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

Optimal. Leaf size=112 \[ -\frac {2 \left (a B e^2-2 A c d e+3 B c d^2\right )}{e^4 \sqrt {d+e x}}+\frac {2 \left (a e^2+c d^2\right ) (B d-A e)}{3 e^4 (d+e x)^{3/2}}-\frac {2 c \sqrt {d+e x} (3 B d-A e)}{e^4}+\frac {2 B c (d+e x)^{3/2}}{3 e^4} \]

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Rubi [A]  time = 0.05, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {772} \begin {gather*} -\frac {2 \left (a B e^2-2 A c d e+3 B c d^2\right )}{e^4 \sqrt {d+e x}}+\frac {2 \left (a e^2+c d^2\right ) (B d-A e)}{3 e^4 (d+e x)^{3/2}}-\frac {2 c \sqrt {d+e x} (3 B d-A e)}{e^4}+\frac {2 B c (d+e x)^{3/2}}{3 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )}{(d+e x)^{5/2}} \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2+a e^2\right )}{e^3 (d+e x)^{5/2}}+\frac {3 B c d^2-2 A c d e+a B e^2}{e^3 (d+e x)^{3/2}}+\frac {c (-3 B d+A e)}{e^3 \sqrt {d+e x}}+\frac {B c \sqrt {d+e x}}{e^3}\right ) \, dx\\ &=\frac {2 (B d-A e) \left (c d^2+a e^2\right )}{3 e^4 (d+e x)^{3/2}}-\frac {2 \left (3 B c d^2-2 A c d e+a B e^2\right )}{e^4 \sqrt {d+e x}}-\frac {2 c (3 B d-A 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 = 0.07, size = 94, normalized size = 0.84 \begin {gather*} -\frac {2 \left (a A e^3+a B e^2 (2 d+3 e x)-A c e \left (8 d^2+12 d e x+3 e^2 x^2\right )+B c \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )\right )}{3 e^4 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a*A*e^3 + a*B*e^2*(2*d + 3*e*x) - A*c*e*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + B*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 = 0.08, size = 114, normalized size = 1.02 \begin {gather*} \frac {2 \left (-a A e^3-3 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 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)*(a + c*x^2))/(d + e*x)^(5/2),x]

[Out]

(2*(B*c*d^3 - A*c*d^2*e + a*B*d*e^2 - a*A*e^3 - 9*B*c*d^2*(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*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 = 120, normalized size = 1.07 \begin {gather*} \frac {2 \, {\left (B c e^{3} x^{3} - 16 \, B c d^{3} + 8 \, A c d^{2} e - 2 \, B a d e^{2} - A a e^{3} - 3 \, {\left (2 \, B c d e^{2} - A c e^{3}\right )} x^{2} - 3 \, {\left (8 \, B c d^{2} e - 4 \, A c d e^{2} + B a e^{3}\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+a)/(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

2/3*(B*c*e^3*x^3 - 16*B*c*d^3 + 8*A*c*d^2*e - 2*B*a*d*e^2 - A*a*e^3 - 3*(2*B*c*d*e^2 - A*c*e^3)*x^2 - 3*(8*B*c
*d^2*e - 4*A*c*d*e^2 + B*a*e^3)*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 = 0.17, size = 126, normalized size = 1.12 \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} 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 )} A c d e + A c d^{2} e + 3 \, {\left (x e + d\right )} B a e^{2} - B a d e^{2} + A a e^{3}\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+a)/(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)*A*c*e^9)*e^(-12) - 2/3*(9*(x*e + d)
*B*c*d^2 - B*c*d^3 - 6*(x*e + d)*A*c*d*e + A*c*d^2*e + 3*(x*e + d)*B*a*e^2 - B*a*d*e^2 + A*a*e^3)*e^(-4)/(x*e
+ d)^(3/2)

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maple [A]  time = 0.05, size = 100, normalized size = 0.89 \begin {gather*} -\frac {2 \left (-B c \,x^{3} e^{3}-3 A c \,e^{3} x^{2}+6 B c d \,e^{2} x^{2}-12 A c d \,e^{2} x +3 B a \,e^{3} x +24 B c \,d^{2} e x +a A \,e^{3}-8 A c \,d^{2} e +2 a B d \,e^{2}+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+a)/(e*x+d)^(5/2),x)

[Out]

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

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maxima [A]  time = 0.50, size = 108, normalized size = 0.96 \begin {gather*} \frac {2 \, {\left (\frac {{\left (e x + d\right )}^{\frac {3}{2}} B c - 3 \, {\left (3 \, B c d - A c e\right )} \sqrt {e x + d}}{e^{3}} + \frac {B c d^{3} - A c d^{2} e + B a d e^{2} - A a e^{3} - 3 \, {\left (3 \, B c d^{2} - 2 \, A c d e + B a e^{2}\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+a)/(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.77, size = 113, normalized size = 1.01 \begin {gather*} \frac {2\,B\,c\,{\left (d+e\,x\right )}^3-2\,A\,a\,e^3+2\,B\,c\,d^3+2\,B\,a\,d\,e^2-2\,A\,c\,d^2\,e-6\,B\,a\,e^2\,\left (d+e\,x\right )+6\,A\,c\,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 )}{3\,e^4\,{\left (d+e\,x\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 1.44, size = 449, normalized size = 4.01 \begin {gather*} \begin {cases} - \frac {2 A a e^{3}}{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 {4 B a d e^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {6 B a e^{3} x}{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 {A a x + \frac {A c x^{3}}{3} + \frac {B a x^{2}}{2} + \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+a)/(e*x+d)**(5/2),x)

[Out]

Piecewise((-2*A*a*e**3/(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*sqrt(d + e*x)) - 4*B*a*d*e**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(
d + e*x)) - 6*B*a*e**3*x/(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*a*x + A*c*x**3/3 + B*a*x**2/2 + B*c*x**4/4)/d**(5/2), True))

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