3.17.65 \(\int \frac {(a+b x)^{7/2}}{(c+d x)^{7/4}} \, dx\) [1665]

Optimal. Leaf size=207 \[ -\frac {4 (a+b x)^{7/2}}{3 d (c+d x)^{3/4}}+\frac {160 b (b c-a d)^2 \sqrt {a+b x} \sqrt [4]{c+d x}}{33 d^4}-\frac {80 b (b c-a d) (a+b x)^{3/2} \sqrt [4]{c+d x}}{33 d^3}+\frac {56 b (a+b x)^{5/2} \sqrt [4]{c+d x}}{33 d^2}-\frac {320 b^{3/4} (b c-a d)^{13/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{33 d^5 \sqrt {a+b x}} \]

[Out]

-4/3*(b*x+a)^(7/2)/d/(d*x+c)^(3/4)-80/33*b*(-a*d+b*c)*(b*x+a)^(3/2)*(d*x+c)^(1/4)/d^3+56/33*b*(b*x+a)^(5/2)*(d
*x+c)^(1/4)/d^2+160/33*b*(-a*d+b*c)^2*(d*x+c)^(1/4)*(b*x+a)^(1/2)/d^4-320/33*b^(3/4)*(-a*d+b*c)^(13/4)*Ellipti
cF(b^(1/4)*(d*x+c)^(1/4)/(-a*d+b*c)^(1/4),I)*(-d*(b*x+a)/(-a*d+b*c))^(1/2)/d^5/(b*x+a)^(1/2)

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Rubi [A]
time = 0.10, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {49, 52, 65, 230, 227} \begin {gather*} -\frac {320 b^{3/4} (b c-a d)^{13/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{33 d^5 \sqrt {a+b x}}+\frac {160 b \sqrt {a+b x} \sqrt [4]{c+d x} (b c-a d)^2}{33 d^4}-\frac {80 b (a+b x)^{3/2} \sqrt [4]{c+d x} (b c-a d)}{33 d^3}+\frac {56 b (a+b x)^{5/2} \sqrt [4]{c+d x}}{33 d^2}-\frac {4 (a+b x)^{7/2}}{3 d (c+d x)^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(7/2)/(c + d*x)^(7/4),x]

[Out]

(-4*(a + b*x)^(7/2))/(3*d*(c + d*x)^(3/4)) + (160*b*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(1/4))/(33*d^4) - (8
0*b*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(1/4))/(33*d^3) + (56*b*(a + b*x)^(5/2)*(c + d*x)^(1/4))/(33*d^2) -
(320*b^(3/4)*(b*c - a*d)^(13/4)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticF[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/
(b*c - a*d)^(1/4)], -1])/(33*d^5*Sqrt[a + b*x])

Rule 49

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 227

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[Rt[-b, 4]*(x/Rt[a, 4])], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 230

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[Sqrt[1 + b*(x^4/a)]/Sqrt[a + b*x^4], Int[1/Sqrt[1 + b*(x^4/
a)], x], x] /; FreeQ[{a, b}, x] && NegQ[b/a] &&  !GtQ[a, 0]

Rubi steps

\begin {align*} \int \frac {(a+b x)^{7/2}}{(c+d x)^{7/4}} \, dx &=-\frac {4 (a+b x)^{7/2}}{3 d (c+d x)^{3/4}}+\frac {(14 b) \int \frac {(a+b x)^{5/2}}{(c+d x)^{3/4}} \, dx}{3 d}\\ &=-\frac {4 (a+b x)^{7/2}}{3 d (c+d x)^{3/4}}+\frac {56 b (a+b x)^{5/2} \sqrt [4]{c+d x}}{33 d^2}-\frac {(140 b (b c-a d)) \int \frac {(a+b x)^{3/2}}{(c+d x)^{3/4}} \, dx}{33 d^2}\\ &=-\frac {4 (a+b x)^{7/2}}{3 d (c+d x)^{3/4}}-\frac {80 b (b c-a d) (a+b x)^{3/2} \sqrt [4]{c+d x}}{33 d^3}+\frac {56 b (a+b x)^{5/2} \sqrt [4]{c+d x}}{33 d^2}+\frac {\left (40 b (b c-a d)^2\right ) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/4}} \, dx}{11 d^3}\\ &=-\frac {4 (a+b x)^{7/2}}{3 d (c+d x)^{3/4}}+\frac {160 b (b c-a d)^2 \sqrt {a+b x} \sqrt [4]{c+d x}}{33 d^4}-\frac {80 b (b c-a d) (a+b x)^{3/2} \sqrt [4]{c+d x}}{33 d^3}+\frac {56 b (a+b x)^{5/2} \sqrt [4]{c+d x}}{33 d^2}-\frac {\left (80 b (b c-a d)^3\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/4}} \, dx}{33 d^4}\\ &=-\frac {4 (a+b x)^{7/2}}{3 d (c+d x)^{3/4}}+\frac {160 b (b c-a d)^2 \sqrt {a+b x} \sqrt [4]{c+d x}}{33 d^4}-\frac {80 b (b c-a d) (a+b x)^{3/2} \sqrt [4]{c+d x}}{33 d^3}+\frac {56 b (a+b x)^{5/2} \sqrt [4]{c+d x}}{33 d^2}-\frac {\left (320 b (b c-a d)^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{33 d^5}\\ &=-\frac {4 (a+b x)^{7/2}}{3 d (c+d x)^{3/4}}+\frac {160 b (b c-a d)^2 \sqrt {a+b x} \sqrt [4]{c+d x}}{33 d^4}-\frac {80 b (b c-a d) (a+b x)^{3/2} \sqrt [4]{c+d x}}{33 d^3}+\frac {56 b (a+b x)^{5/2} \sqrt [4]{c+d x}}{33 d^2}-\frac {\left (320 b (b c-a d)^3 \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {b x^4}{\left (a-\frac {b c}{d}\right ) d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{33 d^5 \sqrt {a+b x}}\\ &=-\frac {4 (a+b x)^{7/2}}{3 d (c+d x)^{3/4}}+\frac {160 b (b c-a d)^2 \sqrt {a+b x} \sqrt [4]{c+d x}}{33 d^4}-\frac {80 b (b c-a d) (a+b x)^{3/2} \sqrt [4]{c+d x}}{33 d^3}+\frac {56 b (a+b x)^{5/2} \sqrt [4]{c+d x}}{33 d^2}-\frac {320 b^{3/4} (b c-a d)^{13/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{33 d^5 \sqrt {a+b x}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.07, size = 73, normalized size = 0.35 \begin {gather*} \frac {2 (a+b x)^{9/2} \left (\frac {b (c+d x)}{b c-a d}\right )^{7/4} \, _2F_1\left (\frac {7}{4},\frac {9}{2};\frac {11}{2};\frac {d (a+b x)}{-b c+a d}\right )}{9 b (c+d x)^{7/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(7/2)/(c + d*x)^(7/4),x]

[Out]

(2*(a + b*x)^(9/2)*((b*(c + d*x))/(b*c - a*d))^(7/4)*Hypergeometric2F1[7/4, 9/2, 11/2, (d*(a + b*x))/(-(b*c) +
a*d)])/(9*b*(c + d*x)^(7/4))

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{\frac {7}{2}}}{\left (d x +c \right )^{\frac {7}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(7/2)/(d*x+c)^(7/4),x)

[Out]

int((b*x+a)^(7/2)/(d*x+c)^(7/4),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(7/2)/(d*x+c)^(7/4),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(7/2)/(d*x + c)^(7/4), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(7/2)/(d*x+c)^(7/4),x, algorithm="fricas")

[Out]

integral((b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)*sqrt(b*x + a)*(d*x + c)^(1/4)/(d^2*x^2 + 2*c*d*x + c^2), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x\right )^{\frac {7}{2}}}{\left (c + d x\right )^{\frac {7}{4}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(7/2)/(d*x+c)**(7/4),x)

[Out]

Integral((a + b*x)**(7/2)/(c + d*x)**(7/4), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(7/2)/(d*x+c)^(7/4),x, algorithm="giac")

[Out]

integrate((b*x + a)^(7/2)/(d*x + c)^(7/4), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{7/2}}{{\left (c+d\,x\right )}^{7/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^(7/2)/(c + d*x)^(7/4),x)

[Out]

int((a + b*x)^(7/2)/(c + d*x)^(7/4), x)

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