3.13.27 \(\int \frac {(A+B x) (b x+c x^2)^2}{(d+e x)^{5/2}} \, dx\) [1227]

Optimal. Leaf size=263 \[ \frac {2 d^2 (B d-A e) (c d-b e)^2}{3 e^6 (d+e x)^{3/2}}-\frac {2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6 \sqrt {d+e x}}+\frac {2 \left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) \sqrt {d+e x}}{e^6}-\frac {2 \left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{3/2}}{3 e^6}-\frac {2 c (5 B c d-2 b B e-A c e) (d+e x)^{5/2}}{5 e^6}+\frac {2 B c^2 (d+e x)^{7/2}}{7 e^6} \]

[Out]

2/3*d^2*(-A*e+B*d)*(-b*e+c*d)^2/e^6/(e*x+d)^(3/2)-2/3*(2*A*c*e*(-b*e+2*c*d)-B*(b^2*e^2-8*b*c*d*e+10*c^2*d^2))*
(e*x+d)^(3/2)/e^6-2/5*c*(-A*c*e-2*B*b*e+5*B*c*d)*(e*x+d)^(5/2)/e^6+2/7*B*c^2*(e*x+d)^(7/2)/e^6-2*d*(-b*e+c*d)*
(B*d*(-3*b*e+5*c*d)-2*A*e*(-b*e+2*c*d))/e^6/(e*x+d)^(1/2)+2*(A*e*(b^2*e^2-6*b*c*d*e+6*c^2*d^2)-B*d*(3*b^2*e^2-
12*b*c*d*e+10*c^2*d^2))*(e*x+d)^(1/2)/e^6

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Rubi [A]
time = 0.10, antiderivative size = 263, 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 = {785} \begin {gather*} -\frac {2 (d+e x)^{3/2} \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 )}{3 e^6}+\frac {2 \sqrt {d+e x} \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 )}{e^6}+\frac {2 d^2 (B d-A e) (c d-b e)^2}{3 e^6 (d+e x)^{3/2}}-\frac {2 c (d+e x)^{5/2} (-A c e-2 b B e+5 B c d)}{5 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))}{e^6 \sqrt {d+e x}}+\frac {2 B c^2 (d+e x)^{7/2}}{7 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 785

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

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Mathematica [A]
time = 0.21, size = 271, normalized size = 1.03 \begin {gather*} \frac {2 \left (7 A e \left (5 b^2 e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+10 b c e \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )+c^2 \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )\right )+B \left (35 b^2 e^2 \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )+14 b c e \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )-5 c^2 \left (256 d^5+384 d^4 e x+96 d^3 e^2 x^2-16 d^2 e^3 x^3+6 d e^4 x^4-3 e^5 x^5\right )\right )\right )}{105 e^6 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(7*A*e*(5*b^2*e^2*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + 10*b*c*e*(-16*d^3 - 24*d^2*e*x - 6*d*e^2*x^2 + e^3*x^3)
+ c^2*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^4)) + B*(35*b^2*e^2*(-16*d^3 - 24*d^2*e*
x - 6*d*e^2*x^2 + e^3*x^3) + 14*b*c*e*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^4) - 5*c
^2*(256*d^5 + 384*d^4*e*x + 96*d^3*e^2*x^2 - 16*d^2*e^3*x^3 + 6*d*e^4*x^4 - 3*e^5*x^5))))/(105*e^6*(d + e*x)^(
3/2))

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Maple [A]
time = 0.70, size = 367, normalized size = 1.40

method	result	size
risch	\(\frac {2 \left (15 B \,c^{2} x^{3} e^{3}+21 A \,c^{2} e^{3} x^{2}+42 B b c \,e^{3} x^{2}-60 B \,c^{2} d \,e^{2} x^{2}+70 A b c \,e^{3} x -98 A \,c^{2} d \,e^{2} x +35 B \,b^{2} e^{3} x -196 B b c d \,e^{2} x +185 B \,c^{2} d^{2} e x +105 A \,b^{2} e^{3}-560 A b c d \,e^{2}+511 A \,c^{2} d^{2} e -280 B \,b^{2} d \,e^{2}+1022 B b c \,d^{2} e -790 B \,c^{2} d^{3}\right ) \sqrt {e x +d}}{105 e^{6}}+\frac {2 \left (6 A b x \,e^{3}-12 A c d \,e^{2} x -9 B b d \,e^{2} x +15 B c \,d^{2} x e +5 A b d \,e^{2}-11 A c \,d^{2} e -8 B b \,d^{2} e +14 B c \,d^{3}\right ) d \left (b e -c d \right )}{3 e^{6} \left (e x +d \right )^{\frac {3}{2}}}\)	\(258\)
gosper	\(\frac {\frac {4}{3} A b c \,e^{5} x^{3}-8 A b c d \,e^{4} x^{2}+\frac {4}{5} B b c \,e^{5} x^{4}-\frac {4}{7} B \,c^{2} d \,e^{4} x^{4}-\frac {32}{15} B b c d \,e^{4} x^{3}+\frac {32}{21} B \,c^{2} d^{2} e^{3} x^{3}+\frac {32}{5} A \,c^{2} d^{2} e^{3} x^{2}+\frac {2}{7} B \,c^{2} x^{5} e^{5}+\frac {512}{15} B b c \,d^{4} e +\frac {128}{5} A \,c^{2} d^{3} e^{2} x +8 A \,b^{2} d \,e^{4} x -\frac {64}{7} B \,c^{2} d^{3} e^{2} x^{2}-\frac {64}{3} A b c \,d^{3} e^{2}-\frac {16}{15} A \,c^{2} d \,e^{4} x^{3}-4 B \,b^{2} d \,e^{4} x^{2}-16 B \,b^{2} d^{2} e^{3} x +\frac {256}{5} B b c \,d^{3} e^{2} x -\frac {256}{7} B \,c^{2} d^{4} e x -\frac {512}{21} B \,c^{2} d^{5}+\frac {2}{5} A \,c^{2} e^{5} x^{4}+\frac {2}{3} B \,b^{2} e^{5} x^{3}+\frac {256}{15} A \,c^{2} d^{4} e -\frac {32}{3} B \,b^{2} d^{3} e^{2}+2 A \,b^{2} e^{5} x^{2}+\frac {16}{3} A \,b^{2} d^{2} e^{3}+\frac {64}{5} B b c \,d^{2} e^{3} x^{2}-32 A b c \,d^{2} e^{3} x}{e^{6} \left (e x +d \right )^{\frac {3}{2}}}\)	\(341\)
trager	\(\frac {\frac {4}{3} A b c \,e^{5} x^{3}-8 A b c d \,e^{4} x^{2}+\frac {4}{5} B b c \,e^{5} x^{4}-\frac {4}{7} B \,c^{2} d \,e^{4} x^{4}-\frac {32}{15} B b c d \,e^{4} x^{3}+\frac {32}{21} B \,c^{2} d^{2} e^{3} x^{3}+\frac {32}{5} A \,c^{2} d^{2} e^{3} x^{2}+\frac {2}{7} B \,c^{2} x^{5} e^{5}+\frac {512}{15} B b c \,d^{4} e +\frac {128}{5} A \,c^{2} d^{3} e^{2} x +8 A \,b^{2} d \,e^{4} x -\frac {64}{7} B \,c^{2} d^{3} e^{2} x^{2}-\frac {64}{3} A b c \,d^{3} e^{2}-\frac {16}{15} A \,c^{2} d \,e^{4} x^{3}-4 B \,b^{2} d \,e^{4} x^{2}-16 B \,b^{2} d^{2} e^{3} x +\frac {256}{5} B b c \,d^{3} e^{2} x -\frac {256}{7} B \,c^{2} d^{4} e x -\frac {512}{21} B \,c^{2} d^{5}+\frac {2}{5} A \,c^{2} e^{5} x^{4}+\frac {2}{3} B \,b^{2} e^{5} x^{3}+\frac {256}{15} A \,c^{2} d^{4} e -\frac {32}{3} B \,b^{2} d^{3} e^{2}+2 A \,b^{2} e^{5} x^{2}+\frac {16}{3} A \,b^{2} d^{2} e^{3}+\frac {64}{5} B b c \,d^{2} e^{3} x^{2}-32 A b c \,d^{2} e^{3} x}{e^{6} \left (e x +d \right )^{\frac {3}{2}}}\)	\(341\)
derivativedivides	\(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 A \,c^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {4 B b c e \left (e x +d \right )^{\frac {5}{2}}}{5}-2 B \,c^{2} d \left (e x +d \right )^{\frac {5}{2}}+\frac {4 A b c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {8 A \,c^{2} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 B \,b^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {16 B b c d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {20 B \,c^{2} d^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+2 A \,b^{2} e^{3} \sqrt {e x +d}-12 A b c d \,e^{2} \sqrt {e x +d}+12 A \,c^{2} d^{2} e \sqrt {e x +d}-6 B \,b^{2} d \,e^{2} \sqrt {e x +d}+24 B b c \,d^{2} e \sqrt {e x +d}-20 B \,c^{2} d^{3} \sqrt {e x +d}+\frac {2 d \left (2 A \,b^{2} e^{3}-6 A b c d \,e^{2}+4 A \,c^{2} d^{2} e -3 B \,b^{2} d \,e^{2}+8 B b c \,d^{2} e -5 B \,c^{2} d^{3}\right )}{\sqrt {e x +d}}-\frac {2 d^{2} \left (A \,b^{2} e^{3}-2 A b c d \,e^{2}+A \,c^{2} d^{2} e -B \,b^{2} d \,e^{2}+2 B b c \,d^{2} e -B \,c^{2} d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{6}}\)	\(367\)
default	\(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 A \,c^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {4 B b c e \left (e x +d \right )^{\frac {5}{2}}}{5}-2 B \,c^{2} d \left (e x +d \right )^{\frac {5}{2}}+\frac {4 A b c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {8 A \,c^{2} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 B \,b^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {16 B b c d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {20 B \,c^{2} d^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+2 A \,b^{2} e^{3} \sqrt {e x +d}-12 A b c d \,e^{2} \sqrt {e x +d}+12 A \,c^{2} d^{2} e \sqrt {e x +d}-6 B \,b^{2} d \,e^{2} \sqrt {e x +d}+24 B b c \,d^{2} e \sqrt {e x +d}-20 B \,c^{2} d^{3} \sqrt {e x +d}+\frac {2 d \left (2 A \,b^{2} e^{3}-6 A b c d \,e^{2}+4 A \,c^{2} d^{2} e -3 B \,b^{2} d \,e^{2}+8 B b c \,d^{2} e -5 B \,c^{2} d^{3}\right )}{\sqrt {e x +d}}-\frac {2 d^{2} \left (A \,b^{2} e^{3}-2 A b c d \,e^{2}+A \,c^{2} d^{2} e -B \,b^{2} d \,e^{2}+2 B b c \,d^{2} e -B \,c^{2} d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{6}}\)	\(367\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 312, normalized size = 1.19 \begin {gather*} \frac {2}{105} \, {\left ({\left (15 \, {\left (x e + d\right )}^{\frac {7}{2}} B c^{2} - 21 \, {\left (5 \, B c^{2} d - 2 \, B b c e - A c^{2} e\right )} {\left (x e + d\right )}^{\frac {5}{2}} + 35 \, {\left (10 \, B c^{2} d^{2} + B b^{2} e^{2} + 2 \, A b c e^{2} - 4 \, {\left (2 \, B b c e + A c^{2} e\right )} d\right )} {\left (x e + d\right )}^{\frac {3}{2}} - 105 \, {\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \, {\left (2 \, B b c e + A c^{2} e\right )} d^{2} + 3 \, {\left (B b^{2} e^{2} + 2 \, A b c e^{2}\right )} d\right )} \sqrt {x e + d}\right )} e^{\left (-5\right )} + \frac {35 \, {\left (B c^{2} d^{5} - A b^{2} d^{2} e^{3} - {\left (2 \, B b c e + A c^{2} e\right )} d^{4} + {\left (B b^{2} e^{2} + 2 \, A b c e^{2}\right )} d^{3} - 3 \, {\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \, {\left (2 \, B b c e + A c^{2} e\right )} d^{3} + 3 \, {\left (B b^{2} e^{2} + 2 \, A b c e^{2}\right )} d^{2}\right )} {\left (x e + d\right )}\right )} e^{\left (-5\right )}}{{\left (x e + d\right )}^{\frac {3}{2}}}\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)^(5/2),x, algorithm="maxima")

[Out]

2/105*((15*(x*e + d)^(7/2)*B*c^2 - 21*(5*B*c^2*d - 2*B*b*c*e - A*c^2*e)*(x*e + d)^(5/2) + 35*(10*B*c^2*d^2 + B
*b^2*e^2 + 2*A*b*c*e^2 - 4*(2*B*b*c*e + A*c^2*e)*d)*(x*e + d)^(3/2) - 105*(10*B*c^2*d^3 - A*b^2*e^3 - 6*(2*B*b
*c*e + A*c^2*e)*d^2 + 3*(B*b^2*e^2 + 2*A*b*c*e^2)*d)*sqrt(x*e + d))*e^(-5) + 35*(B*c^2*d^5 - A*b^2*d^2*e^3 - (
2*B*b*c*e + A*c^2*e)*d^4 + (B*b^2*e^2 + 2*A*b*c*e^2)*d^3 - 3*(5*B*c^2*d^4 - 2*A*b^2*d*e^3 - 4*(2*B*b*c*e + A*c
^2*e)*d^3 + 3*(B*b^2*e^2 + 2*A*b*c*e^2)*d^2)*(x*e + d))*e^(-5)/(x*e + d)^(3/2))*e^(-1)

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Fricas [A]
time = 2.54, size = 296, normalized size = 1.13 \begin {gather*} -\frac {2 \, {\left (1280 \, B c^{2} d^{5} - {\left (15 \, B c^{2} x^{5} + 105 \, A b^{2} x^{2} + 21 \, {\left (2 \, B b c + A c^{2}\right )} x^{4} + 35 \, {\left (B b^{2} + 2 \, A b c\right )} x^{3}\right )} e^{5} + 2 \, {\left (15 \, B c^{2} d x^{4} - 210 \, A b^{2} d x + 28 \, {\left (2 \, B b c + A c^{2}\right )} d x^{3} + 105 \, {\left (B b^{2} + 2 \, A b c\right )} d x^{2}\right )} e^{4} - 8 \, {\left (10 \, B c^{2} d^{2} x^{3} + 35 \, A b^{2} d^{2} + 42 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} x^{2} - 105 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} x\right )} e^{3} + 16 \, {\left (30 \, B c^{2} d^{3} x^{2} - 84 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} x + 35 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3}\right )} e^{2} + 128 \, {\left (15 \, B c^{2} d^{4} x - 7 \, {\left (2 \, B b c + A c^{2}\right )} d^{4}\right )} e\right )} \sqrt {x e + d}}{105 \, {\left (x^{2} e^{8} + 2 \, d x e^{7} + d^{2} e^{6}\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)^(5/2),x, algorithm="fricas")

[Out]

-2/105*(1280*B*c^2*d^5 - (15*B*c^2*x^5 + 105*A*b^2*x^2 + 21*(2*B*b*c + A*c^2)*x^4 + 35*(B*b^2 + 2*A*b*c)*x^3)*
e^5 + 2*(15*B*c^2*d*x^4 - 210*A*b^2*d*x + 28*(2*B*b*c + A*c^2)*d*x^3 + 105*(B*b^2 + 2*A*b*c)*d*x^2)*e^4 - 8*(1
0*B*c^2*d^2*x^3 + 35*A*b^2*d^2 + 42*(2*B*b*c + A*c^2)*d^2*x^2 - 105*(B*b^2 + 2*A*b*c)*d^2*x)*e^3 + 16*(30*B*c^
2*d^3*x^2 - 84*(2*B*b*c + A*c^2)*d^3*x + 35*(B*b^2 + 2*A*b*c)*d^3)*e^2 + 128*(15*B*c^2*d^4*x - 7*(2*B*b*c + A*
c^2)*d^4)*e)*sqrt(x*e + d)/(x^2*e^8 + 2*d*x*e^7 + d^2*e^6)

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Sympy [A]
time = 29.67, size = 292, normalized size = 1.11 \begin {gather*} \frac {2 B c^{2} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{6}} + \frac {2 d^{2} \left (- A e + B d\right ) \left (b e - c d\right )^{2}}{3 e^{6} \left (d + e x\right )^{\frac {3}{2}}} - \frac {2 d \left (b e - c d\right ) \left (- 2 A b e^{2} + 4 A c d e + 3 B b d e - 5 B c d^{2}\right )}{e^{6} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (2 A c^{2} e + 4 B b c e - 10 B c^{2} d\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (4 A b c e^{2} - 8 A c^{2} d e + 2 B b^{2} e^{2} - 16 B b c d e + 20 B c^{2} d^{2}\right )}{3 e^{6}} + \frac {\sqrt {d + e x} \left (2 A b^{2} e^{3} - 12 A b c d e^{2} + 12 A c^{2} d^{2} e - 6 B b^{2} d e^{2} + 24 B b c d^{2} e - 20 B c^{2} d^{3}\right )}{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)**(5/2),x)

[Out]

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

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Giac [A]
time = 0.94, size = 427, normalized size = 1.62 \begin {gather*} \frac {2}{105} \, {\left (15 \, {\left (x e + d\right )}^{\frac {7}{2}} B c^{2} e^{36} - 105 \, {\left (x e + d\right )}^{\frac {5}{2}} B c^{2} d e^{36} + 350 \, {\left (x e + d\right )}^{\frac {3}{2}} B c^{2} d^{2} e^{36} - 1050 \, \sqrt {x e + d} B c^{2} d^{3} e^{36} + 42 \, {\left (x e + d\right )}^{\frac {5}{2}} B b c e^{37} + 21 \, {\left (x e + d\right )}^{\frac {5}{2}} A c^{2} e^{37} - 280 \, {\left (x e + d\right )}^{\frac {3}{2}} B b c d e^{37} - 140 \, {\left (x e + d\right )}^{\frac {3}{2}} A c^{2} d e^{37} + 1260 \, \sqrt {x e + d} B b c d^{2} e^{37} + 630 \, \sqrt {x e + d} A c^{2} d^{2} e^{37} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{2} e^{38} + 70 \, {\left (x e + d\right )}^{\frac {3}{2}} A b c e^{38} - 315 \, \sqrt {x e + d} B b^{2} d e^{38} - 630 \, \sqrt {x e + d} A b c d e^{38} + 105 \, \sqrt {x e + d} A b^{2} e^{39}\right )} e^{\left (-42\right )} - \frac {2 \, {\left (15 \, {\left (x e + d\right )} B c^{2} d^{4} - B c^{2} d^{5} - 24 \, {\left (x e + d\right )} B b c d^{3} e - 12 \, {\left (x e + d\right )} A c^{2} d^{3} e + 2 \, B b c d^{4} e + A c^{2} d^{4} e + 9 \, {\left (x e + d\right )} B b^{2} d^{2} e^{2} + 18 \, {\left (x e + d\right )} A b c d^{2} e^{2} - B b^{2} d^{3} e^{2} - 2 \, A b c d^{3} e^{2} - 6 \, {\left (x e + d\right )} A b^{2} d e^{3} + A b^{2} d^{2} e^{3}\right )} e^{\left (-6\right )}}{3 \, {\left (x e + d\right )}^{\frac {3}{2}}} \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)^(5/2),x, algorithm="giac")

[Out]

2/105*(15*(x*e + d)^(7/2)*B*c^2*e^36 - 105*(x*e + d)^(5/2)*B*c^2*d*e^36 + 350*(x*e + d)^(3/2)*B*c^2*d^2*e^36 -
1050*sqrt(x*e + d)*B*c^2*d^3*e^36 + 42*(x*e + d)^(5/2)*B*b*c*e^37 + 21*(x*e + d)^(5/2)*A*c^2*e^37 - 280*(x*e
+ d)^(3/2)*B*b*c*d*e^37 - 140*(x*e + d)^(3/2)*A*c^2*d*e^37 + 1260*sqrt(x*e + d)*B*b*c*d^2*e^37 + 630*sqrt(x*e
+ d)*A*c^2*d^2*e^37 + 35*(x*e + d)^(3/2)*B*b^2*e^38 + 70*(x*e + d)^(3/2)*A*b*c*e^38 - 315*sqrt(x*e + d)*B*b^2*
d*e^38 - 630*sqrt(x*e + d)*A*b*c*d*e^38 + 105*sqrt(x*e + d)*A*b^2*e^39)*e^(-42) - 2/3*(15*(x*e + d)*B*c^2*d^4
- B*c^2*d^5 - 24*(x*e + d)*B*b*c*d^3*e - 12*(x*e + d)*A*c^2*d^3*e + 2*B*b*c*d^4*e + A*c^2*d^4*e + 9*(x*e + d)*
B*b^2*d^2*e^2 + 18*(x*e + d)*A*b*c*d^2*e^2 - B*b^2*d^3*e^2 - 2*A*b*c*d^3*e^2 - 6*(x*e + d)*A*b^2*d*e^3 + A*b^2
*d^2*e^3)*e^(-6)/(x*e + d)^(3/2)

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Mupad [B]
time = 0.07, size = 316, normalized size = 1.20 \begin {gather*} \frac {{\left (d+e\,x\right )}^{5/2}\,\left (2\,A\,c^2\,e-10\,B\,c^2\,d+4\,B\,b\,c\,e\right )}{5\,e^6}-\frac {\left (d+e\,x\right )\,\left (6\,B\,b^2\,d^2\,e^2-4\,A\,b^2\,d\,e^3-16\,B\,b\,c\,d^3\,e+12\,A\,b\,c\,d^2\,e^2+10\,B\,c^2\,d^4-8\,A\,c^2\,d^3\,e\right )-\frac {2\,B\,c^2\,d^5}{3}+\frac {2\,A\,c^2\,d^4\,e}{3}+\frac {2\,A\,b^2\,d^2\,e^3}{3}-\frac {2\,B\,b^2\,d^3\,e^2}{3}+\frac {4\,B\,b\,c\,d^4\,e}{3}-\frac {4\,A\,b\,c\,d^3\,e^2}{3}}{e^6\,{\left (d+e\,x\right )}^{3/2}}+\frac {\sqrt {d+e\,x}\,\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 )}{e^6}+\frac {{\left (d+e\,x\right )}^{3/2}\,\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 )}{3\,e^6}+\frac {2\,B\,c^2\,{\left (d+e\,x\right )}^{7/2}}{7\,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)^(5/2),x)

[Out]

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

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