3.18.25 \(\int \frac {(a+b x) (A+B x)}{(d+e x)^{7/2}} \, dx\) [1725]

Optimal. Leaf size=81 \[ -\frac {2 (b d-a e) (B d-A e)}{5 e^3 (d+e x)^{5/2}}+\frac {2 (2 b B d-A b e-a B e)}{3 e^3 (d+e x)^{3/2}}-\frac {2 b B}{e^3 \sqrt {d+e x}} \]

[Out]

-2/5*(-a*e+b*d)*(-A*e+B*d)/e^3/(e*x+d)^(5/2)+2/3*(-A*b*e-B*a*e+2*B*b*d)/e^3/(e*x+d)^(3/2)-2*b*B/e^3/(e*x+d)^(1
/2)

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Rubi [A]
time = 0.02, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \begin {gather*} \frac {2 (-a B e-A b e+2 b B d)}{3 e^3 (d+e x)^{3/2}}-\frac {2 (b d-a e) (B d-A e)}{5 e^3 (d+e x)^{5/2}}-\frac {2 b B}{e^3 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

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Mathematica [A]
time = 0.07, size = 68, normalized size = 0.84 \begin {gather*} -\frac {2 \left (A b e (2 d+5 e x)+a e (2 B d+3 A e+5 B e x)+b B \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )}{15 e^3 (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(A*b*e*(2*d + 5*e*x) + a*e*(2*B*d + 3*A*e + 5*B*e*x) + b*B*(8*d^2 + 20*d*e*x + 15*e^2*x^2)))/(15*e^3*(d +
e*x)^(5/2))

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Maple [A]
time = 0.08, size = 75, normalized size = 0.93

method	result	size
gosper	\(-\frac {2 \left (15 b B \,x^{2} e^{2}+5 A b \,e^{2} x +5 B a \,e^{2} x +20 B b d e x +3 A a \,e^{2}+2 A b d e +2 B a d e +8 B b \,d^{2}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{3}}\)	\(73\)
trager	\(-\frac {2 \left (15 b B \,x^{2} e^{2}+5 A b \,e^{2} x +5 B a \,e^{2} x +20 B b d e x +3 A a \,e^{2}+2 A b d e +2 B a d e +8 B b \,d^{2}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{3}}\)	\(73\)
derivativedivides	\(\frac {-\frac {2 \left (A a \,e^{2}-A b d e -B a d e +B b \,d^{2}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 B b}{\sqrt {e x +d}}-\frac {2 \left (A b e +B a e -2 B b d \right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{3}}\)	\(75\)
default	\(\frac {-\frac {2 \left (A a \,e^{2}-A b d e -B a d e +B b \,d^{2}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 B b}{\sqrt {e x +d}}-\frac {2 \left (A b e +B a e -2 B b d \right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{3}}\)	\(75\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/e^3*(-1/5*(A*a*e^2-A*b*d*e-B*a*d*e+B*b*d^2)/(e*x+d)^(5/2)-B*b/(e*x+d)^(1/2)-1/3*(A*b*e+B*a*e-2*B*b*d)/(e*x+d
)^(3/2))

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Maxima [A]
time = 0.55, size = 78, normalized size = 0.96 \begin {gather*} -\frac {2 \, {\left (15 \, {\left (x e + d\right )}^{2} B b + 3 \, B b d^{2} + 3 \, A a e^{2} - 5 \, {\left (2 \, B b d - B a e - A b e\right )} {\left (x e + d\right )} - 3 \, {\left (B a e + A b e\right )} d\right )} e^{\left (-3\right )}}{15 \, {\left (x e + d\right )}^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 1.65, size = 94, normalized size = 1.16 \begin {gather*} -\frac {2 \, {\left (8 \, B b d^{2} + {\left (15 \, B b x^{2} + 3 \, A a + 5 \, {\left (B a + A b\right )} x\right )} e^{2} + 2 \, {\left (10 \, B b d x + {\left (B a + A b\right )} d\right )} e\right )} \sqrt {x e + d}}{15 \, {\left (x^{3} e^{6} + 3 \, d x^{2} e^{5} + 3 \, d^{2} x e^{4} + d^{3} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(8*B*b*d^2 + (15*B*b*x^2 + 3*A*a + 5*(B*a + A*b)*x)*e^2 + 2*(10*B*b*d*x + (B*a + A*b)*d)*e)*sqrt(x*e + d
)/(x^3*e^6 + 3*d*x^2*e^5 + 3*d^2*x*e^4 + d^3*e^3)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 520 vs. \(2 (85) = 170\).
time = 0.76, size = 520, normalized size = 6.42 \begin {gather*} \begin {cases} - \frac {6 A a e^{2}}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {4 A b d e}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {10 A b e^{2} x}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {4 B a d e}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {10 B a e^{2} x}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {16 B b d^{2}}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {40 B b d e x}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {30 B b e^{2} x^{2}}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {A a x + \frac {A b x^{2}}{2} + \frac {B a x^{2}}{2} + \frac {B b x^{3}}{3}}{d^{\frac {7}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-6*A*a*e**2/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) -
4*A*b*d*e/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 10*A*b*e**2
*x/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 4*B*a*d*e/(15*d**2*
e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 10*B*a*e**2*x/(15*d**2*e**3*sqr
t(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 16*B*b*d**2/(15*d**2*e**3*sqrt(d + e*x)
+ 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 40*B*b*d*e*x/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e
**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 30*B*b*e**2*x**2/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x
*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)), Ne(e, 0)), ((A*a*x + A*b*x**2/2 + B*a*x**2/2 + B*b*x**3/3)/d**(7
/2), True))

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Giac [A]
time = 0.58, size = 87, normalized size = 1.07 \begin {gather*} -\frac {2 \, {\left (15 \, {\left (x e + d\right )}^{2} B b - 10 \, {\left (x e + d\right )} B b d + 3 \, B b d^{2} + 5 \, {\left (x e + d\right )} B a e + 5 \, {\left (x e + d\right )} A b e - 3 \, B a d e - 3 \, A b d e + 3 \, A a e^{2}\right )} e^{\left (-3\right )}}{15 \, {\left (x e + d\right )}^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(15*(x*e + d)^2*B*b - 10*(x*e + d)*B*b*d + 3*B*b*d^2 + 5*(x*e + d)*B*a*e + 5*(x*e + d)*A*b*e - 3*B*a*d*e
- 3*A*b*d*e + 3*A*a*e^2)*e^(-3)/(x*e + d)^(5/2)

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Mupad [B]
time = 0.08, size = 72, normalized size = 0.89 \begin {gather*} -\frac {\left (d+e\,x\right )\,\left (\frac {2\,A\,b\,e}{3}+\frac {2\,B\,a\,e}{3}-\frac {4\,B\,b\,d}{3}\right )+2\,B\,b\,{\left (d+e\,x\right )}^2+\frac {2\,A\,a\,e^2}{5}+\frac {2\,B\,b\,d^2}{5}-\frac {2\,A\,b\,d\,e}{5}-\frac {2\,B\,a\,d\,e}{5}}{e^3\,{\left (d+e\,x\right )}^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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