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

Optimal. Leaf size=122 \[ -\frac {2 \sqrt {d+e x} (-A c e-b B e+3 B c d)}{e^4}-\frac {2 (B d (3 c d-2 b e)-A e (2 c d-b e))}{e^4 \sqrt {d+e x}}+\frac {2 d (B d-A e) (c d-b e)}{3 e^4 (d+e x)^{3/2}}+\frac {2 B c (d+e x)^{3/2}}{3 e^4} \]

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Rubi [A]  time = 0.07, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {771} \begin {gather*} -\frac {2 \sqrt {d+e x} (-A c e-b B e+3 B c d)}{e^4}-\frac {2 (B d (3 c d-2 b e)-A e (2 c d-b e))}{e^4 \sqrt {d+e x}}+\frac {2 d (B d-A e) (c d-b e)}{3 e^4 (d+e x)^{3/2}}+\frac {2 B c (d+e x)^{3/2}}{3 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d*(B*d - A*e)*(c*d - b*e))/(3*e^4*(d + e*x)^(3/2)) - (2*(B*d*(3*c*d - 2*b*e) - A*e*(2*c*d - b*e)))/(e^4*Sqr
t[d + e*x]) - (2*(3*B*c*d - b*B*e - A*c*e)*Sqrt[d + e*x])/e^4 + (2*B*c*(d + e*x)^(3/2))/(3*e^4)

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

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Mathematica [A]  time = 0.05, size = 110, normalized size = 0.90 \begin {gather*} \frac {2 \left (A e \left (c \left (8 d^2+12 d e x+3 e^2 x^2\right )-b e (2 d+3 e x)\right )+B \left (b e \left (8 d^2+12 d e x+3 e^2 x^2\right )+c \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )\right )\right )}{3 e^4 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.09, size = 138, normalized size = 1.13 \begin {gather*} \frac {2 \left (-3 A b e^2 (d+e x)+A b d e^2-A c d^2 e+6 A c d e (d+e x)+3 A c e (d+e x)^2-b B d^2 e+6 b B d e (d+e x)+3 b B e (d+e x)^2+B c d^3-9 B c d^2 (d+e x)-9 B c d (d+e x)^2+B c (d+e x)^3\right )}{3 e^4 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(B*c*d^3 - b*B*d^2*e - A*c*d^2*e + A*b*d*e^2 - 9*B*c*d^2*(d + e*x) + 6*b*B*d*e*(d + e*x) + 6*A*c*d*e*(d + e
*x) - 3*A*b*e^2*(d + e*x) - 9*B*c*d*(d + e*x)^2 + 3*b*B*e*(d + e*x)^2 + 3*A*c*e*(d + e*x)^2 + B*c*(d + e*x)^3)
)/(3*e^4*(d + e*x)^(3/2))

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fricas [A]  time = 0.41, size = 128, normalized size = 1.05 \begin {gather*} \frac {2 \, {\left (B c e^{3} x^{3} - 16 \, B c d^{3} - 2 \, A b d e^{2} + 8 \, {\left (B b + A c\right )} d^{2} e - 3 \, {\left (2 \, B c d e^{2} - {\left (B b + A c\right )} e^{3}\right )} x^{2} - 3 \, {\left (8 \, B c d^{2} e + A b e^{3} - 4 \, {\left (B b + A c\right )} d e^{2}\right )} x\right )} \sqrt {e x + d}}{3 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.18, size = 156, normalized size = 1.28 \begin {gather*} \frac {2}{3} \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} B c e^{8} - 9 \, \sqrt {x e + d} B c d e^{8} + 3 \, \sqrt {x e + d} B b e^{9} + 3 \, \sqrt {x e + d} A c e^{9}\right )} e^{\left (-12\right )} - \frac {2 \, {\left (9 \, {\left (x e + d\right )} B c d^{2} - B c d^{3} - 6 \, {\left (x e + d\right )} B b d e - 6 \, {\left (x e + d\right )} A c d e + B b d^{2} e + A c d^{2} e + 3 \, {\left (x e + d\right )} A b e^{2} - A b d e^{2}\right )} e^{\left (-4\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)/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

2/3*((x*e + d)^(3/2)*B*c*e^8 - 9*sqrt(x*e + d)*B*c*d*e^8 + 3*sqrt(x*e + d)*B*b*e^9 + 3*sqrt(x*e + d)*A*c*e^9)*
e^(-12) - 2/3*(9*(x*e + d)*B*c*d^2 - B*c*d^3 - 6*(x*e + d)*B*b*d*e - 6*(x*e + d)*A*c*d*e + B*b*d^2*e + A*c*d^2
*e + 3*(x*e + d)*A*b*e^2 - A*b*d*e^2)*e^(-4)/(x*e + d)^(3/2)

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maple [A]  time = 0.05, size = 121, normalized size = 0.99 \begin {gather*} -\frac {2 \left (-B c \,x^{3} e^{3}-3 A c \,e^{3} x^{2}-3 B b \,e^{3} x^{2}+6 B c d \,e^{2} x^{2}+3 A b \,e^{3} x -12 A c d \,e^{2} x -12 B b d \,e^{2} x +24 B c \,d^{2} e x +2 A b d \,e^{2}-8 A c \,d^{2} e -8 B b \,d^{2} e +16 B c \,d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.64, size = 116, normalized size = 0.95 \begin {gather*} \frac {2 \, {\left (\frac {{\left (e x + d\right )}^{\frac {3}{2}} B c - 3 \, {\left (3 \, B c d - {\left (B b + A c\right )} e\right )} \sqrt {e x + d}}{e^{3}} + \frac {B c d^{3} + A b d e^{2} - {\left (B b + A c\right )} d^{2} e - 3 \, {\left (3 \, B c d^{2} + A b e^{2} - 2 \, {\left (B b + A c\right )} d e\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{3}}\right )}}{3 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.09, size = 137, normalized size = 1.12 \begin {gather*} \frac {2\,B\,c\,{\left (d+e\,x\right )}^3+2\,B\,c\,d^3+2\,A\,b\,d\,e^2-2\,A\,c\,d^2\,e-2\,B\,b\,d^2\,e-6\,A\,b\,e^2\,\left (d+e\,x\right )+6\,A\,c\,e\,{\left (d+e\,x\right )}^2+6\,B\,b\,e\,{\left (d+e\,x\right )}^2-18\,B\,c\,d\,{\left (d+e\,x\right )}^2-18\,B\,c\,d^2\,\left (d+e\,x\right )+12\,A\,c\,d\,e\,\left (d+e\,x\right )+12\,B\,b\,d\,e\,\left (d+e\,x\right )}{3\,e^4\,{\left (d+e\,x\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*B*c*(d + e*x)^3 + 2*B*c*d^3 + 2*A*b*d*e^2 - 2*A*c*d^2*e - 2*B*b*d^2*e - 6*A*b*e^2*(d + e*x) + 6*A*c*e*(d +
e*x)^2 + 6*B*b*e*(d + e*x)^2 - 18*B*c*d*(d + e*x)^2 - 18*B*c*d^2*(d + e*x) + 12*A*c*d*e*(d + e*x) + 12*B*b*d*e
*(d + e*x))/(3*e^4*(d + e*x)^(3/2))

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sympy [A]  time = 1.56, size = 539, normalized size = 4.42 \begin {gather*} \begin {cases} - \frac {4 A b d e^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {6 A b e^{3} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {16 A c d^{2} e}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {24 A c d e^{2} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {6 A c e^{3} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {16 B b d^{2} e}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {24 B b d e^{2} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {6 B b e^{3} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {32 B c d^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {48 B c d^{2} e x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {12 B c d e^{2} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {2 B c e^{3} x^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {\frac {A b x^{2}}{2} + \frac {A c x^{3}}{3} + \frac {B b x^{3}}{3} + \frac {B c x^{4}}{4}}{d^{\frac {5}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-4*A*b*d*e**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 6*A*b*e**3*x/(3*d*e**4*sqrt(d + e
*x) + 3*e**5*x*sqrt(d + e*x)) + 16*A*c*d**2*e/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 24*A*c*d*e**
2*x/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 6*A*c*e**3*x**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqr
t(d + e*x)) + 16*B*b*d**2*e/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 24*B*b*d*e**2*x/(3*d*e**4*sqrt
(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 6*B*b*e**3*x**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 32*B
*c*d**3/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 48*B*c*d**2*e*x/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x
*sqrt(d + e*x)) - 12*B*c*d*e**2*x**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 2*B*c*e**3*x**3/(3*d*
e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)), Ne(e, 0)), ((A*b*x**2/2 + A*c*x**3/3 + B*b*x**3/3 + B*c*x**4/4)/
d**(5/2), True))

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