3.11.80 \(\int \frac {(A+B x) (b x+c x^2)^2}{\sqrt {d+e x}} \, dx\)

Optimal. Leaf size=265 \[ -\frac {2 (d+e x)^{7/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{7 e^6}+\frac {2 (d+e x)^{5/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{5 e^6}-\frac {2 d^2 \sqrt {d+e x} (B d-A e) (c d-b e)^2}{e^6}-\frac {2 c (d+e x)^{9/2} (-A c e-2 b B e+5 B c d)}{9 e^6}+\frac {2 d (d+e x)^{3/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{3 e^6}+\frac {2 B c^2 (d+e x)^{11/2}}{11 e^6} \]

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Rubi [A]  time = 0.16, antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {771} \begin {gather*} -\frac {2 (d+e x)^{7/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{7 e^6}+\frac {2 (d+e x)^{5/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{5 e^6}-\frac {2 d^2 \sqrt {d+e x} (B d-A e) (c d-b e)^2}{e^6}-\frac {2 c (d+e x)^{9/2} (-A c e-2 b B e+5 B c d)}{9 e^6}+\frac {2 d (d+e x)^{3/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{3 e^6}+\frac {2 B c^2 (d+e x)^{11/2}}{11 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d^2*(B*d - A*e)*(c*d - b*e)^2*Sqrt[d + e*x])/e^6 + (2*d*(c*d - b*e)*(B*d*(5*c*d - 3*b*e) - 2*A*e*(2*c*d -
b*e))*(d + e*x)^(3/2))/(3*e^6) + (2*(A*e*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*d*e + 3*
b^2*e^2))*(d + e*x)^(5/2))/(5*e^6) - (2*(2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2))*(d + e*
x)^(7/2))/(7*e^6) - (2*c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(9/2))/(9*e^6) + (2*B*c^2*(d + e*x)^(11/2))/(11
*e^6)

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^2}{\sqrt {d+e x}} \, dx &=\int \left (-\frac {d^2 (B d-A e) (c d-b e)^2}{e^5 \sqrt {d+e x}}+\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) \sqrt {d+e x}}{e^5}+\frac {\left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{3/2}}{e^5}+\frac {\left (-2 A c e (2 c d-b e)+B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{5/2}}{e^5}+\frac {c (-5 B c d+2 b B e+A c e) (d+e x)^{7/2}}{e^5}+\frac {B c^2 (d+e x)^{9/2}}{e^5}\right ) \, dx\\ &=-\frac {2 d^2 (B d-A e) (c d-b e)^2 \sqrt {d+e x}}{e^6}+\frac {2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) (d+e x)^{3/2}}{3 e^6}+\frac {2 \left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{5/2}}{5 e^6}-\frac {2 \left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{7/2}}{7 e^6}-\frac {2 c (5 B c d-2 b B e-A c e) (d+e x)^{9/2}}{9 e^6}+\frac {2 B c^2 (d+e x)^{11/2}}{11 e^6}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 273, normalized size = 1.03 \begin {gather*} \frac {2 \sqrt {d+e x} \left (11 A e \left (21 b^2 e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+18 b c e \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+c^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )+B \left (99 b^2 e^2 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+22 b c e \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )-5 c^2 \left (256 d^5-128 d^4 e x+96 d^3 e^2 x^2-80 d^2 e^3 x^3+70 d e^4 x^4-63 e^5 x^5\right )\right )\right )}{3465 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(11*A*e*(21*b^2*e^2*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 18*b*c*e*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x
^2 + 5*e^3*x^3) + c^2*(128*d^4 - 64*d^3*e*x + 48*d^2*e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4)) + B*(99*b^2*e^2*(-1
6*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3) + 22*b*c*e*(128*d^4 - 64*d^3*e*x + 48*d^2*e^2*x^2 - 40*d*e^3*x^3
+ 35*e^4*x^4) - 5*c^2*(256*d^5 - 128*d^4*e*x + 96*d^3*e^2*x^2 - 80*d^2*e^3*x^3 + 70*d*e^4*x^4 - 63*e^5*x^5))))
/(3465*e^6)

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IntegrateAlgebraic [A]  time = 0.16, size = 399, normalized size = 1.51 \begin {gather*} \frac {2 \sqrt {d+e x} \left (3465 A b^2 d^2 e^3-2310 A b^2 d e^3 (d+e x)+693 A b^2 e^3 (d+e x)^2-6930 A b c d^3 e^2+6930 A b c d^2 e^2 (d+e x)-4158 A b c d e^2 (d+e x)^2+990 A b c e^2 (d+e x)^3+3465 A c^2 d^4 e-4620 A c^2 d^3 e (d+e x)+4158 A c^2 d^2 e (d+e x)^2-1980 A c^2 d e (d+e x)^3+385 A c^2 e (d+e x)^4-3465 b^2 B d^3 e^2+3465 b^2 B d^2 e^2 (d+e x)-2079 b^2 B d e^2 (d+e x)^2+495 b^2 B e^2 (d+e x)^3+6930 b B c d^4 e-9240 b B c d^3 e (d+e x)+8316 b B c d^2 e (d+e x)^2-3960 b B c d e (d+e x)^3+770 b B c e (d+e x)^4-3465 B c^2 d^5+5775 B c^2 d^4 (d+e x)-6930 B c^2 d^3 (d+e x)^2+4950 B c^2 d^2 (d+e x)^3-1925 B c^2 d (d+e x)^4+315 B c^2 (d+e x)^5\right )}{3465 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(-3465*B*c^2*d^5 + 6930*b*B*c*d^4*e + 3465*A*c^2*d^4*e - 3465*b^2*B*d^3*e^2 - 6930*A*b*c*d^3*
e^2 + 3465*A*b^2*d^2*e^3 + 5775*B*c^2*d^4*(d + e*x) - 9240*b*B*c*d^3*e*(d + e*x) - 4620*A*c^2*d^3*e*(d + e*x)
+ 3465*b^2*B*d^2*e^2*(d + e*x) + 6930*A*b*c*d^2*e^2*(d + e*x) - 2310*A*b^2*d*e^3*(d + e*x) - 6930*B*c^2*d^3*(d
+ e*x)^2 + 8316*b*B*c*d^2*e*(d + e*x)^2 + 4158*A*c^2*d^2*e*(d + e*x)^2 - 2079*b^2*B*d*e^2*(d + e*x)^2 - 4158*
A*b*c*d*e^2*(d + e*x)^2 + 693*A*b^2*e^3*(d + e*x)^2 + 4950*B*c^2*d^2*(d + e*x)^3 - 3960*b*B*c*d*e*(d + e*x)^3
- 1980*A*c^2*d*e*(d + e*x)^3 + 495*b^2*B*e^2*(d + e*x)^3 + 990*A*b*c*e^2*(d + e*x)^3 - 1925*B*c^2*d*(d + e*x)^
4 + 770*b*B*c*e*(d + e*x)^4 + 385*A*c^2*e*(d + e*x)^4 + 315*B*c^2*(d + e*x)^5))/(3465*e^6)

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fricas [A]  time = 0.42, size = 290, normalized size = 1.09 \begin {gather*} \frac {2 \, {\left (315 \, B c^{2} e^{5} x^{5} - 1280 \, B c^{2} d^{5} + 1848 \, A b^{2} d^{2} e^{3} + 1408 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e - 1584 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} - 35 \, {\left (10 \, B c^{2} d e^{4} - 11 \, {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 5 \, {\left (80 \, B c^{2} d^{2} e^{3} - 88 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + 99 \, {\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} - 3 \, {\left (160 \, B c^{2} d^{3} e^{2} - 231 \, A b^{2} e^{5} - 176 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 198 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 4 \, {\left (160 \, B c^{2} d^{4} e - 231 \, A b^{2} d e^{4} - 176 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 198 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{3465 \, e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/3465*(315*B*c^2*e^5*x^5 - 1280*B*c^2*d^5 + 1848*A*b^2*d^2*e^3 + 1408*(2*B*b*c + A*c^2)*d^4*e - 1584*(B*b^2 +
2*A*b*c)*d^3*e^2 - 35*(10*B*c^2*d*e^4 - 11*(2*B*b*c + A*c^2)*e^5)*x^4 + 5*(80*B*c^2*d^2*e^3 - 88*(2*B*b*c + A
*c^2)*d*e^4 + 99*(B*b^2 + 2*A*b*c)*e^5)*x^3 - 3*(160*B*c^2*d^3*e^2 - 231*A*b^2*e^5 - 176*(2*B*b*c + A*c^2)*d^2
*e^3 + 198*(B*b^2 + 2*A*b*c)*d*e^4)*x^2 + 4*(160*B*c^2*d^4*e - 231*A*b^2*d*e^4 - 176*(2*B*b*c + A*c^2)*d^3*e^2
+ 198*(B*b^2 + 2*A*b*c)*d^2*e^3)*x)*sqrt(e*x + d)/e^6

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giac [A]  time = 0.19, size = 378, normalized size = 1.43 \begin {gather*} \frac {2}{3465} \, {\left (231 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} A b^{2} e^{\left (-2\right )} + 99 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} B b^{2} e^{\left (-3\right )} + 198 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} A b c e^{\left (-3\right )} + 22 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} B b c e^{\left (-4\right )} + 11 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} A c^{2} e^{\left (-4\right )} + 5 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} B c^{2} e^{\left (-5\right )}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/3465*(231*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*b^2*e^(-2) + 99*(5*(x*e + d)^(
7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*B*b^2*e^(-3) + 198*(5*(x*e + d)^(
7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*b*c*e^(-3) + 22*(35*(x*e + d)^(
9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*B*b*
c*e^(-4) + 11*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3
+ 315*sqrt(x*e + d)*d^4)*A*c^2*e^(-4) + 5*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d
^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*B*c^2*e^(-5))*e^(-1)

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maple [A]  time = 0.05, size = 341, normalized size = 1.29 \begin {gather*} \frac {2 \left (315 B \,c^{2} x^{5} e^{5}+385 A \,c^{2} e^{5} x^{4}+770 B b c \,e^{5} x^{4}-350 B \,c^{2} d \,e^{4} x^{4}+990 A b c \,e^{5} x^{3}-440 A \,c^{2} d \,e^{4} x^{3}+495 B \,b^{2} e^{5} x^{3}-880 B b c d \,e^{4} x^{3}+400 B \,c^{2} d^{2} e^{3} x^{3}+693 A \,b^{2} e^{5} x^{2}-1188 A b c d \,e^{4} x^{2}+528 A \,c^{2} d^{2} e^{3} x^{2}-594 B \,b^{2} d \,e^{4} x^{2}+1056 B b c \,d^{2} e^{3} x^{2}-480 B \,c^{2} d^{3} e^{2} x^{2}-924 A \,b^{2} d \,e^{4} x +1584 A b c \,d^{2} e^{3} x -704 A \,c^{2} d^{3} e^{2} x +792 B \,b^{2} d^{2} e^{3} x -1408 B b c \,d^{3} e^{2} x +640 B \,c^{2} d^{4} e x +1848 A \,b^{2} d^{2} e^{3}-3168 A b c \,d^{3} e^{2}+1408 A \,c^{2} d^{4} e -1584 B \,b^{2} d^{3} e^{2}+2816 B b c \,d^{4} e -1280 B \,c^{2} d^{5}\right ) \sqrt {e x +d}}{3465 e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^(1/2),x)

[Out]

2/3465*(315*B*c^2*e^5*x^5+385*A*c^2*e^5*x^4+770*B*b*c*e^5*x^4-350*B*c^2*d*e^4*x^4+990*A*b*c*e^5*x^3-440*A*c^2*
d*e^4*x^3+495*B*b^2*e^5*x^3-880*B*b*c*d*e^4*x^3+400*B*c^2*d^2*e^3*x^3+693*A*b^2*e^5*x^2-1188*A*b*c*d*e^4*x^2+5
28*A*c^2*d^2*e^3*x^2-594*B*b^2*d*e^4*x^2+1056*B*b*c*d^2*e^3*x^2-480*B*c^2*d^3*e^2*x^2-924*A*b^2*d*e^4*x+1584*A
*b*c*d^2*e^3*x-704*A*c^2*d^3*e^2*x+792*B*b^2*d^2*e^3*x-1408*B*b*c*d^3*e^2*x+640*B*c^2*d^4*e*x+1848*A*b^2*d^2*e
^3-3168*A*b*c*d^3*e^2+1408*A*c^2*d^4*e-1584*B*b^2*d^3*e^2+2816*B*b*c*d^4*e-1280*B*c^2*d^5)*(e*x+d)^(1/2)/e^6

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maxima [A]  time = 0.48, size = 291, normalized size = 1.10 \begin {gather*} \frac {2 \, {\left (315 \, {\left (e x + d\right )}^{\frac {11}{2}} B c^{2} - 385 \, {\left (5 \, B c^{2} d - {\left (2 \, B b c + A c^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 495 \, {\left (10 \, B c^{2} d^{2} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e + {\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 693 \, {\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 1155 \, {\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 3465 \, {\left (B c^{2} d^{5} - A b^{2} d^{2} e^{3} - {\left (2 \, B b c + A c^{2}\right )} d^{4} e + {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2}\right )} \sqrt {e x + d}\right )}}{3465 \, e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/3465*(315*(e*x + d)^(11/2)*B*c^2 - 385*(5*B*c^2*d - (2*B*b*c + A*c^2)*e)*(e*x + d)^(9/2) + 495*(10*B*c^2*d^2
- 4*(2*B*b*c + A*c^2)*d*e + (B*b^2 + 2*A*b*c)*e^2)*(e*x + d)^(7/2) - 693*(10*B*c^2*d^3 - A*b^2*e^3 - 6*(2*B*b
*c + A*c^2)*d^2*e + 3*(B*b^2 + 2*A*b*c)*d*e^2)*(e*x + d)^(5/2) + 1155*(5*B*c^2*d^4 - 2*A*b^2*d*e^3 - 4*(2*B*b*
c + A*c^2)*d^3*e + 3*(B*b^2 + 2*A*b*c)*d^2*e^2)*(e*x + d)^(3/2) - 3465*(B*c^2*d^5 - A*b^2*d^2*e^3 - (2*B*b*c +
A*c^2)*d^4*e + (B*b^2 + 2*A*b*c)*d^3*e^2)*sqrt(e*x + d))/e^6

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mupad [B]  time = 1.51, size = 254, normalized size = 0.96 \begin {gather*} \frac {{\left (d+e\,x\right )}^{9/2}\,\left (2\,A\,c^2\,e-10\,B\,c^2\,d+4\,B\,b\,c\,e\right )}{9\,e^6}+\frac {{\left (d+e\,x\right )}^{5/2}\,\left (-6\,B\,b^2\,d\,e^2+2\,A\,b^2\,e^3+24\,B\,b\,c\,d^2\,e-12\,A\,b\,c\,d\,e^2-20\,B\,c^2\,d^3+12\,A\,c^2\,d^2\,e\right )}{5\,e^6}+\frac {{\left (d+e\,x\right )}^{7/2}\,\left (2\,B\,b^2\,e^2-16\,B\,b\,c\,d\,e+4\,A\,b\,c\,e^2+20\,B\,c^2\,d^2-8\,A\,c^2\,d\,e\right )}{7\,e^6}+\frac {2\,B\,c^2\,{\left (d+e\,x\right )}^{11/2}}{11\,e^6}-\frac {2\,d\,\left (b\,e-c\,d\right )\,{\left (d+e\,x\right )}^{3/2}\,\left (2\,A\,b\,e^2+5\,B\,c\,d^2-4\,A\,c\,d\,e-3\,B\,b\,d\,e\right )}{3\,e^6}+\frac {2\,d^2\,\left (A\,e-B\,d\right )\,{\left (b\,e-c\,d\right )}^2\,\sqrt {d+e\,x}}{e^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x + c*x^2)^2*(A + B*x))/(d + e*x)^(1/2),x)

[Out]

((d + e*x)^(9/2)*(2*A*c^2*e - 10*B*c^2*d + 4*B*b*c*e))/(9*e^6) + ((d + e*x)^(5/2)*(2*A*b^2*e^3 - 20*B*c^2*d^3
+ 12*A*c^2*d^2*e - 6*B*b^2*d*e^2 - 12*A*b*c*d*e^2 + 24*B*b*c*d^2*e))/(5*e^6) + ((d + e*x)^(7/2)*(2*B*b^2*e^2 +
20*B*c^2*d^2 + 4*A*b*c*e^2 - 8*A*c^2*d*e - 16*B*b*c*d*e))/(7*e^6) + (2*B*c^2*(d + e*x)^(11/2))/(11*e^6) - (2*
d*(b*e - c*d)*(d + e*x)^(3/2)*(2*A*b*e^2 + 5*B*c*d^2 - 4*A*c*d*e - 3*B*b*d*e))/(3*e^6) + (2*d^2*(A*e - B*d)*(b
*e - c*d)^2*(d + e*x)^(1/2))/e^6

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sympy [A]  time = 106.60, size = 944, normalized size = 3.56

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**2/(e*x+d)**(1/2),x)

[Out]

Piecewise(((-2*A*b**2*d*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 - 2*A*b**2*(-d**3/s
qrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2 - 4*A*b*c*d*(-d**3/sqrt(d
+ e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**3 - 4*A*b*c*(d**4/sqrt(d + e*x) +
4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**3 - 2*A*c**2*
d*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(
7/2)/7)/e**4 - 2*A*c**2*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d +
e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**4 - 2*B*b**2*d*(-d**3/sqrt(d + e*x) - 3*d**2*sq
rt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**3 - 2*B*b**2*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e
*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**3 - 4*B*b*c*d*(d**4/sqrt(d + e
*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**4 - 4*B
*b*c*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*
(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**4 - 2*B*c**2*d*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d*
*3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**5 - 2*B*c**2
*(d**6/sqrt(d + e*x) + 6*d**5*sqrt(d + e*x) - 5*d**4*(d + e*x)**(3/2) + 4*d**3*(d + e*x)**(5/2) - 15*d**2*(d +
e*x)**(7/2)/7 + 2*d*(d + e*x)**(9/2)/3 - (d + e*x)**(11/2)/11)/e**5)/e, Ne(e, 0)), ((A*b**2*x**3/3 + B*c**2*x
**6/6 + x**5*(A*c**2 + 2*B*b*c)/5 + x**4*(2*A*b*c + B*b**2)/4)/sqrt(d), True))

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