3.2233 \(\int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{15/2}} \, dx\)

Optimal. Leaf size=201 \[ \frac {16 b^2 (a+b x)^{7/2} (-13 a B e+6 A b e+7 b B d)}{9009 e (d+e x)^{7/2} (b d-a e)^4}+\frac {8 b (a+b x)^{7/2} (-13 a B e+6 A b e+7 b B d)}{1287 e (d+e x)^{9/2} (b d-a e)^3}+\frac {2 (a+b x)^{7/2} (-13 a B e+6 A b e+7 b B d)}{143 e (d+e x)^{11/2} (b d-a e)^2}-\frac {2 (a+b x)^{7/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)} \]

[Out]

-2/13*(-A*e+B*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)/(e*x+d)^(13/2)+2/143*(6*A*b*e-13*B*a*e+7*B*b*d)*(b*x+a)^(7/2)/e/(-
a*e+b*d)^2/(e*x+d)^(11/2)+8/1287*b*(6*A*b*e-13*B*a*e+7*B*b*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)^3/(e*x+d)^(9/2)+16/90
09*b^2*(6*A*b*e-13*B*a*e+7*B*b*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)^4/(e*x+d)^(7/2)

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Rubi [A]  time = 0.13, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {78, 45, 37} \[ \frac {16 b^2 (a+b x)^{7/2} (-13 a B e+6 A b e+7 b B d)}{9009 e (d+e x)^{7/2} (b d-a e)^4}+\frac {8 b (a+b x)^{7/2} (-13 a B e+6 A b e+7 b B d)}{1287 e (d+e x)^{9/2} (b d-a e)^3}+\frac {2 (a+b x)^{7/2} (-13 a B e+6 A b e+7 b B d)}{143 e (d+e x)^{11/2} (b d-a e)^2}-\frac {2 (a+b x)^{7/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(15/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(7/2))/(13*e*(b*d - a*e)*(d + e*x)^(13/2)) + (2*(7*b*B*d + 6*A*b*e - 13*a*B*e)*(a +
b*x)^(7/2))/(143*e*(b*d - a*e)^2*(d + e*x)^(11/2)) + (8*b*(7*b*B*d + 6*A*b*e - 13*a*B*e)*(a + b*x)^(7/2))/(128
7*e*(b*d - a*e)^3*(d + e*x)^(9/2)) + (16*b^2*(7*b*B*d + 6*A*b*e - 13*a*B*e)*(a + b*x)^(7/2))/(9009*e*(b*d - a*
e)^4*(d + e*x)^(7/2))

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 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 78

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] || LtQ
[p, n]))))

Rubi steps

\begin {align*} \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{15/2}} \, dx &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {(7 b B d+6 A b e-13 a B e) \int \frac {(a+b x)^{5/2}}{(d+e x)^{13/2}} \, dx}{13 e (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {2 (7 b B d+6 A b e-13 a B e) (a+b x)^{7/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac {(4 b (7 b B d+6 A b e-13 a B e)) \int \frac {(a+b x)^{5/2}}{(d+e x)^{11/2}} \, dx}{143 e (b d-a e)^2}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {2 (7 b B d+6 A b e-13 a B e) (a+b x)^{7/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac {8 b (7 b B d+6 A b e-13 a B e) (a+b x)^{7/2}}{1287 e (b d-a e)^3 (d+e x)^{9/2}}+\frac {\left (8 b^2 (7 b B d+6 A b e-13 a B e)\right ) \int \frac {(a+b x)^{5/2}}{(d+e x)^{9/2}} \, dx}{1287 e (b d-a e)^3}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {2 (7 b B d+6 A b e-13 a B e) (a+b x)^{7/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac {8 b (7 b B d+6 A b e-13 a B e) (a+b x)^{7/2}}{1287 e (b d-a e)^3 (d+e x)^{9/2}}+\frac {16 b^2 (7 b B d+6 A b e-13 a B e) (a+b x)^{7/2}}{9009 e (b d-a e)^4 (d+e x)^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.41, size = 114, normalized size = 0.57 \[ \frac {2 (a+b x)^{7/2} \left (693 (B d-A e)-\frac {(d+e x) \left (4 b (d+e x) (-7 a e+9 b d+2 b e x)+63 (b d-a e)^2\right ) (-13 a B e+6 A b e+7 b B d)}{(b d-a e)^3}\right )}{9009 e (d+e x)^{13/2} (a e-b d)} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(15/2),x]

[Out]

(2*(a + b*x)^(7/2)*(693*(B*d - A*e) - ((7*b*B*d + 6*A*b*e - 13*a*B*e)*(d + e*x)*(63*(b*d - a*e)^2 + 4*b*(d + e
*x)*(9*b*d - 7*a*e + 2*b*e*x)))/(b*d - a*e)^3))/(9009*e*(-(b*d) + a*e)*(d + e*x)^(13/2))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(15/2),x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 8.99, size = 938, normalized size = 4.67 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(15/2),x, algorithm="giac")

[Out]

2/9009*((4*(b*x + a)*(2*(7*B*b^16*d^3*abs(b)*e^8 - 27*B*a*b^15*d^2*abs(b)*e^9 + 6*A*b^16*d^2*abs(b)*e^9 + 33*B
*a^2*b^14*d*abs(b)*e^10 - 12*A*a*b^15*d*abs(b)*e^10 - 13*B*a^3*b^13*abs(b)*e^11 + 6*A*a^2*b^14*abs(b)*e^11)*(b
*x + a)/(b^8*d^6*e^6 - 6*a*b^7*d^5*e^7 + 15*a^2*b^6*d^4*e^8 - 20*a^3*b^5*d^3*e^9 + 15*a^4*b^4*d^2*e^10 - 6*a^5
*b^3*d*e^11 + a^6*b^2*e^12) + 13*(7*B*b^17*d^4*abs(b)*e^7 - 34*B*a*b^16*d^3*abs(b)*e^8 + 6*A*b^17*d^3*abs(b)*e
^8 + 60*B*a^2*b^15*d^2*abs(b)*e^9 - 18*A*a*b^16*d^2*abs(b)*e^9 - 46*B*a^3*b^14*d*abs(b)*e^10 + 18*A*a^2*b^15*d
*abs(b)*e^10 + 13*B*a^4*b^13*abs(b)*e^11 - 6*A*a^3*b^14*abs(b)*e^11)/(b^8*d^6*e^6 - 6*a*b^7*d^5*e^7 + 15*a^2*b
^6*d^4*e^8 - 20*a^3*b^5*d^3*e^9 + 15*a^4*b^4*d^2*e^10 - 6*a^5*b^3*d*e^11 + a^6*b^2*e^12)) + 143*(7*B*b^18*d^5*
abs(b)*e^6 - 41*B*a*b^17*d^4*abs(b)*e^7 + 6*A*b^18*d^4*abs(b)*e^7 + 94*B*a^2*b^16*d^3*abs(b)*e^8 - 24*A*a*b^17
*d^3*abs(b)*e^8 - 106*B*a^3*b^15*d^2*abs(b)*e^9 + 36*A*a^2*b^16*d^2*abs(b)*e^9 + 59*B*a^4*b^14*d*abs(b)*e^10 -
24*A*a^3*b^15*d*abs(b)*e^10 - 13*B*a^5*b^13*abs(b)*e^11 + 6*A*a^4*b^14*abs(b)*e^11)/(b^8*d^6*e^6 - 6*a*b^7*d^
5*e^7 + 15*a^2*b^6*d^4*e^8 - 20*a^3*b^5*d^3*e^9 + 15*a^4*b^4*d^2*e^10 - 6*a^5*b^3*d*e^11 + a^6*b^2*e^12))*(b*x
+ a) - 1287*(B*a*b^18*d^5*abs(b)*e^6 - A*b^19*d^5*abs(b)*e^6 - 5*B*a^2*b^17*d^4*abs(b)*e^7 + 5*A*a*b^18*d^4*a
bs(b)*e^7 + 10*B*a^3*b^16*d^3*abs(b)*e^8 - 10*A*a^2*b^17*d^3*abs(b)*e^8 - 10*B*a^4*b^15*d^2*abs(b)*e^9 + 10*A*
a^3*b^16*d^2*abs(b)*e^9 + 5*B*a^5*b^14*d*abs(b)*e^10 - 5*A*a^4*b^15*d*abs(b)*e^10 - B*a^6*b^13*abs(b)*e^11 + A
*a^5*b^14*abs(b)*e^11)/(b^8*d^6*e^6 - 6*a*b^7*d^5*e^7 + 15*a^2*b^6*d^4*e^8 - 20*a^3*b^5*d^3*e^9 + 15*a^4*b^4*d
^2*e^10 - 6*a^5*b^3*d*e^11 + a^6*b^2*e^12))*(b*x + a)^(7/2)/(b^2*d + (b*x + a)*b*e - a*b*e)^(13/2)

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maple [A]  time = 0.01, size = 322, normalized size = 1.60 \[ -\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (-48 A \,b^{3} e^{3} x^{3}+104 B a \,b^{2} e^{3} x^{3}-56 B \,b^{3} d \,e^{2} x^{3}+168 A a \,b^{2} e^{3} x^{2}-312 A \,b^{3} d \,e^{2} x^{2}-364 B \,a^{2} b \,e^{3} x^{2}+872 B a \,b^{2} d \,e^{2} x^{2}-364 B \,b^{3} d^{2} e \,x^{2}-378 A \,a^{2} b \,e^{3} x +1092 A a \,b^{2} d \,e^{2} x -858 A \,b^{3} d^{2} e x +819 B \,a^{3} e^{3} x -2807 B \,a^{2} b d \,e^{2} x +3133 B a \,b^{2} d^{2} e x -1001 B \,b^{3} d^{3} x +693 A \,a^{3} e^{3}-2457 A \,a^{2} b d \,e^{2}+3003 A a \,b^{2} d^{2} e -1287 A \,b^{3} d^{3}+126 B \,a^{3} d \,e^{2}-364 B \,a^{2} b \,d^{2} e +286 B a \,b^{2} d^{3}\right )}{9009 \left (e x +d \right )^{\frac {13}{2}} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/9009*(b*x+a)^(7/2)*(-48*A*b^3*e^3*x^3+104*B*a*b^2*e^3*x^3-56*B*b^3*d*e^2*x^3+168*A*a*b^2*e^3*x^2-312*A*b^3*
d*e^2*x^2-364*B*a^2*b*e^3*x^2+872*B*a*b^2*d*e^2*x^2-364*B*b^3*d^2*e*x^2-378*A*a^2*b*e^3*x+1092*A*a*b^2*d*e^2*x
-858*A*b^3*d^2*e*x+819*B*a^3*e^3*x-2807*B*a^2*b*d*e^2*x+3133*B*a*b^2*d^2*e*x-1001*B*b^3*d^3*x+693*A*a^3*e^3-24
57*A*a^2*b*d*e^2+3003*A*a*b^2*d^2*e-1287*A*b^3*d^3+126*B*a^3*d*e^2-364*B*a^2*b*d^2*e+286*B*a*b^2*d^3)/(e*x+d)^
(13/2)/(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(15/2),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(a*e-b*d>0)', see `assume?` for
more details)Is a*e-b*d zero or nonzero?

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mupad [B]  time = 3.08, size = 706, normalized size = 3.51 \[ \frac {\sqrt {d+e\,x}\,\left (\frac {x^3\,\sqrt {a+b\,x}\,\left (-2938\,B\,a^4\,b^2\,e^3+11470\,B\,a^3\,b^3\,d\,e^2-30\,A\,a^3\,b^3\,e^3-15886\,B\,a^2\,b^4\,d^2\,e+234\,A\,a^2\,b^4\,d\,e^2+5434\,B\,a\,b^5\,d^3-858\,A\,a\,b^5\,d^2\,e+2574\,A\,b^6\,d^3\right )}{9009\,e^7\,{\left (a\,e-b\,d\right )}^4}-\frac {\sqrt {a+b\,x}\,\left (252\,B\,a^6\,d\,e^2+1386\,A\,a^6\,e^3-728\,B\,a^5\,b\,d^2\,e-4914\,A\,a^5\,b\,d\,e^2+572\,B\,a^4\,b^2\,d^3+6006\,A\,a^4\,b^2\,d^2\,e-2574\,A\,a^3\,b^3\,d^3\right )}{9009\,e^7\,{\left (a\,e-b\,d\right )}^4}-\frac {x\,\sqrt {a+b\,x}\,\left (1638\,B\,a^6\,e^3-4858\,B\,a^5\,b\,d\,e^2+3402\,A\,a^5\,b\,e^3+4082\,B\,a^4\,b^2\,d^2\,e-12558\,A\,a^4\,b^2\,d\,e^2-286\,B\,a^3\,b^3\,d^3+16302\,A\,a^3\,b^3\,d^2\,e-7722\,A\,a^2\,b^4\,d^3\right )}{9009\,e^7\,{\left (a\,e-b\,d\right )}^4}+\frac {x^2\,\sqrt {a+b\,x}\,\left (-4186\,B\,a^5\,b\,e^3+14342\,B\,a^4\,b^2\,d\,e^2-2226\,A\,a^4\,b^2\,e^3-15886\,B\,a^3\,b^3\,d^2\,e+8814\,A\,a^3\,b^3\,d\,e^2+4290\,B\,a^2\,b^4\,d^3-12870\,A\,a^2\,b^4\,d^2\,e+7722\,A\,a\,b^5\,d^3\right )}{9009\,e^7\,{\left (a\,e-b\,d\right )}^4}+\frac {16\,b^5\,x^6\,\sqrt {a+b\,x}\,\left (6\,A\,b\,e-13\,B\,a\,e+7\,B\,b\,d\right )}{9009\,e^5\,{\left (a\,e-b\,d\right )}^4}-\frac {8\,b^4\,x^5\,\left (a\,e-13\,b\,d\right )\,\sqrt {a+b\,x}\,\left (6\,A\,b\,e-13\,B\,a\,e+7\,B\,b\,d\right )}{9009\,e^6\,{\left (a\,e-b\,d\right )}^4}+\frac {2\,b^3\,x^4\,\sqrt {a+b\,x}\,\left (3\,a^2\,e^2-26\,a\,b\,d\,e+143\,b^2\,d^2\right )\,\left (6\,A\,b\,e-13\,B\,a\,e+7\,B\,b\,d\right )}{9009\,e^7\,{\left (a\,e-b\,d\right )}^4}\right )}{x^7+\frac {d^7}{e^7}+\frac {7\,d\,x^6}{e}+\frac {7\,d^6\,x}{e^6}+\frac {21\,d^2\,x^5}{e^2}+\frac {35\,d^3\,x^4}{e^3}+\frac {35\,d^4\,x^3}{e^4}+\frac {21\,d^5\,x^2}{e^5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d + e*x)^(1/2)*((x^3*(a + b*x)^(1/2)*(2574*A*b^6*d^3 + 5434*B*a*b^5*d^3 - 30*A*a^3*b^3*e^3 - 2938*B*a^4*b^2*
e^3 + 234*A*a^2*b^4*d*e^2 - 15886*B*a^2*b^4*d^2*e + 11470*B*a^3*b^3*d*e^2 - 858*A*a*b^5*d^2*e))/(9009*e^7*(a*e
- b*d)^4) - ((a + b*x)^(1/2)*(1386*A*a^6*e^3 + 252*B*a^6*d*e^2 - 2574*A*a^3*b^3*d^3 + 572*B*a^4*b^2*d^3 + 600
6*A*a^4*b^2*d^2*e - 4914*A*a^5*b*d*e^2 - 728*B*a^5*b*d^2*e))/(9009*e^7*(a*e - b*d)^4) - (x*(a + b*x)^(1/2)*(16
38*B*a^6*e^3 + 3402*A*a^5*b*e^3 - 7722*A*a^2*b^4*d^3 - 286*B*a^3*b^3*d^3 + 16302*A*a^3*b^3*d^2*e - 12558*A*a^4
*b^2*d*e^2 + 4082*B*a^4*b^2*d^2*e - 4858*B*a^5*b*d*e^2))/(9009*e^7*(a*e - b*d)^4) + (x^2*(a + b*x)^(1/2)*(7722
*A*a*b^5*d^3 - 4186*B*a^5*b*e^3 - 2226*A*a^4*b^2*e^3 + 4290*B*a^2*b^4*d^3 - 12870*A*a^2*b^4*d^2*e + 8814*A*a^3
*b^3*d*e^2 - 15886*B*a^3*b^3*d^2*e + 14342*B*a^4*b^2*d*e^2))/(9009*e^7*(a*e - b*d)^4) + (16*b^5*x^6*(a + b*x)^
(1/2)*(6*A*b*e - 13*B*a*e + 7*B*b*d))/(9009*e^5*(a*e - b*d)^4) - (8*b^4*x^5*(a*e - 13*b*d)*(a + b*x)^(1/2)*(6*
A*b*e - 13*B*a*e + 7*B*b*d))/(9009*e^6*(a*e - b*d)^4) + (2*b^3*x^4*(a + b*x)^(1/2)*(3*a^2*e^2 + 143*b^2*d^2 -
26*a*b*d*e)*(6*A*b*e - 13*B*a*e + 7*B*b*d))/(9009*e^7*(a*e - b*d)^4)))/(x^7 + d^7/e^7 + (7*d*x^6)/e + (7*d^6*x
)/e^6 + (21*d^2*x^5)/e^2 + (35*d^3*x^4)/e^3 + (35*d^4*x^3)/e^4 + (21*d^5*x^2)/e^5)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(5/2)*(B*x+A)/(e*x+d)**(15/2),x)

[Out]

Timed out

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