3.21.93 \(\int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{13/2}} \, dx\)

Optimal. Leaf size=147 \[ \frac {4 b (a+b x)^{7/2} (-11 a B e+4 A b e+7 b B d)}{693 e (d+e x)^{7/2} (b d-a e)^3}+\frac {2 (a+b x)^{7/2} (-11 a B e+4 A b e+7 b B d)}{99 e (d+e x)^{9/2} (b d-a e)^2}-\frac {2 (a+b x)^{7/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)} \]

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Rubi [A]  time = 0.09, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {78, 45, 37} \begin {gather*} \frac {4 b (a+b x)^{7/2} (-11 a B e+4 A b e+7 b B d)}{693 e (d+e x)^{7/2} (b d-a e)^3}+\frac {2 (a+b x)^{7/2} (-11 a B e+4 A b e+7 b B d)}{99 e (d+e x)^{9/2} (b d-a e)^2}-\frac {2 (a+b x)^{7/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(13/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(7/2))/(11*e*(b*d - a*e)*(d + e*x)^(11/2)) + (2*(7*b*B*d + 4*A*b*e - 11*a*B*e)*(a +
b*x)^(7/2))/(99*e*(b*d - a*e)^2*(d + e*x)^(9/2)) + (4*b*(7*b*B*d + 4*A*b*e - 11*a*B*e)*(a + b*x)^(7/2))/(693*e
*(b*d - a*e)^3*(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)^{13/2}} \, dx &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac {(7 b B d+4 A b e-11 a B e) \int \frac {(a+b x)^{5/2}}{(d+e x)^{11/2}} \, dx}{11 e (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac {2 (7 b B d+4 A b e-11 a B e) (a+b x)^{7/2}}{99 e (b d-a e)^2 (d+e x)^{9/2}}+\frac {(2 b (7 b B d+4 A b e-11 a B e)) \int \frac {(a+b x)^{5/2}}{(d+e x)^{9/2}} \, dx}{99 e (b d-a e)^2}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{11 e (b d-a e) (d+e x)^{11/2}}+\frac {2 (7 b B d+4 A b e-11 a B e) (a+b x)^{7/2}}{99 e (b d-a e)^2 (d+e x)^{9/2}}+\frac {4 b (7 b B d+4 A b e-11 a B e) (a+b x)^{7/2}}{693 e (b d-a e)^3 (d+e x)^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 135, normalized size = 0.92 \begin {gather*} \frac {2 (a+b x)^{7/2} \left (A \left (63 a^2 e^2-14 a b e (11 d+2 e x)+b^2 \left (99 d^2+44 d e x+8 e^2 x^2\right )\right )+B \left (7 a^2 e (2 d+11 e x)-2 a b \left (11 d^2+85 d e x+11 e^2 x^2\right )+7 b^2 d x (11 d+2 e x)\right )\right )}{693 (d+e x)^{11/2} (b d-a e)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(13/2),x]

[Out]

(2*(a + b*x)^(7/2)*(A*(63*a^2*e^2 - 14*a*b*e*(11*d + 2*e*x) + b^2*(99*d^2 + 44*d*e*x + 8*e^2*x^2)) + B*(7*b^2*
d*x*(11*d + 2*e*x) + 7*a^2*e*(2*d + 11*e*x) - 2*a*b*(11*d^2 + 85*d*e*x + 11*e^2*x^2))))/(693*(b*d - a*e)^3*(d
+ e*x)^(11/2))

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IntegrateAlgebraic [A]  time = 0.22, size = 134, normalized size = 0.91 \begin {gather*} -\frac {2 (a+b x)^{11/2} \left (-\frac {99 A b^2 (d+e x)^2}{(a+b x)^2}+\frac {154 A b e (d+e x)}{a+b x}-\frac {77 a B e (d+e x)}{a+b x}+\frac {99 a b B (d+e x)^2}{(a+b x)^2}-\frac {77 b B d (d+e x)}{a+b x}-63 A e^2+63 B d e\right )}{693 (d+e x)^{11/2} (b d-a e)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(13/2),x]

[Out]

(-2*(a + b*x)^(11/2)*(63*B*d*e - 63*A*e^2 - (77*b*B*d*(d + e*x))/(a + b*x) + (154*A*b*e*(d + e*x))/(a + b*x) -
(77*a*B*e*(d + e*x))/(a + b*x) - (99*A*b^2*(d + e*x)^2)/(a + b*x)^2 + (99*a*b*B*(d + e*x)^2)/(a + b*x)^2))/(6
93*(b*d - a*e)^3*(d + e*x)^(11/2))

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fricas [F(-1)]  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)^(5/2)*(B*x+A)/(e*x+d)^(13/2),x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 6.44, size = 621, normalized size = 4.22 \begin {gather*} \frac {2 \, {\left ({\left (b x + a\right )} {\left (\frac {2 \, {\left (7 \, B b^{14} d^{3} {\left | b \right |} e^{6} - 25 \, B a b^{13} d^{2} {\left | b \right |} e^{7} + 4 \, A b^{14} d^{2} {\left | b \right |} e^{7} + 29 \, B a^{2} b^{12} d {\left | b \right |} e^{8} - 8 \, A a b^{13} d {\left | b \right |} e^{8} - 11 \, B a^{3} b^{11} {\left | b \right |} e^{9} + 4 \, A a^{2} b^{12} {\left | b \right |} e^{9}\right )} {\left (b x + a\right )}}{b^{7} d^{5} e^{5} - 5 \, a b^{6} d^{4} e^{6} + 10 \, a^{2} b^{5} d^{3} e^{7} - 10 \, a^{3} b^{4} d^{2} e^{8} + 5 \, a^{4} b^{3} d e^{9} - a^{5} b^{2} e^{10}} + \frac {11 \, {\left (7 \, B b^{15} d^{4} {\left | b \right |} e^{5} - 32 \, B a b^{14} d^{3} {\left | b \right |} e^{6} + 4 \, A b^{15} d^{3} {\left | b \right |} e^{6} + 54 \, B a^{2} b^{13} d^{2} {\left | b \right |} e^{7} - 12 \, A a b^{14} d^{2} {\left | b \right |} e^{7} - 40 \, B a^{3} b^{12} d {\left | b \right |} e^{8} + 12 \, A a^{2} b^{13} d {\left | b \right |} e^{8} + 11 \, B a^{4} b^{11} {\left | b \right |} e^{9} - 4 \, A a^{3} b^{12} {\left | b \right |} e^{9}\right )}}{b^{7} d^{5} e^{5} - 5 \, a b^{6} d^{4} e^{6} + 10 \, a^{2} b^{5} d^{3} e^{7} - 10 \, a^{3} b^{4} d^{2} e^{8} + 5 \, a^{4} b^{3} d e^{9} - a^{5} b^{2} e^{10}}\right )} - \frac {99 \, {\left (B a b^{15} d^{4} {\left | b \right |} e^{5} - A b^{16} d^{4} {\left | b \right |} e^{5} - 4 \, B a^{2} b^{14} d^{3} {\left | b \right |} e^{6} + 4 \, A a b^{15} d^{3} {\left | b \right |} e^{6} + 6 \, B a^{3} b^{13} d^{2} {\left | b \right |} e^{7} - 6 \, A a^{2} b^{14} d^{2} {\left | b \right |} e^{7} - 4 \, B a^{4} b^{12} d {\left | b \right |} e^{8} + 4 \, A a^{3} b^{13} d {\left | b \right |} e^{8} + B a^{5} b^{11} {\left | b \right |} e^{9} - A a^{4} b^{12} {\left | b \right |} e^{9}\right )}}{b^{7} d^{5} e^{5} - 5 \, a b^{6} d^{4} e^{6} + 10 \, a^{2} b^{5} d^{3} e^{7} - 10 \, a^{3} b^{4} d^{2} e^{8} + 5 \, a^{4} b^{3} d e^{9} - a^{5} b^{2} e^{10}}\right )} {\left (b x + a\right )}^{\frac {7}{2}}}{693 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(13/2),x, algorithm="giac")

[Out]

2/693*((b*x + a)*(2*(7*B*b^14*d^3*abs(b)*e^6 - 25*B*a*b^13*d^2*abs(b)*e^7 + 4*A*b^14*d^2*abs(b)*e^7 + 29*B*a^2
*b^12*d*abs(b)*e^8 - 8*A*a*b^13*d*abs(b)*e^8 - 11*B*a^3*b^11*abs(b)*e^9 + 4*A*a^2*b^12*abs(b)*e^9)*(b*x + a)/(
b^7*d^5*e^5 - 5*a*b^6*d^4*e^6 + 10*a^2*b^5*d^3*e^7 - 10*a^3*b^4*d^2*e^8 + 5*a^4*b^3*d*e^9 - a^5*b^2*e^10) + 11
*(7*B*b^15*d^4*abs(b)*e^5 - 32*B*a*b^14*d^3*abs(b)*e^6 + 4*A*b^15*d^3*abs(b)*e^6 + 54*B*a^2*b^13*d^2*abs(b)*e^
7 - 12*A*a*b^14*d^2*abs(b)*e^7 - 40*B*a^3*b^12*d*abs(b)*e^8 + 12*A*a^2*b^13*d*abs(b)*e^8 + 11*B*a^4*b^11*abs(b
)*e^9 - 4*A*a^3*b^12*abs(b)*e^9)/(b^7*d^5*e^5 - 5*a*b^6*d^4*e^6 + 10*a^2*b^5*d^3*e^7 - 10*a^3*b^4*d^2*e^8 + 5*
a^4*b^3*d*e^9 - a^5*b^2*e^10)) - 99*(B*a*b^15*d^4*abs(b)*e^5 - A*b^16*d^4*abs(b)*e^5 - 4*B*a^2*b^14*d^3*abs(b)
*e^6 + 4*A*a*b^15*d^3*abs(b)*e^6 + 6*B*a^3*b^13*d^2*abs(b)*e^7 - 6*A*a^2*b^14*d^2*abs(b)*e^7 - 4*B*a^4*b^12*d*
abs(b)*e^8 + 4*A*a^3*b^13*d*abs(b)*e^8 + B*a^5*b^11*abs(b)*e^9 - A*a^4*b^12*abs(b)*e^9)/(b^7*d^5*e^5 - 5*a*b^6
*d^4*e^6 + 10*a^2*b^5*d^3*e^7 - 10*a^3*b^4*d^2*e^8 + 5*a^4*b^3*d*e^9 - a^5*b^2*e^10))*(b*x + a)^(7/2)/(b^2*d +
(b*x + a)*b*e - a*b*e)^(11/2)

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maple [A]  time = 0.01, size = 177, normalized size = 1.20 \begin {gather*} -\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (8 A \,b^{2} e^{2} x^{2}-22 B a b \,e^{2} x^{2}+14 B \,b^{2} d e \,x^{2}-28 A a b \,e^{2} x +44 A \,b^{2} d e x +77 B \,a^{2} e^{2} x -170 B a b d e x +77 B \,b^{2} d^{2} x +63 A \,a^{2} e^{2}-154 A a b d e +99 A \,b^{2} d^{2}+14 B \,a^{2} d e -22 B a b \,d^{2}\right )}{693 \left (e x +d \right )^{\frac {11}{2}} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(13/2),x)

[Out]

-2/693*(b*x+a)^(7/2)*(8*A*b^2*e^2*x^2-22*B*a*b*e^2*x^2+14*B*b^2*d*e*x^2-28*A*a*b*e^2*x+44*A*b^2*d*e*x+77*B*a^2
*e^2*x-170*B*a*b*d*e*x+77*B*b^2*d^2*x+63*A*a^2*e^2-154*A*a*b*d*e+99*A*b^2*d^2+14*B*a^2*d*e-22*B*a*b*d^2)/(e*x+
d)^(11/2)/(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)

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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)^(5/2)*(B*x+A)/(e*x+d)^(13/2),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 = 2.73, size = 509, normalized size = 3.46 \begin {gather*} -\frac {\sqrt {d+e\,x}\,\left (\frac {\sqrt {a+b\,x}\,\left (28\,B\,a^5\,d\,e+126\,A\,a^5\,e^2-44\,B\,a^4\,b\,d^2-308\,A\,a^4\,b\,d\,e+198\,A\,a^3\,b^2\,d^2\right )}{693\,e^6\,{\left (a\,e-b\,d\right )}^3}+\frac {x\,\sqrt {a+b\,x}\,\left (154\,B\,a^5\,e^2-256\,B\,a^4\,b\,d\,e+322\,A\,a^4\,b\,e^2+22\,B\,a^3\,b^2\,d^2-836\,A\,a^3\,b^2\,d\,e+594\,A\,a^2\,b^3\,d^2\right )}{693\,e^6\,{\left (a\,e-b\,d\right )}^3}+\frac {x^2\,\sqrt {a+b\,x}\,\left (418\,B\,a^4\,b\,e^2-908\,B\,a^3\,b^2\,d\,e+226\,A\,a^3\,b^2\,e^2+330\,B\,a^2\,b^3\,d^2-660\,A\,a^2\,b^3\,d\,e+594\,A\,a\,b^4\,d^2\right )}{693\,e^6\,{\left (a\,e-b\,d\right )}^3}+\frac {x^3\,\sqrt {a+b\,x}\,\left (330\,B\,a^3\,b^2\,e^2-908\,B\,a^2\,b^3\,d\,e+6\,A\,a^2\,b^3\,e^2+418\,B\,a\,b^4\,d^2-44\,A\,a\,b^4\,d\,e+198\,A\,b^5\,d^2\right )}{693\,e^6\,{\left (a\,e-b\,d\right )}^3}+\frac {4\,b^4\,x^5\,\sqrt {a+b\,x}\,\left (4\,A\,b\,e-11\,B\,a\,e+7\,B\,b\,d\right )}{693\,e^5\,{\left (a\,e-b\,d\right )}^3}-\frac {2\,b^3\,x^4\,\left (a\,e-11\,b\,d\right )\,\sqrt {a+b\,x}\,\left (4\,A\,b\,e-11\,B\,a\,e+7\,B\,b\,d\right )}{693\,e^6\,{\left (a\,e-b\,d\right )}^3}\right )}{x^6+\frac {d^6}{e^6}+\frac {6\,d\,x^5}{e}+\frac {6\,d^5\,x}{e^5}+\frac {15\,d^2\,x^4}{e^2}+\frac {20\,d^3\,x^3}{e^3}+\frac {15\,d^4\,x^2}{e^4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^(5/2))/(d + e*x)^(13/2),x)

[Out]

-((d + e*x)^(1/2)*(((a + b*x)^(1/2)*(126*A*a^5*e^2 + 28*B*a^5*d*e - 44*B*a^4*b*d^2 + 198*A*a^3*b^2*d^2 - 308*A
*a^4*b*d*e))/(693*e^6*(a*e - b*d)^3) + (x*(a + b*x)^(1/2)*(154*B*a^5*e^2 + 322*A*a^4*b*e^2 + 594*A*a^2*b^3*d^2
+ 22*B*a^3*b^2*d^2 - 256*B*a^4*b*d*e - 836*A*a^3*b^2*d*e))/(693*e^6*(a*e - b*d)^3) + (x^2*(a + b*x)^(1/2)*(59
4*A*a*b^4*d^2 + 418*B*a^4*b*e^2 + 226*A*a^3*b^2*e^2 + 330*B*a^2*b^3*d^2 - 660*A*a^2*b^3*d*e - 908*B*a^3*b^2*d*
e))/(693*e^6*(a*e - b*d)^3) + (x^3*(a + b*x)^(1/2)*(198*A*b^5*d^2 + 418*B*a*b^4*d^2 + 6*A*a^2*b^3*e^2 + 330*B*
a^3*b^2*e^2 - 44*A*a*b^4*d*e - 908*B*a^2*b^3*d*e))/(693*e^6*(a*e - b*d)^3) + (4*b^4*x^5*(a + b*x)^(1/2)*(4*A*b
*e - 11*B*a*e + 7*B*b*d))/(693*e^5*(a*e - b*d)^3) - (2*b^3*x^4*(a*e - 11*b*d)*(a + b*x)^(1/2)*(4*A*b*e - 11*B*
a*e + 7*B*b*d))/(693*e^6*(a*e - b*d)^3)))/(x^6 + d^6/e^6 + (6*d*x^5)/e + (6*d^5*x)/e^5 + (15*d^2*x^4)/e^2 + (2
0*d^3*x^3)/e^3 + (15*d^4*x^2)/e^4)

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sympy [F(-1)]  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)**(5/2)*(B*x+A)/(e*x+d)**(13/2),x)

[Out]

Timed out

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