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

Optimal. Leaf size=111 \[ \frac {2 (b+2 c x) \left (-4 b c (A e+B d)+8 A c^2 d+b^2 B e\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {2 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}} \]

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Rubi [A]  time = 0.10, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {777, 613} \begin {gather*} \frac {2 (b+2 c x) \left (-4 b c (A e+B d)+8 A c^2 d+b^2 B e\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {2 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.05, size = 107, normalized size = 0.96 \begin {gather*} -\frac {2 \left (A \left (b^3 (d+3 e x)-6 b^2 c x (d-2 e x)+8 b c^2 x^2 (e x-3 d)-16 c^3 d x^3\right )+b B x \left (3 b^2 (d-e x)-2 b c x (e x-6 d)+8 c^2 d x^2\right )\right )}{3 b^4 (x (b+c x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.46, size = 149, normalized size = 1.34 \begin {gather*} \frac {2 \sqrt {b x+c x^2} \left (-A b^3 d-3 A b^3 e x+6 A b^2 c d x-12 A b^2 c e x^2+24 A b c^2 d x^2-8 A b c^2 e x^3+16 A c^3 d x^3-3 b^3 B d x+3 b^3 B e x^2-12 b^2 B c d x^2+2 b^2 B c e x^3-8 b B c^2 d x^3\right )}{3 b^4 x^2 (b+c x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[b*x + c*x^2]*(-(A*b^3*d) - 3*b^3*B*d*x + 6*A*b^2*c*d*x - 3*A*b^3*e*x - 12*b^2*B*c*d*x^2 + 24*A*b*c^2*d
*x^2 + 3*b^3*B*e*x^2 - 12*A*b^2*c*e*x^2 - 8*b*B*c^2*d*x^3 + 16*A*c^3*d*x^3 + 2*b^2*B*c*e*x^3 - 8*A*b*c^2*e*x^3
))/(3*b^4*x^2*(b + c*x)^2)

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fricas [A]  time = 0.41, size = 152, normalized size = 1.37 \begin {gather*} -\frac {2 \, {\left (A b^{3} d + 2 \, {\left (4 \, {\left (B b c^{2} - 2 \, A c^{3}\right )} d - {\left (B b^{2} c - 4 \, A b c^{2}\right )} e\right )} x^{3} + 3 \, {\left (4 \, {\left (B b^{2} c - 2 \, A b c^{2}\right )} d - {\left (B b^{3} - 4 \, A b^{2} c\right )} e\right )} x^{2} + 3 \, {\left (A b^{3} e + {\left (B b^{3} - 2 \, A b^{2} c\right )} d\right )} x\right )} \sqrt {c x^{2} + b x}}{3 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(A*b^3*d + 2*(4*(B*b*c^2 - 2*A*c^3)*d - (B*b^2*c - 4*A*b*c^2)*e)*x^3 + 3*(4*(B*b^2*c - 2*A*b*c^2)*d - (B*
b^3 - 4*A*b^2*c)*e)*x^2 + 3*(A*b^3*e + (B*b^3 - 2*A*b^2*c)*d)*x)*sqrt(c*x^2 + b*x)/(b^4*c^2*x^4 + 2*b^5*c*x^3
+ b^6*x^2)

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giac [A]  time = 0.30, size = 132, normalized size = 1.19 \begin {gather*} -\frac {2 \, {\left ({\left (x {\left (\frac {2 \, {\left (4 \, B b c^{2} d - 8 \, A c^{3} d - B b^{2} c e + 4 \, A b c^{2} e\right )} x}{b^{4}} + \frac {3 \, {\left (4 \, B b^{2} c d - 8 \, A b c^{2} d - B b^{3} e + 4 \, A b^{2} c e\right )}}{b^{4}}\right )} + \frac {3 \, {\left (B b^{3} d - 2 \, A b^{2} c d + A b^{3} e\right )}}{b^{4}}\right )} x + \frac {A d}{b}\right )}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 141, normalized size = 1.27 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (8 A b \,c^{2} e \,x^{3}-16 A \,c^{3} d \,x^{3}-2 B \,b^{2} c e \,x^{3}+8 B b \,c^{2} d \,x^{3}+12 A \,b^{2} c e \,x^{2}-24 A b \,c^{2} d \,x^{2}-3 B \,b^{3} e \,x^{2}+12 B \,b^{2} c d \,x^{2}+3 A \,b^{3} e x -6 A \,b^{2} c d x +3 B \,b^{3} d x +A d \,b^{3}\right ) x}{3 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.54, size = 211, normalized size = 1.90 \begin {gather*} -\frac {4 \, A c d x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}} + \frac {32 \, A c^{2} d x}{3 \, \sqrt {c x^{2} + b x} b^{4}} + \frac {4 \, B e x}{3 \, \sqrt {c x^{2} + b x} b^{2}} - \frac {2 \, B e x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} - \frac {2 \, A d}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} + \frac {16 \, A c d}{3 \, \sqrt {c x^{2} + b x} b^{3}} + \frac {2 \, B e}{3 \, \sqrt {c x^{2} + b x} b c} + \frac {2 \, {\left (B d + A e\right )} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} - \frac {16 \, {\left (B d + A e\right )} c x}{3 \, \sqrt {c x^{2} + b x} b^{3}} - \frac {8 \, {\left (B d + A e\right )}}{3 \, \sqrt {c x^{2} + b x} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)/(c*x^2+b*x)^(5/2),x, algorithm="maxima")

[Out]

-4/3*A*c*d*x/((c*x^2 + b*x)^(3/2)*b^2) + 32/3*A*c^2*d*x/(sqrt(c*x^2 + b*x)*b^4) + 4/3*B*e*x/(sqrt(c*x^2 + b*x)
*b^2) - 2/3*B*e*x/((c*x^2 + b*x)^(3/2)*c) - 2/3*A*d/((c*x^2 + b*x)^(3/2)*b) + 16/3*A*c*d/(sqrt(c*x^2 + b*x)*b^
3) + 2/3*B*e/(sqrt(c*x^2 + b*x)*b*c) + 2/3*(B*d + A*e)*x/((c*x^2 + b*x)^(3/2)*b) - 16/3*(B*d + A*e)*c*x/(sqrt(
c*x^2 + b*x)*b^3) - 8/3*(B*d + A*e)/(sqrt(c*x^2 + b*x)*b^2)

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mupad [B]  time = 1.73, size = 134, normalized size = 1.21 \begin {gather*} -\frac {2\,\left (A\,b^3\,d+3\,A\,b^3\,e\,x+3\,B\,b^3\,d\,x-16\,A\,c^3\,d\,x^3-3\,B\,b^3\,e\,x^2-24\,A\,b\,c^2\,d\,x^2+12\,A\,b^2\,c\,e\,x^2+12\,B\,b^2\,c\,d\,x^2+8\,A\,b\,c^2\,e\,x^3+8\,B\,b\,c^2\,d\,x^3-2\,B\,b^2\,c\,e\,x^3-6\,A\,b^2\,c\,d\,x\right )}{3\,b^4\,{\left (c\,x^2+b\,x\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*(A*b^3*d + 3*A*b^3*e*x + 3*B*b^3*d*x - 16*A*c^3*d*x^3 - 3*B*b^3*e*x^2 - 24*A*b*c^2*d*x^2 + 12*A*b^2*c*e*x^
2 + 12*B*b^2*c*d*x^2 + 8*A*b*c^2*e*x^3 + 8*B*b*c^2*d*x^3 - 2*B*b^2*c*e*x^3 - 6*A*b^2*c*d*x))/(3*b^4*(b*x + c*x
^2)^(3/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x\right ) \left (d + e x\right )}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)/(c*x**2+b*x)**(5/2),x)

[Out]

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

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