Optimal. Leaf size=254 \[ -\frac {4 (a+b x)^{5/2}}{d \sqrt [4]{c+d x}}-\frac {16 b (b c-a d) \sqrt {a+b x} (c+d x)^{3/4}}{3 d^3}+\frac {40 b (a+b x)^{3/2} (c+d x)^{3/4}}{9 d^2}+\frac {32 \sqrt [4]{b} (b c-a d)^{11/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 d^4 \sqrt {a+b x}}-\frac {32 \sqrt [4]{b} (b c-a d)^{11/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 d^4 \sqrt {a+b x}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.18, antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {49, 52, 65,
313, 230, 227, 1214, 1213, 435} \begin {gather*} -\frac {32 \sqrt [4]{b} (b c-a d)^{11/4} \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 )}{3 d^4 \sqrt {a+b x}}+\frac {32 \sqrt [4]{b} (b c-a d)^{11/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 d^4 \sqrt {a+b x}}-\frac {16 b \sqrt {a+b x} (c+d x)^{3/4} (b c-a d)}{3 d^3}+\frac {40 b (a+b x)^{3/2} (c+d x)^{3/4}}{9 d^2}-\frac {4 (a+b x)^{5/2}}{d \sqrt [4]{c+d x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 49
Rule 52
Rule 65
Rule 227
Rule 230
Rule 313
Rule 435
Rule 1213
Rule 1214
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2}}{(c+d x)^{5/4}} \, dx &=-\frac {4 (a+b x)^{5/2}}{d \sqrt [4]{c+d x}}+\frac {(10 b) \int \frac {(a+b x)^{3/2}}{\sqrt [4]{c+d x}} \, dx}{d}\\ &=-\frac {4 (a+b x)^{5/2}}{d \sqrt [4]{c+d x}}+\frac {40 b (a+b x)^{3/2} (c+d x)^{3/4}}{9 d^2}-\frac {(20 b (b c-a d)) \int \frac {\sqrt {a+b x}}{\sqrt [4]{c+d x}} \, dx}{3 d^2}\\ &=-\frac {4 (a+b x)^{5/2}}{d \sqrt [4]{c+d x}}-\frac {16 b (b c-a d) \sqrt {a+b x} (c+d x)^{3/4}}{3 d^3}+\frac {40 b (a+b x)^{3/2} (c+d x)^{3/4}}{9 d^2}+\frac {\left (8 b (b c-a d)^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt [4]{c+d x}} \, dx}{3 d^3}\\ &=-\frac {4 (a+b x)^{5/2}}{d \sqrt [4]{c+d x}}-\frac {16 b (b c-a d) \sqrt {a+b x} (c+d x)^{3/4}}{3 d^3}+\frac {40 b (a+b x)^{3/2} (c+d x)^{3/4}}{9 d^2}+\frac {\left (32 b (b c-a d)^2\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{3 d^4}\\ &=-\frac {4 (a+b x)^{5/2}}{d \sqrt [4]{c+d x}}-\frac {16 b (b c-a d) \sqrt {a+b x} (c+d x)^{3/4}}{3 d^3}+\frac {40 b (a+b x)^{3/2} (c+d x)^{3/4}}{9 d^2}-\frac {\left (32 \sqrt {b} (b c-a d)^{5/2}\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 )}{3 d^4}+\frac {\left (32 \sqrt {b} (b c-a d)^{5/2}\right ) \text {Subst}\left (\int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{3 d^4}\\ &=-\frac {4 (a+b x)^{5/2}}{d \sqrt [4]{c+d x}}-\frac {16 b (b c-a d) \sqrt {a+b x} (c+d x)^{3/4}}{3 d^3}+\frac {40 b (a+b x)^{3/2} (c+d x)^{3/4}}{9 d^2}-\frac {\left (32 \sqrt {b} (b c-a d)^{5/2} \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 )}{3 d^4 \sqrt {a+b x}}+\frac {\left (32 \sqrt {b} (b c-a d)^{5/2} \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \text {Subst}\left (\int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}{\sqrt {1+\frac {b x^4}{\left (a-\frac {b c}{d}\right ) d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{3 d^4 \sqrt {a+b x}}\\ &=-\frac {4 (a+b x)^{5/2}}{d \sqrt [4]{c+d x}}-\frac {16 b (b c-a d) \sqrt {a+b x} (c+d x)^{3/4}}{3 d^3}+\frac {40 b (a+b x)^{3/2} (c+d x)^{3/4}}{9 d^2}-\frac {32 \sqrt [4]{b} (b c-a d)^{11/4} \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 )}{3 d^4 \sqrt {a+b x}}+\frac {\left (32 \sqrt {b} (b c-a d)^{5/2} \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}}{\sqrt {1-\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}} \, dx,x,\sqrt [4]{c+d x}\right )}{3 d^4 \sqrt {a+b x}}\\ &=-\frac {4 (a+b x)^{5/2}}{d \sqrt [4]{c+d x}}-\frac {16 b (b c-a d) \sqrt {a+b x} (c+d x)^{3/4}}{3 d^3}+\frac {40 b (a+b x)^{3/2} (c+d x)^{3/4}}{9 d^2}+\frac {32 \sqrt [4]{b} (b c-a d)^{11/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 d^4 \sqrt {a+b x}}-\frac {32 \sqrt [4]{b} (b c-a d)^{11/4} \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 )}{3 d^4 \sqrt {a+b x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.05, size = 73, normalized size = 0.29 \begin {gather*} \frac {2 (a+b x)^{7/2} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/4} \, _2F_1\left (\frac {5}{4},\frac {7}{2};\frac {9}{2};\frac {d (a+b x)}{-b c+a d}\right )}{7 b (c+d x)^{5/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{\frac {5}{2}}}{\left (d x +c \right )^{\frac {5}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x\right )^{\frac {5}{2}}}{\left (c + d x\right )^{\frac {5}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{5/2}}{{\left (c+d\,x\right )}^{5/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________