b*c^6)*d*e - (9*B*b^3*c^4 - 14*A*b^2*c^5)*e^2)*x^3 + 2520*(B*b^4*c^3 - 2*A*b^3*c^4)*d^2 - 420*(7*B*b^5*c^
2 - 12*A*b^4*c^3)*d*e + 105*(9*B*b^6*c - 14*A*b^5*c^2)*e^2 + 56*(24*(B*b^2*c^5 + 30*A*b*c^6)*d^2 - 4*(7*B*b^3*
c^4 - 12*A*b^2*c^5)*d*e + (9*B*b^4*c^3 - 14*A*b^3*c^4)*e^2)*x^2 - 70*(24*(B*b^3*c^4 - 2*A*b^2*c^5)*d^2 - 4*(7*
B*b^4*c^3 - 12*A*b^3*c^4)*d*e + (9*B*b^5*c^2 - 14*A*b^4*c^3)*e^2)*x)*sqrt(c*x^2 + b*x))/c^6]

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giac [A]  time = 0.25, size = 518, normalized size = 1.50 \begin {gather*} \frac {1}{107520} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (12 \, B c x e^{2} + \frac {28 \, B c^{7} d e + 15 \, B b c^{6} e^{2} + 14 \, A c^{7} e^{2}}{c^{6}}\right )} x + \frac {168 \, B c^{7} d^{2} + 364 \, B b c^{6} d e + 336 \, A c^{7} d e + 3 \, B b^{2} c^{5} e^{2} + 182 \, A b c^{6} e^{2}}{c^{6}}\right )} x + \frac {3 \, {\left (616 \, B b c^{6} d^{2} + 560 \, A c^{7} d^{2} + 28 \, B b^{2} c^{5} d e + 1232 \, A b c^{6} d e - 9 \, B b^{3} c^{4} e^{2} + 14 \, A b^{2} c^{5} e^{2}\right )}}{c^{6}}\right )} x + \frac {7 \, {\left (24 \, B b^{2} c^{5} d^{2} + 720 \, A b c^{6} d^{2} - 28 \, B b^{3} c^{4} d e + 48 \, A b^{2} c^{5} d e + 9 \, B b^{4} c^{3} e^{2} - 14 \, A b^{3} c^{4} e^{2}\right )}}{c^{6}}\right )} x - \frac {35 \, {\left (24 \, B b^{3} c^{4} d^{2} - 48 \, A b^{2} c^{5} d^{2} - 28 \, B b^{4} c^{3} d e + 48 \, A b^{3} c^{4} d e + 9 \, B b^{5} c^{2} e^{2} - 14 \, A b^{4} c^{3} e^{2}\right )}}{c^{6}}\right )} x + \frac {105 \, {\left (24 \, B b^{4} c^{3} d^{2} - 48 \, A b^{3} c^{4} d^{2} - 28 \, B b^{5} c^{2} d e + 48 \, A b^{4} c^{3} d e + 9 \, B b^{6} c e^{2} - 14 \, A b^{5} c^{2} e^{2}\right )}}{c^{6}}\right )} + \frac {{\left (24 \, B b^{5} c^{2} d^{2} - 48 \, A b^{4} c^{3} d^{2} - 28 \, B b^{6} c d e + 48 \, A b^{5} c^{2} d e + 9 \, B b^{7} e^{2} - 14 \, A b^{6} c e^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{2048 \, c^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/107520*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(10*(12*B*c*x*e^2 + (28*B*c^7*d*e + 15*B*b*c^6*e^2 + 14*A*c^7*e^2)/c^6)
*x + (168*B*c^7*d^2 + 364*B*b*c^6*d*e + 336*A*c^7*d*e + 3*B*b^2*c^5*e^2 + 182*A*b*c^6*e^2)/c^6)*x + 3*(616*B*b
*c^6*d^2 + 560*A*c^7*d^2 + 28*B*b^2*c^5*d*e + 1232*A*b*c^6*d*e - 9*B*b^3*c^4*e^2 + 14*A*b^2*c^5*e^2)/c^6)*x +
7*(24*B*b^2*c^5*d^2 + 720*A*b*c^6*d^2 - 28*B*b^3*c^4*d*e + 48*A*b^2*c^5*d*e + 9*B*b^4*c^3*e^2 - 14*A*b^3*c^4*e
^2)/c^6)*x - 35*(24*B*b^3*c^4*d^2 - 48*A*b^2*c^5*d^2 - 28*B*b^4*c^3*d*e + 48*A*b^3*c^4*d*e + 9*B*b^5*c^2*e^2 -
14*A*b^4*c^3*e^2)/c^6)*x + 105*(24*B*b^4*c^3*d^2 - 48*A*b^3*c^4*d^2 - 28*B*b^5*c^2*d*e + 48*A*b^4*c^3*d*e + 9
*B*b^6*c*e^2 - 14*A*b^5*c^2*e^2)/c^6) + 1/2048*(24*B*b^5*c^2*d^2 - 48*A*b^4*c^3*d^2 - 28*B*b^6*c*d*e + 48*A*b^
5*c^2*d*e + 9*B*b^7*e^2 - 14*A*b^6*c*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 = 0.06, size = 949, normalized size = 2.75 \begin {gather*} \frac {7 A \,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 A \,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 A \,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 {9 B \,b^{7} e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2048 c^{\frac {11}{2}}}+\frac {7 B \,b^{6} d e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{512 c^{\frac {9}{2}}}-\frac {3 B \,b^{5} d^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{256 c^{\frac {7}{2}}}-\frac {7 \sqrt {c \,x^{2}+b x}\, A \,b^{4} e^{2} x}{256 c^{3}}+\frac {3 \sqrt {c \,x^{2}+b x}\, A \,b^{3} d e x}{32 c^{2}}-\frac {3 \sqrt {c \,x^{2}+b x}\, A \,b^{2} d^{2} x}{32 c}+\frac {9 \sqrt {c \,x^{2}+b x}\, B \,b^{5} e^{2} x}{512 c^{4}}-\frac {7 \sqrt {c \,x^{2}+b x}\, B \,b^{4} d e x}{128 c^{3}}+\frac {3 \sqrt {c \,x^{2}+b x}\, B \,b^{3} d^{2} x}{64 c^{2}}-\frac {7 \sqrt {c \,x^{2}+b x}\, A \,b^{5} e^{2}}{512 c^{4}}+\frac {3 \sqrt {c \,x^{2}+b x}\, A \,b^{4} d e}{64 c^{3}}-\frac {3 \sqrt {c \,x^{2}+b x}\, A \,b^{3} d^{2}}{64 c^{2}}+\frac {7 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} A \,b^{2} e^{2} x}{96 c^{2}}-\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} A b d e x}{4 c}+\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} A \,d^{2} x}{4}+\frac {9 \sqrt {c \,x^{2}+b x}\, B \,b^{6} e^{2}}{1024 c^{5}}-\frac {7 \sqrt {c \,x^{2}+b x}\, B \,b^{5} d e}{256 c^{4}}+\frac {3 \sqrt {c \,x^{2}+b x}\, B \,b^{4} d^{2}}{128 c^{3}}-\frac {3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{3} e^{2} x}{64 c^{3}}+\frac {7 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{2} d e x}{48 c^{2}}-\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} B b \,d^{2} x}{8 c}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} B \,e^{2} x^{2}}{7 c}+\frac {7 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} A \,b^{3} e^{2}}{192 c^{3}}-\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} A \,b^{2} d e}{8 c^{2}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} A b \,d^{2}}{8 c}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} A \,e^{2} x}{6 c}-\frac {3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{4} e^{2}}{128 c^{4}}+\frac {7 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{3} d e}{96 c^{3}}-\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{2} d^{2}}{16 c^{2}}-\frac {3 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} B b \,e^{2} x}{28 c^{2}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} B d e x}{3 c}-\frac {7 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} A b \,e^{2}}{60 c^{2}}+\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} A d e}{5 c}+\frac {3 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} B \,b^{2} e^{2}}{40 c^{3}}-\frac {7 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} B b d e}{30 c^{2}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} B \,d^{2}}{5 c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/128*b^5/c^(7/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2))*A*d*e-3/32*A*d^2*b^2/c*(c*x^2+b*x)^(1/2)*x+7/512*
b^6/c^(9/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2))*B*d*e+1/5*(c*x^2+b*x)^(5/2)/c*B*d^2+1/4*A*d^2*x*(c*x^2+b
*x)^(3/2)-9/2048*B*e^2*b^7/c^(11/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2))+9/1024*B*e^2*b^6/c^5*(c*x^2+b*x)
^(1/2)-3/256*b^5/c^(7/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2))*B*d^2+7/1024*b^6/c^(9/2)*ln((c*x+1/2*b)/c^(
1/2)+(c*x^2+b*x)^(1/2))*A*e^2-7/60*b/c^2*(c*x^2+b*x)^(5/2)*A*e^2+7/192*b^3/c^3*(c*x^2+b*x)^(3/2)*A*e^2-7/512*b
^5/c^4*(c*x^2+b*x)^(1/2)*A*e^2+3/128*A*d^2*b^4/c^(5/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2))+7/48*b^2/c^2*
x*(c*x^2+b*x)^(3/2)*B*d*e-7/128*b^4/c^3*(c*x^2+b*x)^(1/2)*x*B*d*e-1/4*b/c*x*(c*x^2+b*x)^(3/2)*A*d*e+2/5*(c*x^2
+b*x)^(5/2)/c*A*d*e+3/128*b^4/c^3*(c*x^2+b*x)^(1/2)*B*d^2+1/7*B*e^2*x^2*(c*x^2+b*x)^(5/2)/c+3/32*b^3/c^2*(c*x^
2+b*x)^(1/2)*x*A*d*e-3/128*B*e^2*b^4/c^4*(c*x^2+b*x)^(3/2)+3/40*B*e^2*b^2/c^3*(c*x^2+b*x)^(5/2)+1/8*A*d^2/c*(c
*x^2+b*x)^(3/2)*b+1/6*x*(c*x^2+b*x)^(5/2)/c*A*e^2-1/16*b^2/c^2*(c*x^2+b*x)^(3/2)*B*d^2-3/64*A*d^2*b^3/c^2*(c*x
^2+b*x)^(1/2)+3/64*b^3/c^2*(c*x^2+b*x)^(1/2)*x*B*d^2+3/64*b^4/c^3*(c*x^2+b*x)^(1/2)*A*d*e-3/64*B*e^2*b^3/c^3*x
*(c*x^2+b*x)^(3/2)-1/8*b/c*x*(c*x^2+b*x)^(3/2)*B*d^2-1/8*b^2/c^2*(c*x^2+b*x)^(3/2)*A*d*e-7/256*b^5/c^4*(c*x^2+
b*x)^(1/2)*B*d*e+7/96*b^3/c^3*(c*x^2+b*x)^(3/2)*B*d*e+1/3*x*(c*x^2+b*x)^(5/2)/c*B*d*e-7/30*b/c^2*(c*x^2+b*x)^(
5/2)*B*d*e+7/96*b^2/c^2*x*(c*x^2+b*x)^(3/2)*A*e^2+9/512*B*e^2*b^5/c^4*(c*x^2+b*x)^(1/2)*x-3/28*B*e^2*b/c^2*x*(
c*x^2+b*x)^(5/2)-7/256*b^4/c^3*(c*x^2+b*x)^(1/2)*x*A*e^2

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maxima [B]  time = 0.60, size = 728, normalized size = 2.11 \begin {gather*} \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} B e^{2} x^{2}}{7 \, c} + \frac {1}{4} \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A d^{2} x - \frac {3 \, \sqrt {c x^{2} + b x} A b^{2} d^{2} x}{32 \, c} + \frac {9 \, \sqrt {c x^{2} + b x} B b^{5} e^{2} x}{512 \, c^{4}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{3} e^{2} x}{64 \, c^{3}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b e^{2} x}{28 \, c^{2}} + \frac {3 \, A 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 {9 \, B b^{7} e^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2048 \, c^{\frac {11}{2}}} - \frac {3 \, \sqrt {c x^{2} + b x} A b^{3} d^{2}}{64 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b d^{2}}{8 \, c} + \frac {9 \, \sqrt {c x^{2} + b x} B b^{6} e^{2}}{1024 \, c^{5}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{4} e^{2}}{128 \, c^{4}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b^{2} e^{2}}{40 \, c^{3}} - \frac {7 \, {\left (2 \, B d e + A e^{2}\right )} \sqrt {c x^{2} + b x} b^{4} x}{256 \, c^{3}} + \frac {7 \, {\left (2 \, B d e + A e^{2}\right )} {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2} x}{96 \, c^{2}} + \frac {3 \, {\left (B d^{2} + 2 \, A d e\right )} \sqrt {c x^{2} + b x} b^{3} x}{64 \, c^{2}} + \frac {{\left (2 \, B d e + A e^{2}\right )} {\left (c x^{2} + b x\right )}^{\frac {5}{2}} x}{6 \, c} - \frac {{\left (B d^{2} + 2 \, A d e\right )} {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b x}{8 \, c} + \frac {7 \, {\left (2 \, B d e + A e^{2}\right )} b^{6} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{1024 \, c^{\frac {9}{2}}} - \frac {3 \, {\left (B d^{2} + 2 \, A d e\right )} b^{5} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{256 \, c^{\frac {7}{2}}} - \frac {7 \, {\left (2 \, B d e + A e^{2}\right )} \sqrt {c x^{2} + b x} b^{5}}{512 \, c^{4}} + \frac {7 \, {\left (2 \, B d e + A e^{2}\right )} {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3}}{192 \, c^{3}} + \frac {3 \, {\left (B d^{2} + 2 \, A d e\right )} \sqrt {c x^{2} + b x} b^{4}}{128 \, c^{3}} - \frac {7 \, {\left (2 \, B d e + A e^{2}\right )} {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b}{60 \, c^{2}} - \frac {{\left (B d^{2} + 2 \, A d e\right )} {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}}{16 \, c^{2}} + \frac {{\left (B d^{2} + 2 \, A d e\right )} {\left (c x^{2} + b x\right )}^{\frac {5}{2}}}{5 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*(c*x^2 + b*x)^(5/2)*B*e^2*x^2/c + 1/4*(c*x^2 + b*x)^(3/2)*A*d^2*x - 3/32*sqrt(c*x^2 + b*x)*A*b^2*d^2*x/c +
9/512*sqrt(c*x^2 + b*x)*B*b^5*e^2*x/c^4 - 3/64*(c*x^2 + b*x)^(3/2)*B*b^3*e^2*x/c^3 - 3/28*(c*x^2 + b*x)^(5/2)
*B*b*e^2*x/c^2 + 3/128*A*b^4*d^2*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(5/2) - 9/2048*B*b^7*e^2*log(2
*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(11/2) - 3/64*sqrt(c*x^2 + b*x)*A*b^3*d^2/c^2 + 1/8*(c*x^2 + b*x)^(3
/2)*A*b*d^2/c + 9/1024*sqrt(c*x^2 + b*x)*B*b^6*e^2/c^5 - 3/128*(c*x^2 + b*x)^(3/2)*B*b^4*e^2/c^4 + 3/40*(c*x^2
+ b*x)^(5/2)*B*b^2*e^2/c^3 - 7/256*(2*B*d*e + A*e^2)*sqrt(c*x^2 + b*x)*b^4*x/c^3 + 7/96*(2*B*d*e + A*e^2)*(c*
x^2 + b*x)^(3/2)*b^2*x/c^2 + 3/64*(B*d^2 + 2*A*d*e)*sqrt(c*x^2 + b*x)*b^3*x/c^2 + 1/6*(2*B*d*e + A*e^2)*(c*x^2
+ b*x)^(5/2)*x/c - 1/8*(B*d^2 + 2*A*d*e)*(c*x^2 + b*x)^(3/2)*b*x/c + 7/1024*(2*B*d*e + A*e^2)*b^6*log(2*c*x +
b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(9/2) - 3/256*(B*d^2 + 2*A*d*e)*b^5*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sq
rt(c))/c^(7/2) - 7/512*(2*B*d*e + A*e^2)*sqrt(c*x^2 + b*x)*b^5/c^4 + 7/192*(2*B*d*e + A*e^2)*(c*x^2 + b*x)^(3/
2)*b^3/c^3 + 3/128*(B*d^2 + 2*A*d*e)*sqrt(c*x^2 + b*x)*b^4/c^3 - 7/60*(2*B*d*e + A*e^2)*(c*x^2 + b*x)^(5/2)*b/
c^2 - 1/16*(B*d^2 + 2*A*d*e)*(c*x^2 + b*x)^(3/2)*b^2/c^2 + 1/5*(B*d^2 + 2*A*d*e)*(c*x^2 + b*x)^(5/2)/c

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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 (A+B\,x\right )\,{\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)*(A + B*x)*(d + e*x)^2,x)

[Out]

int((b*x + c*x^2)^(3/2)*(A + B*x)*(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 (A + B x\right ) \left (d + e x\right )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**2*(c*x**2+b*x)**(3/2),x)

[Out]

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

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