3.23.18 \(\int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{9/2}} \, dx\) [2218]

Optimal. Leaf size=95 \[ -\frac {2 (B d-A e) (a+b x)^{5/2}}{7 e (b d-a e) (d+e x)^{7/2}}+\frac {2 (5 b B d+2 A b e-7 a B e) (a+b x)^{5/2}}{35 e (b d-a e)^2 (d+e x)^{5/2}} \]

[Out]

-2/7*(-A*e+B*d)*(b*x+a)^(5/2)/e/(-a*e+b*d)/(e*x+d)^(7/2)+2/35*(2*A*b*e-7*B*a*e+5*B*b*d)*(b*x+a)^(5/2)/e/(-a*e+
b*d)^2/(e*x+d)^(5/2)

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Rubi [A]
time = 0.03, antiderivative size = 95, 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 = {79, 37} \begin {gather*} \frac {2 (a+b x)^{5/2} (-7 a B e+2 A b e+5 b B d)}{35 e (d+e x)^{5/2} (b d-a e)^2}-\frac {2 (a+b x)^{5/2} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(9/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(5/2))/(7*e*(b*d - a*e)*(d + e*x)^(7/2)) + (2*(5*b*B*d + 2*A*b*e - 7*a*B*e)*(a + b*x
)^(5/2))/(35*e*(b*d - a*e)^2*(d + e*x)^(5/2))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rubi steps

\begin {align*} \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{9/2}} \, dx &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{7 e (b d-a e) (d+e x)^{7/2}}+\frac {(5 b B d+2 A b e-7 a B e) \int \frac {(a+b x)^{3/2}}{(d+e x)^{7/2}} \, dx}{7 e (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{5/2}}{7 e (b d-a e) (d+e x)^{7/2}}+\frac {2 (5 b B d+2 A b e-7 a B e) (a+b x)^{5/2}}{35 e (b d-a e)^2 (d+e x)^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 66, normalized size = 0.69 \begin {gather*} \frac {2 (a+b x)^{5/2} (B (-2 a d+5 b d x-7 a e x)+A (7 b d-5 a e+2 b e x))}{35 (b d-a e)^2 (d+e x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(9/2),x]

[Out]

(2*(a + b*x)^(5/2)*(B*(-2*a*d + 5*b*d*x - 7*a*e*x) + A*(7*b*d - 5*a*e + 2*b*e*x)))/(35*(b*d - a*e)^2*(d + e*x)
^(7/2))

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Maple [A]
time = 0.10, size = 107, normalized size = 1.13

method	result	size
gosper	\(-\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (-2 A b e x +7 B a e x -5 B b d x +5 A a e -7 A b d +2 B a d \right )}{35 \left (e x +d \right )^{\frac {7}{2}} \left (a^{2} e^{2}-2 b e a d +b^{2} d^{2}\right )}\)	\(74\)
default	\(-\frac {2 \left (-2 A \,b^{2} e \,x^{2}+7 B a b e \,x^{2}-5 B \,b^{2} d \,x^{2}+3 A a b e x -7 A \,b^{2} d x +7 B \,a^{2} e x -3 B a b d x +5 a^{2} A e -7 A a b d +2 B \,a^{2} d \right ) \left (b x +a \right )^{\frac {3}{2}}}{35 \left (e x +d \right )^{\frac {7}{2}} \left (a e -b d \right )^{2}}\)	\(107\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(9/2),x,method=_RETURNVERBOSE)

[Out]

-2/35*(-2*A*b^2*e*x^2+7*B*a*b*e*x^2-5*B*b^2*d*x^2+3*A*a*b*e*x-7*A*b^2*d*x+7*B*a^2*e*x-3*B*a*b*d*x+5*A*a^2*e-7*
A*a*b*d+2*B*a^2*d)*(b*x+a)^(3/2)/(e*x+d)^(7/2)/(a*e-b*d)^2

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*d-%e*a>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (88) = 176\).
time = 9.41, size = 303, normalized size = 3.19 \begin {gather*} \frac {2 \, {\left (5 \, B b^{3} d x^{3} + {\left (8 \, B a b^{2} + 7 \, A b^{3}\right )} d x^{2} + {\left (B a^{2} b + 14 \, A a b^{2}\right )} d x - {\left (2 \, B a^{3} - 7 \, A a^{2} b\right )} d - {\left (5 \, A a^{3} + {\left (7 \, B a b^{2} - 2 \, A b^{3}\right )} x^{3} + {\left (14 \, B a^{2} b + A a b^{2}\right )} x^{2} + {\left (7 \, B a^{3} + 8 \, A a^{2} b\right )} x\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{35 \, {\left (b^{2} d^{6} + a^{2} x^{4} e^{6} - 2 \, {\left (a b d x^{4} - 2 \, a^{2} d x^{3}\right )} e^{5} + {\left (b^{2} d^{2} x^{4} - 8 \, a b d^{2} x^{3} + 6 \, a^{2} d^{2} x^{2}\right )} e^{4} + 4 \, {\left (b^{2} d^{3} x^{3} - 3 \, a b d^{3} x^{2} + a^{2} d^{3} x\right )} e^{3} + {\left (6 \, b^{2} d^{4} x^{2} - 8 \, a b d^{4} x + a^{2} d^{4}\right )} e^{2} + 2 \, {\left (2 \, b^{2} d^{5} x - a b d^{5}\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(9/2),x, algorithm="fricas")

[Out]

2/35*(5*B*b^3*d*x^3 + (8*B*a*b^2 + 7*A*b^3)*d*x^2 + (B*a^2*b + 14*A*a*b^2)*d*x - (2*B*a^3 - 7*A*a^2*b)*d - (5*
A*a^3 + (7*B*a*b^2 - 2*A*b^3)*x^3 + (14*B*a^2*b + A*a*b^2)*x^2 + (7*B*a^3 + 8*A*a^2*b)*x)*e)*sqrt(b*x + a)*sqr
t(x*e + d)/(b^2*d^6 + a^2*x^4*e^6 - 2*(a*b*d*x^4 - 2*a^2*d*x^3)*e^5 + (b^2*d^2*x^4 - 8*a*b*d^2*x^3 + 6*a^2*d^2
*x^2)*e^4 + 4*(b^2*d^3*x^3 - 3*a*b*d^3*x^2 + a^2*d^3*x)*e^3 + (6*b^2*d^4*x^2 - 8*a*b*d^4*x + a^2*d^4)*e^2 + 2*
(2*b^2*d^5*x - a*b*d^5)*e)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(3/2)*(B*x+A)/(e*x+d)**(9/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (88) = 176\).
time = 1.64, size = 268, normalized size = 2.82 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {5}{2}} {\left (\frac {{\left (5 \, B b^{9} d^{2} {\left | b \right |} e^{3} - 12 \, B a b^{8} d {\left | b \right |} e^{4} + 2 \, A b^{9} d {\left | b \right |} e^{4} + 7 \, B a^{2} b^{7} {\left | b \right |} e^{5} - 2 \, A a b^{8} {\left | b \right |} e^{5}\right )} {\left (b x + a\right )}}{b^{5} d^{3} e^{3} - 3 \, a b^{4} d^{2} e^{4} + 3 \, a^{2} b^{3} d e^{5} - a^{3} b^{2} e^{6}} - \frac {7 \, {\left (B a b^{9} d^{2} {\left | b \right |} e^{3} - A b^{10} d^{2} {\left | b \right |} e^{3} - 2 \, B a^{2} b^{8} d {\left | b \right |} e^{4} + 2 \, A a b^{9} d {\left | b \right |} e^{4} + B a^{3} b^{7} {\left | b \right |} e^{5} - A a^{2} b^{8} {\left | b \right |} e^{5}\right )}}{b^{5} d^{3} e^{3} - 3 \, a b^{4} d^{2} e^{4} + 3 \, a^{2} b^{3} d e^{5} - a^{3} b^{2} e^{6}}\right )}}{35 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(9/2),x, algorithm="giac")

[Out]

2/35*(b*x + a)^(5/2)*((5*B*b^9*d^2*abs(b)*e^3 - 12*B*a*b^8*d*abs(b)*e^4 + 2*A*b^9*d*abs(b)*e^4 + 7*B*a^2*b^7*a
bs(b)*e^5 - 2*A*a*b^8*abs(b)*e^5)*(b*x + a)/(b^5*d^3*e^3 - 3*a*b^4*d^2*e^4 + 3*a^2*b^3*d*e^5 - a^3*b^2*e^6) -
7*(B*a*b^9*d^2*abs(b)*e^3 - A*b^10*d^2*abs(b)*e^3 - 2*B*a^2*b^8*d*abs(b)*e^4 + 2*A*a*b^9*d*abs(b)*e^4 + B*a^3*
b^7*abs(b)*e^5 - A*a^2*b^8*abs(b)*e^5)/(b^5*d^3*e^3 - 3*a*b^4*d^2*e^4 + 3*a^2*b^3*d*e^5 - a^3*b^2*e^6))/(b^2*d
+ (b*x + a)*b*e - a*b*e)^(7/2)

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Mupad [B]
time = 2.14, size = 257, normalized size = 2.71 \begin {gather*} -\frac {\sqrt {d+e\,x}\,\left (\frac {\sqrt {a+b\,x}\,\left (10\,A\,a^3\,e+4\,B\,a^3\,d-14\,A\,a^2\,b\,d\right )}{35\,e^4\,{\left (a\,e-b\,d\right )}^2}-\frac {x^3\,\sqrt {a+b\,x}\,\left (4\,A\,b^3\,e+10\,B\,b^3\,d-14\,B\,a\,b^2\,e\right )}{35\,e^4\,{\left (a\,e-b\,d\right )}^2}+\frac {x\,\sqrt {a+b\,x}\,\left (14\,B\,a^3\,e-28\,A\,a\,b^2\,d+16\,A\,a^2\,b\,e-2\,B\,a^2\,b\,d\right )}{35\,e^4\,{\left (a\,e-b\,d\right )}^2}-\frac {x^2\,\sqrt {a+b\,x}\,\left (14\,A\,b^3\,d-2\,A\,a\,b^2\,e+16\,B\,a\,b^2\,d-28\,B\,a^2\,b\,e\right )}{35\,e^4\,{\left (a\,e-b\,d\right )}^2}\right )}{x^4+\frac {d^4}{e^4}+\frac {4\,d\,x^3}{e}+\frac {4\,d^3\,x}{e^3}+\frac {6\,d^2\,x^2}{e^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^(3/2))/(d + e*x)^(9/2),x)

[Out]

-((d + e*x)^(1/2)*(((a + b*x)^(1/2)*(10*A*a^3*e + 4*B*a^3*d - 14*A*a^2*b*d))/(35*e^4*(a*e - b*d)^2) - (x^3*(a
+ b*x)^(1/2)*(4*A*b^3*e + 10*B*b^3*d - 14*B*a*b^2*e))/(35*e^4*(a*e - b*d)^2) + (x*(a + b*x)^(1/2)*(14*B*a^3*e
- 28*A*a*b^2*d + 16*A*a^2*b*e - 2*B*a^2*b*d))/(35*e^4*(a*e - b*d)^2) - (x^2*(a + b*x)^(1/2)*(14*A*b^3*d - 2*A*
a*b^2*e + 16*B*a*b^2*d - 28*B*a^2*b*e))/(35*e^4*(a*e - b*d)^2)))/(x^4 + d^4/e^4 + (4*d*x^3)/e + (4*d^3*x)/e^3
+ (6*d^2*x^2)/e^2)

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