3.18.41 \(\int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{7/2}} \, dx\) [1741]

Optimal. Leaf size=169 \[ -\frac {2 (b d-a e)^3 (B d-A e)}{5 e^5 (d+e x)^{5/2}}+\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e)}{3 e^5 (d+e x)^{3/2}}-\frac {6 b (b d-a e) (2 b B d-A b e-a B e)}{e^5 \sqrt {d+e x}}-\frac {2 b^2 (4 b B d-A b e-3 a B e) \sqrt {d+e x}}{e^5}+\frac {2 b^3 B (d+e x)^{3/2}}{3 e^5} \]

[Out]

-2/5*(-a*e+b*d)^3*(-A*e+B*d)/e^5/(e*x+d)^(5/2)+2/3*(-a*e+b*d)^2*(-3*A*b*e-B*a*e+4*B*b*d)/e^5/(e*x+d)^(3/2)+2/3
*b^3*B*(e*x+d)^(3/2)/e^5-6*b*(-a*e+b*d)*(-A*b*e-B*a*e+2*B*b*d)/e^5/(e*x+d)^(1/2)-2*b^2*(-A*b*e-3*B*a*e+4*B*b*d
)*(e*x+d)^(1/2)/e^5

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Rubi [A]
time = 0.05, antiderivative size = 169, 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 = {78} \begin {gather*} -\frac {2 b^2 \sqrt {d+e x} (-3 a B e-A b e+4 b B d)}{e^5}-\frac {6 b (b d-a e) (-a B e-A b e+2 b B d)}{e^5 \sqrt {d+e x}}+\frac {2 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{3 e^5 (d+e x)^{3/2}}-\frac {2 (b d-a e)^3 (B d-A e)}{5 e^5 (d+e x)^{5/2}}+\frac {2 b^3 B (d+e x)^{3/2}}{3 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^3*(A + B*x))/(d + e*x)^(7/2),x]

[Out]

(-2*(b*d - a*e)^3*(B*d - A*e))/(5*e^5*(d + e*x)^(5/2)) + (2*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e))/(3*e^5*
(d + e*x)^(3/2)) - (6*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e))/(e^5*Sqrt[d + e*x]) - (2*b^2*(4*b*B*d - A*b*e -
3*a*B*e)*Sqrt[d + e*x])/e^5 + (2*b^3*B*(d + e*x)^(3/2))/(3*e^5)

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)^3 (A+B x)}{(d+e x)^{7/2}} \, dx &=\int \left (\frac {(-b d+a e)^3 (-B d+A e)}{e^4 (d+e x)^{7/2}}+\frac {(-b d+a e)^2 (-4 b B d+3 A b e+a B e)}{e^4 (d+e x)^{5/2}}-\frac {3 b (b d-a e) (-2 b B d+A b e+a B e)}{e^4 (d+e x)^{3/2}}+\frac {b^2 (-4 b B d+A b e+3 a B e)}{e^4 \sqrt {d+e x}}+\frac {b^3 B \sqrt {d+e x}}{e^4}\right ) \, dx\\ &=-\frac {2 (b d-a e)^3 (B d-A e)}{5 e^5 (d+e x)^{5/2}}+\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e)}{3 e^5 (d+e x)^{3/2}}-\frac {6 b (b d-a e) (2 b B d-A b e-a B e)}{e^5 \sqrt {d+e x}}-\frac {2 b^2 (4 b B d-A b e-3 a B e) \sqrt {d+e x}}{e^5}+\frac {2 b^3 B (d+e x)^{3/2}}{3 e^5}\\ \end {align*}

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Mathematica [A]
time = 0.21, size = 226, normalized size = 1.34 \begin {gather*} -\frac {2 \left (a^3 e^3 (2 B d+3 A e+5 B e x)+3 a^2 b e^2 \left (A e (2 d+5 e x)+B \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )-3 a b^2 e \left (-A e \left (8 d^2+20 d e x+15 e^2 x^2\right )+3 B \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )\right )+b^3 \left (-3 A e \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )+B \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )\right )\right )}{15 e^5 (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^3*(A + B*x))/(d + e*x)^(7/2),x]

[Out]

(-2*(a^3*e^3*(2*B*d + 3*A*e + 5*B*e*x) + 3*a^2*b*e^2*(A*e*(2*d + 5*e*x) + B*(8*d^2 + 20*d*e*x + 15*e^2*x^2)) -
3*a*b^2*e*(-(A*e*(8*d^2 + 20*d*e*x + 15*e^2*x^2)) + 3*B*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3)) + b
^3*(-3*A*e*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3) + B*(128*d^4 + 320*d^3*e*x + 240*d^2*e^2*x^2 + 40*
d*e^3*x^3 - 5*e^4*x^4))))/(15*e^5*(d + e*x)^(5/2))

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Maple [A]
time = 0.10, size = 286, normalized size = 1.69

method	result	size
risch	\(\frac {2 b^{2} \left (b B x e +3 A b e +9 B a e -11 B b d \right ) \sqrt {e x +d}}{3 e^{5}}-\frac {2 \left (45 A \,b^{2} e^{3} x^{2}+45 B a b \,e^{3} x^{2}-90 B \,b^{2} d \,e^{2} x^{2}+15 A a b \,e^{3} x +75 A \,b^{2} d \,e^{2} x +5 B \,a^{2} e^{3} x +65 B a b d \,e^{2} x -160 B \,b^{2} d^{2} e x +3 a^{2} A \,e^{3}+9 A a b d \,e^{2}+33 A \,b^{2} d^{2} e +2 B \,a^{2} d \,e^{2}+26 B a b \,d^{2} e -73 b^{2} B \,d^{3}\right ) \left (a e -b d \right )}{15 e^{5} \sqrt {e x +d}\, \left (e^{2} x^{2}+2 d x e +d^{2}\right )}\)	\(220\)
derivativedivides	\(\frac {\frac {2 b^{3} B \left (e x +d \right )^{\frac {3}{2}}}{3}+2 A \,b^{3} e \sqrt {e x +d}+6 B a \,b^{2} e \sqrt {e x +d}-8 B \,b^{3} d \sqrt {e x +d}-\frac {2 \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +B \,a^{3} e^{3}-6 B \,a^{2} b d \,e^{2}+9 B a \,b^{2} d^{2} e -4 b^{3} B \,d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (a^{3} A \,e^{4}-3 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}-A \,b^{3} d^{3} e -B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}-3 B a \,b^{2} d^{3} e +b^{3} B \,d^{4}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {6 b \left (A a b \,e^{2}-A \,b^{2} d e +B \,a^{2} e^{2}-3 B a b d e +2 b^{2} B \,d^{2}\right )}{\sqrt {e x +d}}}{e^{5}}\)	\(286\)
default	\(\frac {\frac {2 b^{3} B \left (e x +d \right )^{\frac {3}{2}}}{3}+2 A \,b^{3} e \sqrt {e x +d}+6 B a \,b^{2} e \sqrt {e x +d}-8 B \,b^{3} d \sqrt {e x +d}-\frac {2 \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +B \,a^{3} e^{3}-6 B \,a^{2} b d \,e^{2}+9 B a \,b^{2} d^{2} e -4 b^{3} B \,d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (a^{3} A \,e^{4}-3 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}-A \,b^{3} d^{3} e -B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}-3 B a \,b^{2} d^{3} e +b^{3} B \,d^{4}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {6 b \left (A a b \,e^{2}-A \,b^{2} d e +B \,a^{2} e^{2}-3 B a b d e +2 b^{2} B \,d^{2}\right )}{\sqrt {e x +d}}}{e^{5}}\)	\(286\)
gosper	\(-\frac {2 \left (-5 b^{3} B \,x^{4} e^{4}-15 A \,b^{3} e^{4} x^{3}-45 B a \,b^{2} e^{4} x^{3}+40 B \,b^{3} d \,e^{3} x^{3}+45 A a \,b^{2} e^{4} x^{2}-90 A \,b^{3} d \,e^{3} x^{2}+45 B \,a^{2} b \,e^{4} x^{2}-270 B a \,b^{2} d \,e^{3} x^{2}+240 B \,b^{3} d^{2} e^{2} x^{2}+15 A \,a^{2} b \,e^{4} x +60 A a \,b^{2} d \,e^{3} x -120 A \,b^{3} d^{2} e^{2} x +5 B \,a^{3} e^{4} x +60 B \,a^{2} b d \,e^{3} x -360 B a \,b^{2} d^{2} e^{2} x +320 B \,b^{3} d^{3} e x +3 a^{3} A \,e^{4}+6 A \,a^{2} b d \,e^{3}+24 A a \,b^{2} d^{2} e^{2}-48 A \,b^{3} d^{3} e +2 B \,a^{3} d \,e^{3}+24 B \,a^{2} b \,d^{2} e^{2}-144 B a \,b^{2} d^{3} e +128 b^{3} B \,d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{5}}\)	\(301\)
trager	\(-\frac {2 \left (-5 b^{3} B \,x^{4} e^{4}-15 A \,b^{3} e^{4} x^{3}-45 B a \,b^{2} e^{4} x^{3}+40 B \,b^{3} d \,e^{3} x^{3}+45 A a \,b^{2} e^{4} x^{2}-90 A \,b^{3} d \,e^{3} x^{2}+45 B \,a^{2} b \,e^{4} x^{2}-270 B a \,b^{2} d \,e^{3} x^{2}+240 B \,b^{3} d^{2} e^{2} x^{2}+15 A \,a^{2} b \,e^{4} x +60 A a \,b^{2} d \,e^{3} x -120 A \,b^{3} d^{2} e^{2} x +5 B \,a^{3} e^{4} x +60 B \,a^{2} b d \,e^{3} x -360 B a \,b^{2} d^{2} e^{2} x +320 B \,b^{3} d^{3} e x +3 a^{3} A \,e^{4}+6 A \,a^{2} b d \,e^{3}+24 A a \,b^{2} d^{2} e^{2}-48 A \,b^{3} d^{3} e +2 B \,a^{3} d \,e^{3}+24 B \,a^{2} b \,d^{2} e^{2}-144 B a \,b^{2} d^{3} e +128 b^{3} B \,d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{5}}\)	\(301\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(B*x+A)/(e*x+d)^(7/2),x,method=_RETURNVERBOSE)

[Out]

2/e^5*(1/3*b^3*B*(e*x+d)^(3/2)+A*b^3*e*(e*x+d)^(1/2)+3*B*a*b^2*e*(e*x+d)^(1/2)-4*B*b^3*d*(e*x+d)^(1/2)-1/3*(3*
A*a^2*b*e^3-6*A*a*b^2*d*e^2+3*A*b^3*d^2*e+B*a^3*e^3-6*B*a^2*b*d*e^2+9*B*a*b^2*d^2*e-4*B*b^3*d^3)/(e*x+d)^(3/2)
-1/5*(A*a^3*e^4-3*A*a^2*b*d*e^3+3*A*a*b^2*d^2*e^2-A*b^3*d^3*e-B*a^3*d*e^3+3*B*a^2*b*d^2*e^2-3*B*a*b^2*d^3*e+B*
b^3*d^4)/(e*x+d)^(5/2)-3*b*(A*a*b*e^2-A*b^2*d*e+B*a^2*e^2-3*B*a*b*d*e+2*B*b^2*d^2)/(e*x+d)^(1/2))

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Maxima [A]
time = 0.38, size = 285, normalized size = 1.69 \begin {gather*} \frac {2}{15} \, {\left (5 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} B b^{3} - 3 \, {\left (4 \, B b^{3} d - 3 \, B a b^{2} e - A b^{3} e\right )} \sqrt {x e + d}\right )} e^{\left (-4\right )} - \frac {{\left (3 \, B b^{3} d^{4} + 3 \, A a^{3} e^{4} - 3 \, {\left (3 \, B a b^{2} e + A b^{3} e\right )} d^{3} + 45 \, {\left (2 \, B b^{3} d^{2} + B a^{2} b e^{2} + A a b^{2} e^{2} - {\left (3 \, B a b^{2} e + A b^{3} e\right )} d\right )} {\left (x e + d\right )}^{2} + 9 \, {\left (B a^{2} b e^{2} + A a b^{2} e^{2}\right )} d^{2} - 5 \, {\left (4 \, B b^{3} d^{3} - B a^{3} e^{3} - 3 \, A a^{2} b e^{3} - 3 \, {\left (3 \, B a b^{2} e + A b^{3} e\right )} d^{2} + 6 \, {\left (B a^{2} b e^{2} + A a b^{2} e^{2}\right )} d\right )} {\left (x e + d\right )} - 3 \, {\left (B a^{3} e^{3} + 3 \, A a^{2} b e^{3}\right )} d\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{\frac {5}{2}}}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

2/15*(5*((x*e + d)^(3/2)*B*b^3 - 3*(4*B*b^3*d - 3*B*a*b^2*e - A*b^3*e)*sqrt(x*e + d))*e^(-4) - (3*B*b^3*d^4 +
3*A*a^3*e^4 - 3*(3*B*a*b^2*e + A*b^3*e)*d^3 + 45*(2*B*b^3*d^2 + B*a^2*b*e^2 + A*a*b^2*e^2 - (3*B*a*b^2*e + A*b
^3*e)*d)*(x*e + d)^2 + 9*(B*a^2*b*e^2 + A*a*b^2*e^2)*d^2 - 5*(4*B*b^3*d^3 - B*a^3*e^3 - 3*A*a^2*b*e^3 - 3*(3*B
*a*b^2*e + A*b^3*e)*d^2 + 6*(B*a^2*b*e^2 + A*a*b^2*e^2)*d)*(x*e + d) - 3*(B*a^3*e^3 + 3*A*a^2*b*e^3)*d)*e^(-4)
/(x*e + d)^(5/2))*e^(-1)

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Fricas [A]
time = 0.99, size = 275, normalized size = 1.63 \begin {gather*} -\frac {2 \, {\left (128 \, B b^{3} d^{4} - {\left (5 \, B b^{3} x^{4} - 3 \, A a^{3} + 15 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} - 45 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} - 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )} e^{4} + 2 \, {\left (20 \, B b^{3} d x^{3} - 45 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d x^{2} + 30 \, {\left (B a^{2} b + A a b^{2}\right )} d x + {\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} e^{3} + 24 \, {\left (10 \, B b^{3} d^{2} x^{2} - 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} x + {\left (B a^{2} b + A a b^{2}\right )} d^{2}\right )} e^{2} + 16 \, {\left (20 \, B b^{3} d^{3} x - 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3}\right )} e\right )} \sqrt {x e + d}}{15 \, {\left (x^{3} e^{8} + 3 \, d x^{2} e^{7} + 3 \, d^{2} x e^{6} + d^{3} e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

-2/15*(128*B*b^3*d^4 - (5*B*b^3*x^4 - 3*A*a^3 + 15*(3*B*a*b^2 + A*b^3)*x^3 - 45*(B*a^2*b + A*a*b^2)*x^2 - 5*(B
*a^3 + 3*A*a^2*b)*x)*e^4 + 2*(20*B*b^3*d*x^3 - 45*(3*B*a*b^2 + A*b^3)*d*x^2 + 30*(B*a^2*b + A*a*b^2)*d*x + (B*
a^3 + 3*A*a^2*b)*d)*e^3 + 24*(10*B*b^3*d^2*x^2 - 5*(3*B*a*b^2 + A*b^3)*d^2*x + (B*a^2*b + A*a*b^2)*d^2)*e^2 +
16*(20*B*b^3*d^3*x - 3*(3*B*a*b^2 + A*b^3)*d^3)*e)*sqrt(x*e + d)/(x^3*e^8 + 3*d*x^2*e^7 + 3*d^2*x*e^6 + d^3*e^
5)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1654 vs. \(2 (167) = 334\).
time = 0.80, size = 1654, normalized size = 9.79 \begin {gather*} \begin {cases} - \frac {6 A a^{3} e^{4}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {12 A a^{2} b d e^{3}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {30 A a^{2} b e^{4} x}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {48 A a b^{2} d^{2} e^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {120 A a b^{2} d e^{3} x}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {90 A a b^{2} e^{4} x^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {96 A b^{3} d^{3} e}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {240 A b^{3} d^{2} e^{2} x}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {180 A b^{3} d e^{3} x^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {30 A b^{3} e^{4} x^{3}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {4 B a^{3} d e^{3}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {10 B a^{3} e^{4} x}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {48 B a^{2} b d^{2} e^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {120 B a^{2} b d e^{3} x}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {90 B a^{2} b e^{4} x^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {288 B a b^{2} d^{3} e}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {720 B a b^{2} d^{2} e^{2} x}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {540 B a b^{2} d e^{3} x^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {90 B a b^{2} e^{4} x^{3}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {256 B b^{3} d^{4}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {640 B b^{3} d^{3} e x}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {480 B b^{3} d^{2} e^{2} x^{2}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} - \frac {80 B b^{3} d e^{3} x^{3}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} + \frac {10 B b^{3} e^{4} x^{4}}{15 d^{2} e^{5} \sqrt {d + e x} + 30 d e^{6} x \sqrt {d + e x} + 15 e^{7} x^{2} \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {A a^{3} x + \frac {3 A a^{2} b x^{2}}{2} + A a b^{2} x^{3} + \frac {A b^{3} x^{4}}{4} + \frac {B a^{3} x^{2}}{2} + B a^{2} b x^{3} + \frac {3 B a b^{2} x^{4}}{4} + \frac {B b^{3} x^{5}}{5}}{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)**3*(B*x+A)/(e*x+d)**(7/2),x)

[Out]

Piecewise((-6*A*a**3*e**4/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)
) - 12*A*a**2*b*d*e**3/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) -
30*A*a**2*b*e**4*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 48
*A*a*b**2*d**2*e**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 12
0*A*a*b**2*d*e**3*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 90
*A*a*b**2*e**4*x**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 96
*A*b**3*d**3*e/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 240*A*b
**3*d**2*e**2*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 180*A*
b**3*d*e**3*x**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 30*A*
b**3*e**4*x**3/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 4*B*a**
3*d*e**3/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 10*B*a**3*e**
4*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 48*B*a**2*b*d**2*e
**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 120*B*a**2*b*d*e**
3*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 90*B*a**2*b*e**4*x
**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 288*B*a*b**2*d**3*
e/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 720*B*a*b**2*d**2*e*
*2*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 540*B*a*b**2*d*e*
*3*x**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 90*B*a*b**2*e*
*4*x**3/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 256*B*b**3*d**
4/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 640*B*b**3*d**3*e*x/
(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 480*B*b**3*d**2*e**2*x
**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 80*B*b**3*d*e**3*x
**3/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 10*B*b**3*e**4*x**
4/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)), Ne(e, 0)), ((A*a**3*x
+ 3*A*a**2*b*x**2/2 + A*a*b**2*x**3 + A*b**3*x**4/4 + B*a**3*x**2/2 + B*a**2*b*x**3 + 3*B*a*b**2*x**4/4 + B*b
**3*x**5/5)/d**(7/2), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 364 vs. \(2 (163) = 326\).
time = 0.71, size = 364, normalized size = 2.15 \begin {gather*} \frac {2}{3} \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} B b^{3} e^{10} - 12 \, \sqrt {x e + d} B b^{3} d e^{10} + 9 \, \sqrt {x e + d} B a b^{2} e^{11} + 3 \, \sqrt {x e + d} A b^{3} e^{11}\right )} e^{\left (-15\right )} - \frac {2 \, {\left (90 \, {\left (x e + d\right )}^{2} B b^{3} d^{2} - 20 \, {\left (x e + d\right )} B b^{3} d^{3} + 3 \, B b^{3} d^{4} - 135 \, {\left (x e + d\right )}^{2} B a b^{2} d e - 45 \, {\left (x e + d\right )}^{2} A b^{3} d e + 45 \, {\left (x e + d\right )} B a b^{2} d^{2} e + 15 \, {\left (x e + d\right )} A b^{3} d^{2} e - 9 \, B a b^{2} d^{3} e - 3 \, A b^{3} d^{3} e + 45 \, {\left (x e + d\right )}^{2} B a^{2} b e^{2} + 45 \, {\left (x e + d\right )}^{2} A a b^{2} e^{2} - 30 \, {\left (x e + d\right )} B a^{2} b d e^{2} - 30 \, {\left (x e + d\right )} A a b^{2} d e^{2} + 9 \, B a^{2} b d^{2} e^{2} + 9 \, A a b^{2} d^{2} e^{2} + 5 \, {\left (x e + d\right )} B a^{3} e^{3} + 15 \, {\left (x e + d\right )} A a^{2} b e^{3} - 3 \, B a^{3} d e^{3} - 9 \, A a^{2} b d e^{3} + 3 \, A a^{3} e^{4}\right )} e^{\left (-5\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)^3*(B*x+A)/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

2/3*((x*e + d)^(3/2)*B*b^3*e^10 - 12*sqrt(x*e + d)*B*b^3*d*e^10 + 9*sqrt(x*e + d)*B*a*b^2*e^11 + 3*sqrt(x*e +
d)*A*b^3*e^11)*e^(-15) - 2/15*(90*(x*e + d)^2*B*b^3*d^2 - 20*(x*e + d)*B*b^3*d^3 + 3*B*b^3*d^4 - 135*(x*e + d)
^2*B*a*b^2*d*e - 45*(x*e + d)^2*A*b^3*d*e + 45*(x*e + d)*B*a*b^2*d^2*e + 15*(x*e + d)*A*b^3*d^2*e - 9*B*a*b^2*
d^3*e - 3*A*b^3*d^3*e + 45*(x*e + d)^2*B*a^2*b*e^2 + 45*(x*e + d)^2*A*a*b^2*e^2 - 30*(x*e + d)*B*a^2*b*d*e^2 -
30*(x*e + d)*A*a*b^2*d*e^2 + 9*B*a^2*b*d^2*e^2 + 9*A*a*b^2*d^2*e^2 + 5*(x*e + d)*B*a^3*e^3 + 15*(x*e + d)*A*a
^2*b*e^3 - 3*B*a^3*d*e^3 - 9*A*a^2*b*d*e^3 + 3*A*a^3*e^4)*e^(-5)/(x*e + d)^(5/2)

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Mupad [B]
time = 0.13, size = 300, normalized size = 1.78 \begin {gather*} -\frac {2\,\left (2\,B\,a^3\,d\,e^3+5\,B\,a^3\,e^4\,x+3\,A\,a^3\,e^4+24\,B\,a^2\,b\,d^2\,e^2+60\,B\,a^2\,b\,d\,e^3\,x+6\,A\,a^2\,b\,d\,e^3+45\,B\,a^2\,b\,e^4\,x^2+15\,A\,a^2\,b\,e^4\,x-144\,B\,a\,b^2\,d^3\,e-360\,B\,a\,b^2\,d^2\,e^2\,x+24\,A\,a\,b^2\,d^2\,e^2-270\,B\,a\,b^2\,d\,e^3\,x^2+60\,A\,a\,b^2\,d\,e^3\,x-45\,B\,a\,b^2\,e^4\,x^3+45\,A\,a\,b^2\,e^4\,x^2+128\,B\,b^3\,d^4+320\,B\,b^3\,d^3\,e\,x-48\,A\,b^3\,d^3\,e+240\,B\,b^3\,d^2\,e^2\,x^2-120\,A\,b^3\,d^2\,e^2\,x+40\,B\,b^3\,d\,e^3\,x^3-90\,A\,b^3\,d\,e^3\,x^2-5\,B\,b^3\,e^4\,x^4-15\,A\,b^3\,e^4\,x^3\right )}{15\,e^5\,{\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)^3)/(d + e*x)^(7/2),x)

[Out]

-(2*(3*A*a^3*e^4 + 128*B*b^3*d^4 - 48*A*b^3*d^3*e + 2*B*a^3*d*e^3 + 5*B*a^3*e^4*x - 15*A*b^3*e^4*x^3 - 5*B*b^3
*e^4*x^4 + 320*B*b^3*d^3*e*x + 24*A*a*b^2*d^2*e^2 + 24*B*a^2*b*d^2*e^2 + 45*A*a*b^2*e^4*x^2 + 45*B*a^2*b*e^4*x
^2 - 45*B*a*b^2*e^4*x^3 - 120*A*b^3*d^2*e^2*x - 90*A*b^3*d*e^3*x^2 + 40*B*b^3*d*e^3*x^3 + 240*B*b^3*d^2*e^2*x^
2 + 6*A*a^2*b*d*e^3 - 144*B*a*b^2*d^3*e + 15*A*a^2*b*e^4*x + 60*A*a*b^2*d*e^3*x + 60*B*a^2*b*d*e^3*x - 360*B*a
*b^2*d^2*e^2*x - 270*B*a*b^2*d*e^3*x^2))/(15*e^5*(d + e*x)^(5/2))

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