3.2492 \(\int \frac {(A+B x) (d+e x)}{(a+b x+c x^2)^{7/2}} \, dx\)

Optimal. Leaf size=225 \[ -\frac {16 (b+2 c x) \left (4 c (a B e+4 A c d)-8 b c (A e+B d)+3 b^2 B e\right )}{15 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}}+\frac {2 (b+2 c x) \left (4 c (a B e+4 A c d)-8 b c (A e+B d)+3 b^2 B e\right )}{15 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}} \]

[Out]

2/5*(2*a*c*(A*e+B*d)-b*(A*c*d+B*a*e)-(b^2*B*e-b*c*(A*e+B*d)+2*c*(A*c*d-B*a*e))*x)/c/(-4*a*c+b^2)/(c*x^2+b*x+a)
^(5/2)+2/15*(3*b^2*B*e-8*b*c*(A*e+B*d)+4*c*(4*A*c*d+B*a*e))*(2*c*x+b)/c/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^(3/2)-16/
15*(3*b^2*B*e-8*b*c*(A*e+B*d)+4*c*(4*A*c*d+B*a*e))*(2*c*x+b)/(-4*a*c+b^2)^3/(c*x^2+b*x+a)^(1/2)

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Rubi [A]  time = 0.18, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {777, 614, 613} \[ -\frac {16 (b+2 c x) \left (4 c (a B e+4 A c d)-8 b c (A e+B d)+3 b^2 B e\right )}{15 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}}+\frac {2 (b+2 c x) \left (4 c (a B e+4 A c d)-8 b c (A e+B d)+3 b^2 B e\right )}{15 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x))/(a + b*x + c*x^2)^(7/2),x]

[Out]

(2*(2*a*c*(B*d + A*e) - b*(A*c*d + a*B*e) - (b^2*B*e - b*c*(B*d + A*e) + 2*c*(A*c*d - a*B*e))*x))/(5*c*(b^2 -
4*a*c)*(a + b*x + c*x^2)^(5/2)) + (2*(3*b^2*B*e - 8*b*c*(B*d + A*e) + 4*c*(4*A*c*d + a*B*e))*(b + 2*c*x))/(15*
c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^(3/2)) - (16*(3*b^2*B*e - 8*b*c*(B*d + A*e) + 4*c*(4*A*c*d + a*B*e))*(b +
2*c*x))/(15*(b^2 - 4*a*c)^3*Sqrt[a + b*x + c*x^2])

Rule 613

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2]), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 777

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

Rubi steps

\begin {align*} \int \frac {(A+B x) (d+e x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx &=\frac {2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {\left (3 b^2 B e-8 b c (B d+A e)+4 c (4 A c d+a B e)\right ) \int \frac {1}{\left (a+b x+c x^2\right )^{5/2}} \, dx}{5 c \left (b^2-4 a c\right )}\\ &=\frac {2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac {2 \left (3 b^2 B e-8 b c (B d+A e)+4 c (4 A c d+a B e)\right ) (b+2 c x)}{15 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {\left (8 \left (3 b^2 B e-8 b c (B d+A e)+4 c (4 A c d+a B e)\right )\right ) \int \frac {1}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{15 \left (b^2-4 a c\right )^2}\\ &=\frac {2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac {2 \left (3 b^2 B e-8 b c (B d+A e)+4 c (4 A c d+a B e)\right ) (b+2 c x)}{15 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {16 \left (3 b^2 B e-8 b c (B d+A e)+4 c (4 A c d+a B e)\right ) (b+2 c x)}{15 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.59, size = 200, normalized size = 0.89 \[ \frac {2 \left (-3 \left (b^2-4 a c\right )^2 (A c (-2 a e+b (d-e x)+2 c d x)+B (a b e-2 a c (d+e x)+b x (b e-c d)))+\left (b^2-4 a c\right ) (b+2 c x) (a+x (b+c x)) \left (4 c (a B e+4 A c d)-8 b c (A e+B d)+3 b^2 B e\right )-8 c (b+2 c x) (a+x (b+c x))^2 \left (4 c (a B e+4 A c d)-8 b c (A e+B d)+3 b^2 B e\right )\right )}{15 c \left (b^2-4 a c\right )^3 (a+x (b+c x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x))/(a + b*x + c*x^2)^(7/2),x]

[Out]

(2*((b^2 - 4*a*c)*(3*b^2*B*e - 8*b*c*(B*d + A*e) + 4*c*(4*A*c*d + a*B*e))*(b + 2*c*x)*(a + x*(b + c*x)) - 8*c*
(3*b^2*B*e - 8*b*c*(B*d + A*e) + 4*c*(4*A*c*d + a*B*e))*(b + 2*c*x)*(a + x*(b + c*x))^2 - 3*(b^2 - 4*a*c)^2*(A
*c*(-2*a*e + 2*c*d*x + b*(d - e*x)) + B*(a*b*e + b*(-(c*d) + b*e)*x - 2*a*c*(d + e*x)))))/(15*c*(b^2 - 4*a*c)^
3*(a + x*(b + c*x))^(5/2))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)/(c*x^2+b*x+a)^(7/2),x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 0.31, size = 727, normalized size = 3.23 \[ \frac {2 \, {\left ({\left ({\left (2 \, {\left (4 \, {\left (\frac {2 \, {\left (8 \, B b c^{4} d - 16 \, A c^{5} d - 3 \, B b^{2} c^{3} e - 4 \, B a c^{4} e + 8 \, A b c^{4} e\right )} x}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}} + \frac {5 \, {\left (8 \, B b^{2} c^{3} d - 16 \, A b c^{4} d - 3 \, B b^{3} c^{2} e - 4 \, B a b c^{3} e + 8 \, A b^{2} c^{3} e\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x + \frac {5 \, {\left (24 \, B b^{3} c^{2} d + 32 \, B a b c^{3} d - 48 \, A b^{2} c^{3} d - 64 \, A a c^{4} d - 9 \, B b^{4} c e - 24 \, B a b^{2} c^{2} e + 24 \, A b^{3} c^{2} e - 16 \, B a^{2} c^{3} e + 32 \, A a b c^{3} e\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x + \frac {5 \, {\left (8 \, B b^{4} c d + 96 \, B a b^{2} c^{2} d - 16 \, A b^{3} c^{2} d - 192 \, A a b c^{3} d - 3 \, B b^{5} e - 40 \, B a b^{3} c e + 8 \, A b^{4} c e - 48 \, B a^{2} b c^{2} e + 96 \, A a b^{2} c^{2} e\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x - \frac {5 \, {\left (B b^{5} d - 24 \, B a b^{3} c d - 2 \, A b^{4} c d - 48 \, B a^{2} b c^{2} d + 48 \, A a b^{2} c^{2} d + 96 \, A a^{2} c^{3} d + 4 \, B a b^{4} e + A b^{5} e + 48 \, B a^{2} b^{2} c e - 24 \, A a b^{3} c e - 48 \, A a^{2} b c^{2} e\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x - \frac {2 \, B a b^{4} d + 3 \, A b^{5} d - 48 \, B a^{2} b^{2} c d - 40 \, A a b^{3} c d - 96 \, B a^{3} c^{2} d + 240 \, A a^{2} b c^{2} d + 8 \, B a^{2} b^{3} e + 2 \, A a b^{4} e + 96 \, B a^{3} b c e - 48 \, A a^{2} b^{2} c e - 96 \, A a^{3} c^{2} e}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )}}{15 \, {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)/(c*x^2+b*x+a)^(7/2),x, algorithm="giac")

[Out]

2/15*(((2*(4*(2*(8*B*b*c^4*d - 16*A*c^5*d - 3*B*b^2*c^3*e - 4*B*a*c^4*e + 8*A*b*c^4*e)*x/(b^6 - 12*a*b^4*c + 4
8*a^2*b^2*c^2 - 64*a^3*c^3) + 5*(8*B*b^2*c^3*d - 16*A*b*c^4*d - 3*B*b^3*c^2*e - 4*B*a*b*c^3*e + 8*A*b^2*c^3*e)
/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))*x + 5*(24*B*b^3*c^2*d + 32*B*a*b*c^3*d - 48*A*b^2*c^3*d - 6
4*A*a*c^4*d - 9*B*b^4*c*e - 24*B*a*b^2*c^2*e + 24*A*b^3*c^2*e - 16*B*a^2*c^3*e + 32*A*a*b*c^3*e)/(b^6 - 12*a*b
^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))*x + 5*(8*B*b^4*c*d + 96*B*a*b^2*c^2*d - 16*A*b^3*c^2*d - 192*A*a*b*c^3*d
- 3*B*b^5*e - 40*B*a*b^3*c*e + 8*A*b^4*c*e - 48*B*a^2*b*c^2*e + 96*A*a*b^2*c^2*e)/(b^6 - 12*a*b^4*c + 48*a^2*b
^2*c^2 - 64*a^3*c^3))*x - 5*(B*b^5*d - 24*B*a*b^3*c*d - 2*A*b^4*c*d - 48*B*a^2*b*c^2*d + 48*A*a*b^2*c^2*d + 96
*A*a^2*c^3*d + 4*B*a*b^4*e + A*b^5*e + 48*B*a^2*b^2*c*e - 24*A*a*b^3*c*e - 48*A*a^2*b*c^2*e)/(b^6 - 12*a*b^4*c
+ 48*a^2*b^2*c^2 - 64*a^3*c^3))*x - (2*B*a*b^4*d + 3*A*b^5*d - 48*B*a^2*b^2*c*d - 40*A*a*b^3*c*d - 96*B*a^3*c
^2*d + 240*A*a^2*b*c^2*d + 8*B*a^2*b^3*e + 2*A*a*b^4*e + 96*B*a^3*b*c*e - 48*A*a^2*b^2*c*e - 96*A*a^3*c^2*e)/(
b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))/(c*x^2 + b*x + a)^(5/2)

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maple [B]  time = 0.01, size = 608, normalized size = 2.70 \[ -\frac {2 \left (128 A b \,c^{4} e \,x^{5}-256 A \,c^{5} d \,x^{5}-64 B a \,c^{4} e \,x^{5}-48 B \,b^{2} c^{3} e \,x^{5}+128 B b \,c^{4} d \,x^{5}+320 A \,b^{2} c^{3} e \,x^{4}-640 A b \,c^{4} d \,x^{4}-160 B a b \,c^{3} e \,x^{4}-120 B \,b^{3} c^{2} e \,x^{4}+320 B \,b^{2} c^{3} d \,x^{4}+320 A a b \,c^{3} e \,x^{3}-640 A a \,c^{4} d \,x^{3}+240 A \,b^{3} c^{2} e \,x^{3}-480 A \,b^{2} c^{3} d \,x^{3}-160 B \,a^{2} c^{3} e \,x^{3}-240 B a \,b^{2} c^{2} e \,x^{3}+320 B a b \,c^{3} d \,x^{3}-90 B \,b^{4} c e \,x^{3}+240 B \,b^{3} c^{2} d \,x^{3}+480 A a \,b^{2} c^{2} e \,x^{2}-960 A a b \,c^{3} d \,x^{2}+40 A \,b^{4} c e \,x^{2}-80 A \,b^{3} c^{2} d \,x^{2}-240 B \,a^{2} b \,c^{2} e \,x^{2}-200 B a \,b^{3} c e \,x^{2}+480 B a \,b^{2} c^{2} d \,x^{2}-15 B \,b^{5} e \,x^{2}+40 B \,b^{4} c d \,x^{2}+240 A \,a^{2} b \,c^{2} e x -480 A \,a^{2} c^{3} d x +120 A a \,b^{3} c e x -240 A a \,b^{2} c^{2} d x -5 A \,b^{5} e x +10 A \,b^{4} c d x -240 B \,a^{2} b^{2} c e x +240 B \,a^{2} b \,c^{2} d x -20 B a \,b^{4} e x +120 B a \,b^{3} c d x -5 B \,b^{5} d x +96 A \,a^{3} c^{2} e +48 A \,a^{2} b^{2} c e -240 A \,a^{2} b \,c^{2} d -2 A a \,b^{4} e +40 A a \,b^{3} c d -3 A \,b^{5} d -96 B \,a^{3} b c e +96 B \,a^{3} c^{2} d -8 B \,a^{2} b^{3} e +48 B \,a^{2} b^{2} c d -2 B a \,b^{4} d \right )}{15 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)/(c*x^2+b*x+a)^(7/2),x)

[Out]

-2/15/(c*x^2+b*x+a)^(5/2)*(128*A*b*c^4*e*x^5-256*A*c^5*d*x^5-64*B*a*c^4*e*x^5-48*B*b^2*c^3*e*x^5+128*B*b*c^4*d
*x^5+320*A*b^2*c^3*e*x^4-640*A*b*c^4*d*x^4-160*B*a*b*c^3*e*x^4-120*B*b^3*c^2*e*x^4+320*B*b^2*c^3*d*x^4+320*A*a
*b*c^3*e*x^3-640*A*a*c^4*d*x^3+240*A*b^3*c^2*e*x^3-480*A*b^2*c^3*d*x^3-160*B*a^2*c^3*e*x^3-240*B*a*b^2*c^2*e*x
^3+320*B*a*b*c^3*d*x^3-90*B*b^4*c*e*x^3+240*B*b^3*c^2*d*x^3+480*A*a*b^2*c^2*e*x^2-960*A*a*b*c^3*d*x^2+40*A*b^4
*c*e*x^2-80*A*b^3*c^2*d*x^2-240*B*a^2*b*c^2*e*x^2-200*B*a*b^3*c*e*x^2+480*B*a*b^2*c^2*d*x^2-15*B*b^5*e*x^2+40*
B*b^4*c*d*x^2+240*A*a^2*b*c^2*e*x-480*A*a^2*c^3*d*x+120*A*a*b^3*c*e*x-240*A*a*b^2*c^2*d*x-5*A*b^5*e*x+10*A*b^4
*c*d*x-240*B*a^2*b^2*c*e*x+240*B*a^2*b*c^2*d*x-20*B*a*b^4*e*x+120*B*a*b^3*c*d*x-5*B*b^5*d*x+96*A*a^3*c^2*e+48*
A*a^2*b^2*c*e-240*A*a^2*b*c^2*d-2*A*a*b^4*e+40*A*a*b^3*c*d-3*A*b^5*d-96*B*a^3*b*c*e+96*B*a^3*c^2*d-8*B*a^2*b^3
*e+48*B*a^2*b^2*c*d-2*B*a*b^4*d)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)/(c*x^2+b*x+a)^(7/2),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 zero or nonzero?

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mupad [B]  time = 3.26, size = 892, normalized size = 3.96 \[ \frac {\frac {16\,B\,c^2\,e\,x}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}+\frac {8\,B\,b\,c\,e}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}}{\sqrt {c\,x^2+b\,x+a}}-\frac {x\,\left (\frac {b\,\left (\frac {2\,c^2\,\left (\frac {2\,A\,e}{5}+\frac {2\,B\,d}{5}\right )}{4\,a\,c^2-b^2\,c}-\frac {2\,B\,b\,c\,e}{5\,\left (4\,a\,c^2-b^2\,c\right )}\right )}{c}-\frac {b\,c\,\left (\frac {2\,A\,e}{5}+\frac {2\,B\,d}{5}\right )}{4\,a\,c^2-b^2\,c}-\frac {4\,A\,c^2\,d}{5\,\left (4\,a\,c^2-b^2\,c\right )}+\frac {4\,B\,a\,c\,e}{5\,\left (4\,a\,c^2-b^2\,c\right )}\right )+\frac {a\,\left (\frac {2\,c^2\,\left (\frac {2\,A\,e}{5}+\frac {2\,B\,d}{5}\right )}{4\,a\,c^2-b^2\,c}-\frac {2\,B\,b\,c\,e}{5\,\left (4\,a\,c^2-b^2\,c\right )}\right )}{c}-\frac {2\,A\,b\,c\,d}{5\,\left (4\,a\,c^2-b^2\,c\right )}}{{\left (c\,x^2+b\,x+a\right )}^{5/2}}+\frac {x\,\left (\frac {b\,\left (\frac {16\,c^3\,\left (A\,e+B\,d\right )}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}-\frac {8\,B\,b\,c^2\,e}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}\right )}{c}+\frac {2\,c\,\left (32\,A\,c^2\,d+8\,B\,b^2\,e-20\,A\,b\,c\,e+8\,B\,a\,c\,e-20\,B\,b\,c\,d\right )}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}-\frac {8\,b\,c^2\,\left (A\,e+B\,d\right )}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}+\frac {16\,B\,a\,c^2\,e}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}\right )+\frac {a\,\left (\frac {16\,c^3\,\left (A\,e+B\,d\right )}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}-\frac {8\,B\,b\,c^2\,e}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}\right )}{c}+\frac {b\,\left (32\,A\,c^2\,d+8\,B\,b^2\,e-20\,A\,b\,c\,e+8\,B\,a\,c\,e-20\,B\,b\,c\,d\right )}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}}{{\left (c\,x^2+b\,x+a\right )}^{3/2}}+\frac {\frac {b\,c\,\left (256\,A\,c^2\,d+56\,B\,b^2\,e-128\,A\,b\,c\,e+32\,B\,a\,c\,e-128\,B\,b\,c\,d\right )}{15\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^2}+\frac {2\,c^2\,x\,\left (256\,A\,c^2\,d+56\,B\,b^2\,e-128\,A\,b\,c\,e+32\,B\,a\,c\,e-128\,B\,b\,c\,d\right )}{15\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^2}}{\sqrt {c\,x^2+b\,x+a}}-\frac {\frac {4\,A\,c\,e-2\,B\,b\,e+4\,B\,c\,d}{15\,c\,\left (4\,a\,c-b^2\right )}+\frac {4\,B\,e\,x}{15\,\left (4\,a\,c-b^2\right )}}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x))/(a + b*x + c*x^2)^(7/2),x)

[Out]

((16*B*c^2*e*x)/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) + (8*B*b*c*e)/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)))/(a +
b*x + c*x^2)^(1/2) - (x*((b*((2*c^2*((2*A*e)/5 + (2*B*d)/5))/(4*a*c^2 - b^2*c) - (2*B*b*c*e)/(5*(4*a*c^2 - b^2
*c))))/c - (b*c*((2*A*e)/5 + (2*B*d)/5))/(4*a*c^2 - b^2*c) - (4*A*c^2*d)/(5*(4*a*c^2 - b^2*c)) + (4*B*a*c*e)/(
5*(4*a*c^2 - b^2*c))) + (a*((2*c^2*((2*A*e)/5 + (2*B*d)/5))/(4*a*c^2 - b^2*c) - (2*B*b*c*e)/(5*(4*a*c^2 - b^2*
c))))/c - (2*A*b*c*d)/(5*(4*a*c^2 - b^2*c)))/(a + b*x + c*x^2)^(5/2) + (x*((b*((16*c^3*(A*e + B*d))/(15*(4*a*c
^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c^2*e)/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c + (2*c*(32*A*c^2*d + 8*B*b
^2*e - 20*A*b*c*e + 8*B*a*c*e - 20*B*b*c*d))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*b*c^2*(A*e + B*d))/(15*
(4*a*c^2 - b^2*c)*(4*a*c - b^2)) + (16*B*a*c^2*e)/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2))) + (a*((16*c^3*(A*e + B
*d))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c^2*e)/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c + (b*(32*A*
c^2*d + 8*B*b^2*e - 20*A*b*c*e + 8*B*a*c*e - 20*B*b*c*d))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)))/(a + b*x + c*x
^2)^(3/2) + ((b*c*(256*A*c^2*d + 56*B*b^2*e - 128*A*b*c*e + 32*B*a*c*e - 128*B*b*c*d))/(15*(4*a*c^2 - b^2*c)*(
4*a*c - b^2)^2) + (2*c^2*x*(256*A*c^2*d + 56*B*b^2*e - 128*A*b*c*e + 32*B*a*c*e - 128*B*b*c*d))/(15*(4*a*c^2 -
b^2*c)*(4*a*c - b^2)^2))/(a + b*x + c*x^2)^(1/2) - ((4*A*c*e - 2*B*b*e + 4*B*c*d)/(15*c*(4*a*c - b^2)) + (4*B
*e*x)/(15*(4*a*c - b^2)))/(a + b*x + c*x^2)^(3/2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)/(c*x**2+b*x+a)**(7/2),x)

[Out]

Timed out

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